# Automatic calculation of two-loop ELWK corrections to the muon ($g$-2)∗

Tadashi Ishikawa1 Nobuya Nakazawa2 and Yoshiaki Yasui3 1 High Energy
Accelerator Organization(KEK), 1-1 OHO Tsukuba Ibaraki 305-0801,Japan  2
Department of Physics, Kogakuin University, Shinjuku,Tokyo 163-8677,Japan  3
Department of Management, Tokyo Management College,Ichikawa, Chiba 272-0001,
Japan<EMAIL_ADDRESS>
###### Abstract

Two-loop electroweak corrections to the muon anomalous magnetic moment are
automatically calculated by using GRACE-FORM system, as a trial to extend our
system for two-loop calculation. We adopt the non-linear gauge (NLG) to check
the reliability of our calculation. In total 1780 two-loop diagrams consisting
of 14 different topological types and 70 one-loop diagrams composed of counter
terms are calculated. We check UV- and IR-divergences cancellation and the
independence of the results from NLG parameters. As for the numerical
calculation, we adopt trapezoidal rule with Double Exponential method (DE).
Linear extrapolation method (LE) is introduced to regularize UV- and IR-
divergences and to get finite values.

##  1 Introduction

In order to get a sign of beyond the standard model physics from high
precision experimental data, we need higher order radiative corrections within
Standard Model (SM). For this purpose our group has been developing the
automatic calculation system GRACE [1] since the late 1980’s. The measurement
of the muon anomalous magnetic moment ($g$-2) is the one of the most precise
experiments to check the SM. QED correction was calculated by T.Kinoshita et
al. [2] up to tenth-order. The two-loop electroweak (ELWK) correction to
($g$-2) was calculated approximately by Kukhto et al. [3] in 1992.
Surprisingly, the two-loop correction is almost 20% of the one-loop
correction. We started to calculate the full two-loop corrections in 1995 and
presented our formalism at Pisa conference [4]. We also showed that the two-
loop QED value [5] was correctly reproduced within our general formalism.
However, the number of diagrams is huge (1780+70) and the numerical
integration requires the big CPU-power to achieve required accuracy, we must
wait until various environments are improved.

During these days, the several groups did the calculations using leading
log($M^{2}$) approximations. ($M$=heavy particle mass) [6] [7] and the
approximate value of the two-loop ELWK correction is widely accepted [8]. In
2001, BNL-Experiment 821 [9] announced that the precise experimental value
deviates from that of SM around $(2.2\sim 2.7)~{}\sigma$. It brought much
interest in the theoretical value.

∗To appear in the proceedings of the 4th Computational Particle Physics
Workshop, October 2016, Hayama, Japan.

The various theoretical predictions and experimental value are summarized in
the next subsection. The main theoretical concern is shifted to the hadronic
contributions.

###  1.1 present status of the theoretical predictions of ($g$-2) and future
experiments

We summarize the present theoretical predictions of muon ($g$-2) in Table 1.

Table 1: Theoretical predictions of muon ($g$-2) [10] Type of correction | Numerical Value (unit $10^{-11}$) | Error | Reference
---|---|---|---
QED up to tenth-order | 116584718.95 | (0.08) | [2]
Leading Order Hadronic Vac.Pol. | 6923 | (42)(3) | [11]
NLO Vac.Pol.+Hadronic LBL | 7  | (26) | [12]
ELWK (one-loop) | 195.82 | (0.02) | [13]
ELWK (two-loop) | \- 41.2  | (1.0) |  [3][6][7][14]
Theory total | 116591803 | (1)(42)(26) | [10]
Experimental Value | 116592091 | (54)(33) |  [9][15]
|  |  |

The discrepancy between the experimental value and the theoretical value is
still large. As new experiments are scheduled at FNAL-E989 [16] and
J-PARC-E034 [17], we can expect to have new data within a few years.

###  1.2 Purpose of our calculation

Although the two-loop ELWK correction is almost established as summarized in
the Table 1, we try to get the value without any approximation to confirm the
validity of the earlier studies. It is also an important trial to extend
GRACE-system from one-loop to two-loop calculation. It may become a milestone
to construct the framework of Perturbative Numerical Quantum Field Theory
(PNQFT). The concepts of PNQFT are summarized as follows. (a) It is essential
to assume amplitudes as meromorphic functions of space time dimension $n$, for
extracting both UV- and IR-divergences. By adopting dimensional
regularization, gauge invariant renormalized value is obtained directly. (b)
The source for numerical integration is automatically generated by a symbolic
manipulation system. (c) It is crucial to reduce human intervention to avoid
careless mistakes, so that numerical method is fully exploited. The following
calculation shows the outline of this direction.

In section 2 and 3, we briefly explain the flow and foundation of our
calculation. In section 4, we touch on our method of numerical calculation. We
emphasize that the Linear Extrapolation (LE) method [18] is simple and
efficient method to regularize UV- and IR-divergences and also to get finite
values. In numerical calculation, the validity of the results must be
guaranteed by comparing several independent methods. We explain shortly our
consistency conditions to ensure the results. We also show some examples of
calculations and how the consistency conditions are satisfied. In section 5,
we give a part of our results on ($g$-2) for restricted types of diagrams. In
the last section, we summarize the present status and give some comments to
make extensive progress.

##  2 Outline of our frame work

Our calculation is formulated under the following conditions.

  1. 1.

Adopt non-linear gauge (NLG) formulation with ’t Hooft-Feynman propagator.

  2. 2.

Dimensional regularization for both Ultra Violet (UV)- and Infrared (IR)
-divergences.

  3. 3.

On mass shell renormalization scheme is adopted.

  4. 4.

Linear Extrapolation method (LE) is fully used for regularization and getting
finite values.

Next, we briefly explain the flow of our calculation.

  1. 1.

GRACE system generates all the diagrams we need in SM. There are 1780 two-loop
diagrams and 70 one-loop diagrams composed of one-loop order counter term
(CT).

  2. 2.

These 1780 diagrams are classified into 14 types of topology. Types of the
topology are displayed in Fig.1. Among these types, some of them give the same
contribution because of symmetry. (an example: 5-a vs. 5-b ) The diagrams
including CT are essentially two types, namely, vertex and self-energy types.

Figure 1: Types of topology

  1. 3.

In order to distinguish the UV-part from finite part, we prepare a few files
determined by the type of topology, in advance. They represent the following
quantities. [19]

     * •

Internal loop momentum flow$(\ell_{s},\eta(s))$($s=1,2$)

     * •

External momentum flow $(q_{j})$ ($j=$ internal line number)

     * •

Kirchhoff’s law of momentum conservation at each vertex

  2. 4.

Using these tools, both UV-divergent and finite contributions to ($g$-2) are
calculated in turn, according to the formulas given in the next section [19].
We make use of a symbolic manipulation system FORM [20] exhaustively.

##  3 Logic of the calculation

We explain the case where there are six-propagators. Starting formula is the
two-loop muon vertex,

$\displaystyle\Gamma_{\mu}=\int\frac{d^{n}\ell_{1}}{i(2\pi)^{n}}\frac{d^{n}\ell_{2}}{i(2\pi)^{n}}\frac{F_{\mu}(D)}{\prod_{j}(p_{j}^{2}-m_{j}^{2})}=\Gamma(6)\int\prod
dz_{j}\delta(1-\sum_{j}z_{j})\int\frac{d^{n}\ell_{1}}{i(2\pi)^{n}}\frac{d^{n}\ell_{2}}{i(2\pi)^{n}}\frac{F_{\mu}(D)}{\sum_{j}z_{j}(p_{j}^{2}-m_{j}^{2})}$
(1)

where $\displaystyle{p_{j}=\sum_{s=1}^{2}\eta(s)\ell_{s}+q_{j}}$, is the
momentum on the internal line $(j)$. The function $\eta(s)(=\pm 1,0)$ defines
loop momentum on the internal line $(j)$. The $z_{j}$’s are the Feynman
parameters to combine six propagators. The numerator function $F_{\mu}(D)$ is
explained later.

Next we diagonalize the denominator function with respect to loop momenta
$\ell_{1}$ and $\ell_{2}$ and perform integration. The result is,

$\displaystyle\Gamma_{\mu}=\frac{1}{(4\pi)^{n}}\int\prod
dz_{j}\delta(1-\sum_{j}z_{j})\frac{\Gamma(6-n)}{({\rm
detU})^{n/2}}F_{\mu}(D)\frac{1}{(V-i\varepsilon)^{6-n}},~{}~{}U_{s,t}=\sum_{j=1}^{6}z_{j}\eta_{s}(j)\eta_{t}(j).$
(2)

Where ${U}$ is well known 2$\times$2 matrix, composed of Feynman parameters
$(z_{j})$. $V(z_{j},m_{j},q_{j})$ is the denominator function. The arguments
are easily extended to the case with five-propagators (diagrams with four-
point coupling).

In order to generate the numerator function we use the following differential
integral operator $D_{\mu}$.

$\displaystyle\frac{p_{j}^{\mu}}{(p_{j}^{2}-m_{j}^{2})}=D_{j}^{\mu}\frac{1}{(p_{j}^{2}-m_{j}^{2})},~{}~{}~{}~{}{\rm
where}~{}~{}~{}D_{j}^{\mu}\equiv\frac{1}{2}\int\nolimits_{m_{j}^{2}}^{\infty}dm_{j}^{2}\frac{\partial}{\partial
q_{j\mu}}$ (3)

The operator $D_{j}^{\mu}$ generates momentum $p_{j}^{\mu}$ on the internal
line $(j)$. By operating $D_{j}^{\mu}$ to the denominator function $V$, we get
the following expression.

$\displaystyle
D_{j}^{\mu}\frac{1}{V^{m}}=\frac{Q^{\prime}_{j\mu}}{V^{m}},~{}~{}~{}Q^{\prime}_{j}=q_{j}-\frac{1}{\rm
detU}\sum_{i=1}z_{i}B_{ij}q_{j},~{}~{}~{}B_{ij}=\sum_{s,t}\eta_{s}(i)\eta_{t}(j)U^{-1}_{st}{\rm
detU}=B_{ji}$ (4)

Similarly we can generate the higher order term.

$\displaystyle
D_{i}^{\mu}D_{j}^{\nu}\frac{1}{V^{m}}=\frac{Q^{\prime}_{i\mu}Q^{\prime}_{j\nu}}{V^{m}}+\left(-\frac{1}{2\rm
detU}\right)\frac{g_{\mu\nu}}{(m-1)}\frac{B_{ij}}{V^{m-1}}$ (5)

We can also derive higher order terms such as
$D_{i}^{\mu}D_{j}^{\nu}D_{k}^{\lambda}(1/V^{m})$ and so on, however, Eq.(5) is
sufficient in our case. We can verify the equivalence of the above method and
the well known method of shifting momentum in the numerators. The
correspondence between two methods are symbolized as follows.

$\displaystyle\ell^{0}~{}\rightarrow~{}\frac{\\{1,Q^{\prime}_{i\mu},Q^{\prime}_{i,\mu}Q^{\prime}_{j,\nu},\cdots\\}}{V^{m}},~{}~{}~{}~{}\ell_{i}\ell_{j}~{}~{}\rightarrow~{}~{}\left(-\frac{1}{2{\rm
detU}}\right)\frac{g^{\mu\nu}}{(m-1)}\frac{B_{ij}}{V^{m-1}}$ (6)

Next step is to extract the ($g$-2) factor by using projection operator, from
the photon muon vertex $\Gamma_{\mu}$. The quantity ($g$-2) is given as
follows. ( $m_{0}$=muon mass )

$\displaystyle(g{\rm-2})$ $\displaystyle=$
$\displaystyle\lim_{q^{2}\rightarrow 0}\frac{{m_{0}}}{p^{4}q^{2}}{\rm
Tr}\left(\Gamma_{\mu}{\rm Proj}(\mu)\right)$ $\displaystyle{\rm Proj}(\mu)$
$\displaystyle=$ $\displaystyle\frac{1}{4}(\hbox to0.0pt{/\hss}p-{1\over
2}\hbox to0.0pt{/\hss}q+m_{0})\\{{\rm
m_{0}}\gamma_{\mu}(p.p)-({m_{0}}^{2}+\frac{q.q}{2})p_{\mu}\\}(\hbox
to0.0pt{/\hss}p+{1\over 2}\hbox to0.0pt{/\hss}q+{\rm m_{0}}),~{}~{}~{}$ (7)

where we set momentum of incoming $\mu^{-}$, outgoing $\mu^{+}$ and incoming
photon, as $(p-q/2),(p+q/2)$ and $q$, respectively.

Next step is the regularization of UV-divergence. By adopting $n$-dimensional
regularization method, any integrand $F$ of Feynman parameter integration is
regarded as a function of $\varepsilon=4-n/2$. Starting from the function
$F(\varepsilon)$, we adopt two methods for regularization. The first method is
well known one and extract $1/\varepsilon$ singularity from $F(\varepsilon)$
when one of Feynman parameters approaches 0, ($x\rightarrow 0$). Another one
is called LE-method explained below [18]. We compare two methods and conclude
LE-method is simpler and more efficient than the first one if the accuracy of
numerical integration is increased considerably.

First we explain the first method. In order to extract UV-divergence, we use
the following procedure. First we transform the Feynman parameters
$(z_{1},z_{2},\cdots,z_{6})$ into the appropriate [0,1] variables
$(x,y,u,v,w)$ depending on the topology. The key point is to factorize the
function detU=$x\times u(x,\cdots)$, where $u(0,\cdots)\neq 0$ . Then it is
shown that the UV-part in dimensional regularization comes from the following
formula. We show the example of vertex type renormalization. We set
$n=4-2\varepsilon,~{}~{}\varepsilon>0$ for UV-regularization.

$\displaystyle
I=\int_{0}^{1}dxx^{\varepsilon-1}F(x,\varepsilon)=\frac{1}{\varepsilon}F(0,0)+\frac{\partial
F(0,0)}{\partial\varepsilon}+\int_{0}^{1}\frac{F(x,0)-F(0,0)}{x}dx$ (8)

In the case where there is self-energy type diagram, we need the formula
$\int_{0}^{1}dxx^{\varepsilon-2}F(x,\varepsilon)$ in addition to Eq.(8). So it
becomes more complex.

The second method named LE is applicable for both UV- and IR-regularization
and obtaining finite values [18]. Space time dimension $n$ is set to be
$(4-2\varepsilon)$ for UV-case or $(4+2\varepsilon_{R})$ for IR-case. In both
cases, $\varepsilon>0,\varepsilon_{R}>0$.

After dimensional regularization method was introduced [21], the analyticity
with respect to $\varepsilon$ was discussed extensively. It is shown that the
Feynman amplitude is a meromorphic function of $\varepsilon$ [21] [22]. This
is a key point to utilize the LE-method to the Feynman amplitude. Numerically,
we calculate $G(i)=\int F(z_{j},\varepsilon)\prod dz_{j}$ when
$\varepsilon=\varepsilon(i)$. $(i=1,2,\cdots M)$. We set
$\varepsilon(i)=1/\alpha^{i+14}$ by taking relevant value $\alpha$. According
to the analyticity, we can expand $G$ and truncate it at
O($\varepsilon^{M-2}$).

$\displaystyle
G(i)=c_{-1}\frac{1}{\varepsilon(i)}+c_{0}+c_{1}\varepsilon(i)+\cdots+c_{M-2}\varepsilon(i)^{M-2}=\sum_{j=-1}^{M-2}c_{j}\\{\varepsilon(i)\\}^{j}~{}~{}~{}~{}(i=1,2,\cdots
M)$ (9)

From this formula, we get $\\{c_{j}\\}$ by multiplying the inverse of $M\times
M$ matrix $A$, whose element is
$A(i,j)=\\{\varepsilon(i)\\}^{j},~{}(i=1,\cdots M,~{}j=-1,0,\cdots M-2)$, to
M-component vector $G(i)$. Practically we set $M\sim 15$ and $\alpha\sim
1.15$.

As for counter terms, GRACE has a library of renormalization constants at one-
loop level. We make use of this library for 70-diagrams composed of counter
terms. We use FORM [20] to generate Fortran source, according to the above
formulation.

##  4 Numerical Calculation

The final step to get the value ($g$-2) is the numerical integration over
Feynman parameters. We employed trapezoidal rule with Double Exponential (DE)
transformation method [23]. It is very powerful if the integrand has singular
behavior at the edge of the integration domain. Speed of convergence is
accelerated by the DE transformation,

$\displaystyle I=\int_{0}^{1}dxf(x)~{}\rightarrow~{}x=\phi(t)={1\over
2}\left\\{1+\tanh\left(\frac{\pi}{2}\sinh(t)\right)\right\\}$ (10)

The maximum dimension of multiple integration is five. We apply DE-method to
any integration parameter involved. As we need the accuracy of 7$\sim$10
digits to see the cancellation of UV, the adaptive Monte Carlo method is not
suitable for two-loop calculation.

In order to ensure the validity of our results, we impose several conditions
given below.

  1. 1.

Well known QED two-loop value is reproduced.

  2. 2.

UV-divergence is cancelled.

  3. 3.

IR-divergence is cancelled.

  4. 4.

The result is independent of Non-linear gauge parameters.

  5. 5.

In some cases (examples: topology 4,5-a,5-b,7,9-1,9b and 10) , we perform
loop-integrations $(\ell_{1},\ell_{2})$ in turn, (we call it successive
method) and obtain the same results.

In all these cases, if we have plural methods to evaluate, we compare the
numerical values to ascertain the validity. If all these five conditions are
cleared, we can insist on the validity of our calculation.

We demonstrate how the conditions are cleared by showing the examples.

  * •

Reproduction of QED two-loop value. (Unit =$(\alpha/\pi)^{2})$
Analytic expression -0.328478966 : our value -0.328478911

  * •

UV-cancellation in Feynman gauge (FG) among the diagrams.
Examples of a group of diagrams are shown in Fig.2.

In this case, total 13-diagrams and 1-counter term make a group to cancel UV-
divergence. As you can read from the table, the cancellation is marvelous, up
to almost 10 digits.

Figure 2: Sample of diagrams with $W$ Table 2: Sample of UV-cancellation in FG Topology | Diagram No. | Value $(\rm{unit~{}}10^{-11})$
---|---|---
3 | 149 | 1.679393199868
3 | 153 | 0.490572808433
3 | 157 | 0.055640154059
3 | 161 | 0.055640154059
3 | 171 | -0.021530682050
3 | 172 | -0.006289394980
3 | 173 | -0.021530682050
3 | 174 | -0.006289394980
8-a | 1517 | -0.166093750630
8-a | 1518 | -0.048518165797
8-b | 1578 | -0.166093750630
8-b | 1579 | -0.048518165797
7 | 1671 | -2.575344486333
CT-vtx | 1781 | 0.778962156811
| Sum | -0.000000000017

  * •

IR-cancellation is also checked by using LE-method. Among two-loop diagrams,
the 8 diagrams in Fig.3 have IR-divergence. The diagrams with CT also have IR-
divergence through $\delta Z_{W}^{1/2}$ (19-diagrams) and $\delta
Z_{\mu}^{1/2}$ (28-diagrams). In LE-method, the IR-divergence is proportional
to $C_{IR}^{(2)}=(-1/\varepsilon_{R}-2\gamma+2\ln(4\pi))$,
$\varepsilon_{R}=(n/2-2)$. It is easily shown that the IR-divergence coming
from $\delta Z_{W}^{1/2}$ cancels among the 19-CT-diagrams. As for the
diagrams with $\delta Z_{\mu}^{1/2}$, IR-divergence is cancelled by the
corresponding two-loop diagrams. We show coefficients of $C_{IR}^{(2)}$ in
Table • ‣ 4. The correspondence between small photon mass ($\lambda$) method
and LE-method for IR-regularization is checked in the case of QED ladder
diagram. Analytic value of a coefficient of $\ln(\lambda^{2}/m_{\mu}^{2})$ in
unit of $(\alpha/\pi)^{2}$ is $(1/4)$ [5]. It is 0.249999998 by our
calculation using small photon mass. In LE-method, the coefficient of
$C_{IR}^{(2)}$ becomes 0.249999999. We understand the correspondence between
$\ln(\lambda^{2}/m_{\mu}^{2})$ and $C_{IR}^{(2)}$ is established.

Figure 3: IR cancellation among diagrams with $W$ and $\chi^{\pm}$ bosons Table 3: Cancellation of IR-divergence (unit $10^{-11}$) Diagram Type |  $\delta Z_{\mu}^{1/2}$ Vertex CT |  $\delta Z_{\mu}^{1/2}$ Self CT | two-loop diagram | sum
---|---|---|---|---
Neutral Type |  |  | diagram |
$\gamma$ | $-134887.2756\times 3$ | $134887.2756\times 2$ |  (a) $+134887.2755$ | $-1.6\times 10^{-4}$
$Z-boson$ | $0.2152351626\times 3$ | $-0.2152351626\times 2$ | (b) -0.2152351628 | $-1.7\times 10^{-10}$
$Higgs,\chi_{3}$ | $1.66293755\times 10^{-6}\times 3$ | $-1.66293755\times 10^{-6}\times 2$ | $-1.66293757\times 10^{-6}$ | $-2.5\times 10^{-14}$
Charged type |  |  |  diagram(c)$\sim$(h) |
$W^{\pm},\chi^{\pm}$ | -0.4312728113299 | $---$ | 0.4312728113330 | $3.1\times 10^{-12}$
|  |  |  |

As we show in Table • ‣ 4, no IR-divergence remains in the final expression.

  * •

Non-linear Gauge (NLG) parameter independence is also checked. Non linear
gauge was introduced to reduce the number of diagrams, particularly containing
boson-boson couplings [24]. Here, we adopt NLG to check the validity of our
calculation. The gauge fixing Lagrangian is constructed as,

$\displaystyle\mathscr{L}_{GF}=-\frac{1}{\xi_{W}}F^{+}F^{-}-\frac{1}{2\xi_{Z}}(F^{Z})^{2}-\frac{1}{\xi}(F^{A})^{2}$
(11)

where

$\displaystyle F^{\pm}$ $\displaystyle=$ $\displaystyle\left(\partial^{\mu}\mp
ie\tilde{\alpha}A^{\mu}\mp
i\frac{ec_{W}}{s_{W}}\tilde{\beta}Z^{\mu}\right)W_{\mu}^{\pm}+\xi_{W}\left(M_{W}\chi^{\pm}+\frac{e}{2s_{W}}\tilde{\delta}H\chi^{\pm}\pm
i\frac{e}{2s_{W}}\tilde{\kappa}\chi_{3}\chi^{\pm}\right)$ $\displaystyle
F^{Z}$ $\displaystyle=$
$\displaystyle\partial^{\mu}Z_{\mu}+\xi_{z}\left(M_{Z}\chi_{3}+\frac{e}{2s_{W}c_{W}}\tilde{\varepsilon}H\chi_{3}\right)$
$\displaystyle F^{A}$ $\displaystyle=$ $\displaystyle\partial^{\mu}A_{\mu}$
(12)

Here, $\tilde{\alpha},\tilde{\beta},\tilde{\delta},\tilde{\varepsilon}$ and
$\tilde{\kappa}$ are non-linear gauge parameters specific to this formalism.
The parameters $s_{W}$ and $c_{W}$ are the $sine$ and $cosine$ of Weinberg
angle $\theta_{W}$. In our calculation we set $\xi=\xi_{W}=\xi_{Z}=1$ to make
the gauge boson propagators simple. As an example of cancellation of NLG
parameters, we show the UV-divergent part proportional to
$\tilde{\alpha}^{2}$. The 53 diagrams make a group and we show sum of a
coefficient of $C_{UV}^{(2)}=(1/\varepsilon-2\gamma+2\ln(4\pi))$ for the group
of the same topological type in the Table • ‣ 4. By summing up these
contributions, 12 digits cancellation is confirmed even in 4 to 5 dimensional
integration over Feynman parameters.

Table 4: UV-part ($\tilde{\alpha}^{2}$-term) Type | Number of |  Coeff. of $C^{(2)}_{UV}$
---|---|---
| diagrams |  in unit $(\alpha/\pi)^{2}$
2a | 4 | 5.994906222E-08
2b | 4 | 5.994906222E-08
3 | 7 | -9.325381915E-06
5a | 13 | 1.398809747E-07
5b | 13 | 1.398809747E-07
8a | 2 | 4.422894730E-06
8b | 2 | 4.422894730E-06
9a | 4 | 3.996606459E-08
9b | 4 | 3.996606459E-08
sum(*) | 53 | -6.048741686E-18
|  |

(*) Each contribution is calculated in quadruple precision method and has more
effective digit than shown in the table.

  * •

Successive method is also applied to two-loop diagrams with self energy type
two-point function. An example is diagram with $(\gamma-\gamma)$ or
$(\gamma-Z)$ vacuum polarization type diagram. We decompose the
renormalization constants $\delta Z_{AA}^{1/2},\delta Z_{ZA}^{1/2},\delta
Z_{AZ}^{1/2}$ etc. into components according to the particles involved in the
loop. By adding the corresponding counter term to one loop unrenormalized two-
point function, we integrate over the first loop momentum $\ell_{1}$. The
result is finite renormalized two-point function $\Pi_{R}(\ell_{2})$. The
function $\Pi_{R}$ contains the second loop momentum $\ell_{2}$, however, the
divergent part of $\ell_{2}$ integration does not contribute to ($g$-2). We
use this alternative method to reconfirm the results obtained by the methods
given in section 3.

##  5 Finite contribution from restricted type of diagrams

The publicly accepted value of ELWK two-loop correction is classified
according to types of diagrams. For example, fermion loop contribution with or
without Higgs boson and purely bosonic contribution are discussed separately.
As an example, we summarize the naive fermion loop contribution without any
approximation. The diagrams we consider are shown in Fig.4.

Figure 4: Fermion loop diagrams

As is shown in Fig.4, we consider diagrams with both fermion loop and neutral
boson.

The contribution from fermion triangle diagrams Fig.(4-a,b) comes from VVA-
anomaly. If lepton and quark mass within the same generation are degenerate,
VVA-anomaly cancellation works so that the contribution becomes zero. So
fermion mass difference is crucial to get the value. The
$\ln(M_{z}^{2})$-dependence cancels within one generation, because of no-
anomaly condition. Czarnecki et al.[6] give the approximate formula for
diagram Fig.(4-b). Taking the same parameters (see below), we compare their
values and our exact results in Table 5.

Table 5:  Comparison of Fig.(4-b) (unit $10^{-11}$) lepton and quark | Ref.[6] approximate value | Our exact value
---|---|---
electron, up,down | -3.9984 | -4.0272
muon, charm,strange | -4.6524 | -4.6426
$\tau$, top, bottom | -8.1854 | -8.1940
Total | -16.8362 | -16.8638
|  |

Although each fermion contribution is quite different, the coincidence becomes
pretty good by taking the sum in each generation. The difference comes from
the common constant terms which cancels by summing up within each generation,
due to the anomaly free condition. We also show the preliminary value of other
diagrams Fig.(4-a,c,d,e) in Table 5.

Table 6: Finite value Fig(4-a,c,d,e) Type | Value $\\{10^{-11}\\}$
---|---
4-a | -0.0494
4-c | -1.8718
4-d | -0.1803
4-e |  1.5842
|

We use the following fermion and boson masses in the calculation in unit GeV.
$m_{\mu}=105.658389\times 10^{-3},m_{e}=0.51099906\times
10^{-3},m_{\tau}=1.7771,m_{u}=0.3,m_{c}=1.5,m_{t}=173.21,m_{d}=0.3,m_{s}=0.5,m_{b}=4.5,M_{W}=80.22,M_{Z}=91.187,M_{H}=125.09$

##  6 Discussion and Comment

We develop the system to calculate the full ELWK two-loop corrections to
($g$-2). Unfortunately it is not quite completely automatic system yet. We can
produce Fortran source automatically. The work we need beforehand is only to
prepare several files which only depend on the type of the topology of
diagrams as we explained in section 2.

If we adopt the well known first method to extract UV-divergent part, it is
crucial to introduce the most suitable transformations from $(z_{j})$ to [0,1]
integration variables $(x,y,u,v,w)$. In the case of Linear Expansion method
(LE), however, the choice of integration variables is not sensitive to get the
reliable results.

Further comments are given on the regularization method. Firstly, we introduce
$n$-dimensional regularization method and take out UV-divergence
$(1/\varepsilon)$, when one of the integration variables is approaching 0
($x\rightarrow 0$). In order to perform this procedure, we need rather complex
operations including differentiation of the amplitudes, etc. As a result, it
makes Fortran source lengthy so that CPU-time increases extensively. The
second prescription to regularize both UV- and IR-divergences is LE-method,
briefly explained in section 3. This method decreases the number of operation
drastically. It is sufficient to define the quantity as function of
$\varepsilon(=2-n/2)$ . We only need to treat Dirac matrices and various
vectors appeared in the numerator, in $n$-dimension. This is easily done by
using symbolic manipulation system such as FORM. The operation is simple and
we can make use of the resultant short sources for both UV-($\varepsilon>0$)
and IR-($\varepsilon_{R}=-\varepsilon>0$) regularization and also to get
finite results. By comparing the results of various methods mentioned in the
previous sections, we conclude that LE-method is the most simple and reliable
method, at this moment. In order to get reliable physical value by this
method, accurate numerical integration over Feynman parameters is inevitable.
The DE-method introduced in section 4 is the suitable candidate.

By using these technical approaches mentioned above, we clear almost all the
constraints given in section 4. Namely, (i) reproduction of QED values,
(ii)(iii) UV-and IR-divergences are cancelled, (iv) the result is independent
of NLG-parameters. We show some samples how they are cleared. We are now
recalculating and rechecking the constraints under the best computer
circumstances. The final value will be given soon. We expect ongoing work
provides the fruitful foundation to formulate PNQFT (Perturbative Numerical
Quantum Field Theory) mentioned in section 1.2.

We would like to thank Prof.T.Kaneko for his important contribution to
construct the framework of calculation at the early stage of this work. We
also wish to thank Prof.K.Kato and Prof.F.Yuasa for discussion. Last but not
least, we express our deep appreciation to late Prof.Y.Shimizu for his
continual encouragement. This research is partially supported by Grant-in-Aid
for Scientific Research (15H03668,15H03602) of JSPS.

## References

## References

  * [1] Yuasa F et al. 2000 Prog. Theore. Phys. Suppl.  138 18 Belanger G, Boudjema F, Fujimoto J, Ishikawa T, Kaneko T, Kato K and Shimizu Y 2006 Phys.Rept. 430 117
  * [2] Aoyama T, Hayakawa M, Kinoshita T and Nio M 2012 Phys.Rev.Lett. 109 111808
  * [3] Kukhto T V, Kuraev E A, Schiller A and Silagadze Z K 1992 Nucl.Phys. B371 567
  * [4] Kaneko T and Nakazawa N 1995 Proc.of AIHENP(Artificial Intelligence for High Energy and Nuclear Physics,(Pisa,Italy) 173
  * [5] Karplus R and Kroll N M 1950 Phys.Rev.77 536 Sommerfield C M 1957 Phys.Rev.107 328; 1958 Annals of Physics 5 26 Petermann A 1957 Hel.Phys.Acta30 407; 1958 Nucl. Phys. 5 677
  * [6] Czarnecki A, Krause B and Maciano W J 1995 Phys.Rev. D52 R2619 Czarnecki A, Krause B and Marciano W J 1996 Phys.Rev.Lett. 76 3267 Czarnecki A, Marciano W J and Vainshtein A 2003 Phys.Rev. D67 073006 Erratum ibid. 2006 Phys.Rev. D73 119901
  * [7] Gribouk T and Czarnecki A 2005 Phys.Rev. D72 053016
  * [8] Marquard P 2015 Nuclear and Particle Physics Proceedings 260 107
  * [9] Bennett G W et al. 2002 Phys.Rev.Lett. 89 101804 Erratum ibid 2002 Phys.Rev.Lett. 89 129903 Bennett G W et al. 2004 Phys.Rev.Lett. 92 161802 Bennett G W et al. 2006 Phys.Rev. D73 072003
  * [10] Patrignani C et al. (Particle Data Group) 2016 Chin.Phys. C40 100001 pp.715
  * [11] Davier M, Hoecker A, Malaescu B and Zhang Z 2011 Eur.Phys.J. C71 1515
  * [12] Prades J, de Rafael E and Vainshtein A 2009 Advanced series on directions in high energy physics vol. 20 ed Roberts B L and Marciano W pp 303-317
  * [13] Jackiw R and Weinberg S 1972 Phys.Rev. D5 2396 Altarelli G, Cabibbo N and Maiani L 1972 Phys.Lett. B40 415 Bars I and M Yoshimura 1972 Phys.Rev. D6 374 Fujikawa K, Lee B W and Sanda A I 1972 Phys.Rev. D6 2923
  * [14] Gnendiger C, St$\ddot{\rm o}$ckinger D and St$\ddot{\rm o}$ckinger-Kim H 2013 Phys.Rev. D88 053005 Heinemeyer S, St$\ddot{\rm o}$ckinger D and Weiglein S 2004 Nucl.Phys. B699 103
  * [15] Mohr P J, Taylor B N and Newell D B (CODATA Group) 2012 Rwv.Mod.Phys. 84 1527
  * [16] Gohn W 2016 The Muon $g$-2 experiment at Fermilab Preprint arXiv:1611.04964[hep-ex]
  * [17] Otani M (E34 collaboration) 2015 JPS Conf.Proc. 8 Status of the Muon $g$-2/EDM Experiment at J-PARC(E34) 025008
  * [18] de Doncker E, Yuasa F and Kurihara Y 2011 Proc., 3rd CPP Workshop on Computational particle physics (CPP 2010) (Tsukuba, Japan, September 23-25,2010) p 11 de Doncker E, Fujimoto J, Hamaguchi N, Ishikawa T, Kurihara Y, Shimizu Y and Yuasa F 2012 Jounal of Computational Science 3 102 de Doncker E, Yuasa F, Kato K, Ishikawa T,Kapenga J and O.Olagbemi 2017, Preprint arXiv:1702.04904[hep-ph]
  * [19] Civitanovi$\acute{\rm c}$ P and Kinoshita T 1974 Phys.Rev. D10 3978, ibid 3991
  * [20] Vermaseren J A M 2000 preprint arXiv:0010025[math-ph]
  * [21] Bollini C G and Giambiagi J J 1972 IL Nuovo Cimento B 12 20 ’t Hooft G and M Veltman 1972 Nucl.Phys. B44 189
  * [22] Nakanishi N 1975 Quantum Field Theory (Tokyo:Baihukan) p.272 (in Japanese)
  * [23] Takahashi H and Mori M 1974 Publ. RIMS Kyoto Univ. 9 721
  * [24] Boudjema Z and Chopin E 1996 Z.Phys.Rev. C73 85


# Modified RRT* for Path Planning in Autonomous Driving

Sugirtha Ta, Pranav Sb, Nitin Benjamin Dasiahb, Sridevi Mb
aDepartment of CSE, Indian Institute of Information Technology, Kottayam,
India
bDepartment of CSE, National Institute of Technology, Tiruchirappalli-620015,
India

###### Abstract

Essential tasks in autonomous driving includes environment perception,
detection and tracking, path planning and action control. This paper focus on
path planning, which is one of the challenging task as it needs to find
optimal path in highly complex and dynamic environments. Usually, a driving
scenario has large num- ber of obstacles in their route. In this paper, we
propose a two-stage path planning algorithm named Angle-based Directed Rapidly
exploring Random Trees (AD- RRT*) to address the problem of optimal path in
complex environment. The proposed algorithm uses A* algorithm for global path
planning and modifies RRT* to bound the samples using angle. The efficiency of
the proposed algorithm is evaluated through experiments in different scenarios
based on the location and number of obstacles. The proposed algorithm showed
higher rate of convergence with reduced time and less number of nodes than the
base RRT* algorithm.

##  I Introduction

Autonomous driving has emerged as a hot research topic in recent decades.
Hence, many researchers and automotive companies spend billions of dollars in
developing various modules to make Level-5 self driving cars. Deep learning
has achieved state-of-the-art results in both semantic and geometric tasks in
autonomous driving which includes object detection [1, 2, 3, 4], semantic
segmentation [5, 6, 7, 8], depth prediction [9, 10, 11, 12], adverse weather
detection [13, 14, 15, 16], moving object detection [17, 18, 19, 20], SLAM
[21, 22, 23], multi-task learning [24, 25, 26, 27] and sensor fusion [28, 29,
30]. The preliminary task is object detection and tracking, and the next
crucial task in a self-driving car is to find the optimal path from source to
destination point. Path planning helps the ego-vehicle to navigate through the
environment by considering the surrounding obstacles.
Path planning algorithms have widespread applications in autonomous driving
[31], planetary and space missions [32], Unmanned Aerial Vehicles (UAVs) [33]
, computerized robotic surgery [34], , artificial intelligence for video games
[35], and molecular biology [36].
Path planning refers to generation of collision free trajectory from an
initial state to destination state and a velocity profile to the controller by
considering the following factors :

  * •

State of the self driving car which includes velocity, inclination etc.,

  * •

Surrounding environment of the ego-vehicle which includes static and dynamic
obstacles, drivable spaces

  * •

Traffic laws and allowed maneuvers

The path planning task is accomplished as a sequence of four subtasks namely
(i) Global path planning (ii) behavioural planner (iii) local path planning
(iv) local feedback control.
Global path planning refers to finding the collision free optimal path from
starting state to the destination state. Local path planning refers to
planning the intermediate local move by creating the nodes within the
coordinate space.

Various algorithms were proposed over past four decades to solve path-planning
problem in static and dynamic environment which includes grid based [37, 38],
Artificial Potential Field (APC) [39, 40, 41], evolutionary methods [42, 43,
44, 45], geometric and neural networks [46, 47]. An extensive survey that
describes the merits and demerits of these path planning algorithms can be
found in [48, 49, 50].

Figure 1: Tree expansion process of RRT* [51]

Though, there were several techniques employed for the path planning, the
following issues were identified:

  * •

Slow convergence to optimal solution

  * •

Longer paths from source to destination

  * •

High memory requirement by creating large number of nodes

  * •

Not suitable for complex dynamic environment

To address these issues, this paper proposes a variant of RRT*. The major
contributions of this paper is given as follows :

  * •

Proposed an angle based detector in front of the ego-vehicle to quickly find
optimal path

  * •

Produce comparatively shorter path than base RRT*

  * •

Consumes less memory

  * •

Deals with complex environment

The remaining section of the paper is organized as follows : Section 2
describes the detailed explanation of various existing path planning
algorithms. Section 3 explains the proposed Angle-based Directed RRT* in
detail manner. The experimental results and performance analysis of the
proposed method is provided in Section 4. Finally, Section 5 concludes the
paper.

##  II Related Work

This section explains the local path planning methods exists in the
literature. We classify them into two categories namely, (i) Potential field
based path planning methods and (ii) Graph based path planning methods.
Conventional path planning approaches use Potential Field (PF) [52] which
includes attractive and repulsive potential fields. The vehicle is attracted
towards the goal by the attractive field and repelled from obstacles by the
repulsive field. The limitation of conventional Potential Field Method (PFM)
is that it gets stuck at local minima. To overcome the limitation of
conventional PFM, modified Artificial Potential Field (APF) [53] was proposed
where the repulsive field makes use of goal co-ordinates from camera, obstacle
co-ordinates from radar and autonomous vehicle co-ordinate from Global
positioning system (GPS) and computes the improved artificial potential field.
Obstacle dependent Gaussian Potential Field (GPF) proposed in [54] calculates
repulsive potential field by considering the objects information from the
sensor within a threshold and uses yaw angle information and calculate
attractive field to find the path.
In recent years, Sampling Based Planning (SBP) methods are probabilistic
complete and they are most extensively used for finding the optimal path. They
perform random sampling in search space and are computationally efficient in
solving high dimensional real time complex problems. They create a roadmap of
possible trajectories in configuration space without explicitly using the
obstacle information.

Figure 2: Rewiring Process of RRT* [51]. (a) Finding near vertices (b)
Selection of best parent (c) cost check (d) rewiring with minimum cost  Figure
3: Lane changing maneuver [55]

A study and analysis of various SBP techniques can be found in [39].
Probabilistic Road Map (PRM) [56] based approach is widely used in highly
structured static environment and suitable for holonomic constraints. But,
they depend on geometry of obstacles and doesn’t ensure asymptotic optimality.
Rapidly exploring Random Tree (RRT) is one of the fastest path planning
algorithms which supports dynamic cluttered environment and can be extended to
non-holonomic constraints. But, it does not ensure asymptotic optimality. Due
to its advantages, various extensions of RRT were proposed to improve the
efficiency. A major breakthrough was witnessed in high dimensional optimal
path planning algorithms with asymptotic optimality after the introduction of
RRT* [51] in 2010. The major advantage of RRT* is the quick exploration of
initial path and optimizes it in successive iterations. The asymptotic optimal
property of RRT* generates near optimal path when the number of iterations
approach infinity makes it very expedient for real time applications. The tree
expansion process of RRT* is shown in Fig 1. Nearest neighbour search and
rewiring operations are shown in Fig 2. However, RRT* never produce the
optimal path in finite time and the convergence rate is slow. This happens
because it explores and samples the nodes in whole configuration space. Also,
the space complexity is high as the number of nodes in the tree is high.
To address the above drawbacks, a new variant of RRT*, named RRT*-smart [57]
was proposed. It uses intelligent sampling and path optimization for faster
convergence of optimal path. RRT*-smart finds the initial path similar to RRT*
and optimizes the path by performing intelligent sampling at regular intervals
using biasing ratio. RRT*-smart converges faster than RRT*, but adjusting the
biasing ratio causes computational overhead. RRT*-Adjustable Bounds (RRT*-AB)
address the issues of RRT*-smart such as dense tree and high memory
requirements. RRT*-AB finds optimal path by exploring only favorable regions
and focusing on features like (i) connectivity (ii) intelligent bounded
sampling and (iii) path optimization. In order to overcome the limitations of
RRT and RRT*, we propose a two-stage path planning algorithm for autonomous
driving by modifying base RRT*.
In this paper, we focus on the path planning task. To ensure safety, the
surrounding vehicles need to follow correct lane, overtake other vehicles when
it is safe to do so, stop at the intersections and take U-turns if required
and do efficient parking in the allotted space. A lane changing maneuver
scenario of an autonomous vehicle is shown in Fig 3.

##  III Proposed Work

This section explains the proposed two-stage path planning algorithm which
finds global path and local path to plan the maneuvers. The overall pipeline
of various tasks involved in autonomous driving is shown in Fig 4. The
proposed work focus on the decision making task which finds the optimal path
from source to destination points using Algorithm 1.

The proposed algorithm utilizes A* algorithm [58] to create a global path for
deciding which roads to turn. This global path will be used by the local path
to decide the angles. RRT and RRT* algorithms are generally very inefficient
for real-time path planning. They tend to search large portions of the map,
that is not necessarily useful in case of vehicles. We propose Angle-Based
Directed RRT* (AD-RRT*) to not only make the algorithm more efficient for
vehicles, but to add another layer of algorithm on top of it, so the algorithm
can be tuned to required situations with the help of hyper-parameters. The
proposed method uses angle to bound the sample. As the time taken to find a
path increases, the angle increases with expansion of the search area. The
angle uses the global path for the initial angle. If there is an obstacle and
the algorithm is unable to find a path, it expands its search angle, equally
on both sides.

Figure 4: General pipeline of autonomous driving tasks [focus on Decision
making task]

After a local path reaches a given distance towards the global path, the
vehicle will move towards it, and updation of the point from which the angle
is calculated, reducing the search area back to 0°. For most cases, it will
force the vehicle to go towards the goal path reaching the final state. This
method of bounding the search area, will not always produce an optimal path
and in worst case scenarios take more time to search for path. Therefore, the
proposed algorithm is best used for vehicles and needs more optimizations to
work efficiently. It is required to use visibility regions to search in
shorter distances than longer distances when there are more crowd. This allows
for searching through narrow areas if needed.

Notation :
Let $\mathbb{Z}\subset\mathbb{R}^{n},n\in\mathbb{N}$ represents the
configuration space and $\mathbb{Z}_{obs}$ represents the obstacle region and
$\mathbb{Z}_{\mathrm{free}}=\mathbb{Z}/\mathbb{Z}_{\mathrm{obs}}$ denotes the
obstacle free region. $\mathbb{Z}_{src}$ and $\mathbb{Z}_{destn}$ denotes the
starting and destination points respectively. $u:[0:s]$ denotes the path
connecting the states $z_{1}$ and $z_{2}$ such that $u(0)=z_{1}$ and
$u(s)=z_{2}$, where s is the positive scalar length of the path. $\Sigma_{f}$
denotes the set of paths in $\mathbb{Z}_{\mathrm{free}}$ that are collision
free and $\sigma_{f}\in Z_{\text{free }}$ denotes end to end feasible path.
${\mathbb{C}_{\text{Region }}}$ denotes a region with connected nodes near to
the starting node $\mathbb{Z}_{src}$.

Algorithm 1 Angle based Directed RRT*

1:$T\leftarrow\text{ InitializeTree }()$

2:$T\leftarrow\text{ InsertNode }\left(\phi,z_{\text{src }},T\right)$

3:$Angle\leftarrow 0\degree$

4:$A_{Length}\leftarrow collectfrompreviouslayervisibility$

5:$C_{\text{Region }}\leftarrow\text{ ConnectivityRegion }\left(z_{\text{src
}},A_{Length},A_{angle},T\right)$

6:for $i\leftarrow 0\quad\text{to}\quad N$ do

7: $z_{\text{rand }}\leftarrow\text{ Bounded Sample }\left(i,C_{\text{Region
}}\right)$

8: $z_{\text{nearest }}\leftarrow\text{ Nearest }\left(T,z_{\text{rand
}}\right)$

9: $\left(z_{\text{new }},U_{\text{new }}\right)\leftarrow\text{ Steer
}\left(z_{\text{nearest }},z_{\text{rand }}\right)$

10: if $\text{CollisionCheck }\left(z_{\text{new }}\right)$ then

11: $z_{\text{near }}\leftarrow\operatorname{Near}\left(T,z_{\text{new
}},|V|\right)$

12: $z_{\text{min }}\leftarrow\text{ ChooseParent }\left(z_{\text{near
}},z_{\text{nearest }},z_{\text{new }}\right)$

13: $T\leftarrow\text{ Insert Node }\left(z_{\min},z_{\text{new }},T\right)$

14: $T\leftarrow\text{ Rewire }\left(T,z_{\text{near }},z_{\min},z_{\text{new
}}\right)$

15: if $PathFound(T)$ then

16: $C_{\text{Region }}\leftarrow\text{ ConnectivityRegion
}\left(z_{\text{init }},z_{\text{destn }},T\right)$

17: else if $Timetakenhigh\left(\right)$

18: $C_{\text{Region }}\leftarrow\text{ ConnectivityRegion
}\left(z_{\text{init }},z_{\text{goal }},T\right)$

19: $angle\leftarrow angle+increase$

20: $C_{\text{Region }}\leftarrow\text{ ConnectivityRegion }\left(z_{\text{src
}},A_{Length},A_{angle},T\right)$

21: $T\leftarrow\text{ Prune Path }(T)$

22: return

23: end if

24: end if

25:end for

TABLE I: Comparison of base RRT* with Proposed Angle-based Directed RRT*

No. of obstacles | Base RRT* | Angle based Directed RRT* (AD-RRT*)
---|---|---
No. of nodes |  | Time taken
---
(s)
Total path cost | Average path cost | No. of nodes |  | Time taken
---
(s)
Total path cost | Average path cost
50 | 8174 | 21.28 | 41203.74 | 934.01 | 6904 | 20.7 | 39556.83 | 791.13
68 | 9070 | 30.05 | 39226.16 | 784.52 | 8534 | 31.42 | 38502.22 | 770.04
70 | 11803 | 78.74 | 46700.4 | 934.01 | 8656 | 67.18 | 43174.01 | 863.48
80 | 15353 | 59.5 | 44426.36 | 888.52 | 10461 | 47.55 | 41802.47 | 836.04
85 | 13198 | 56.48 | 44419.73 | 888.39 | 11834 | 56.27 | 41530.99 | 830.62
100 | 16785 | 99.72 | 47707.62 | 954.15 | 12567 | 88.23 | 44661.59 | 893.23

The proposed algorithm finds a feasible path $\sigma_{f}\in\Sigma_{f}$ from
$\sigma_{f(0)}=z_{\text{src }}$ to $\sigma_{f(s)}=z_{\text{destn}}$ by
building a Tree $T=(V,E)$, that has the vertices ($V$) sampled from obstacle
free region $\mathbb{Z}_{\mathrm{free}}$ and ($E$), the edges that connect
these vertices.

Figure 5: Visual results of Path planning for Scenarios S1, S2, S3, S4, S5 and
S6. (a) and (b) under each scenario denotes the intermediate and final step in
path planning. Figure 6: Visual results of proposed method on CARLA Simulator
(a) Plot of global path planning, Local path planning - (b) Initial step (c)
Intermediate step and (d) Final step where the ego-vehicle takes the right
turn

The major procedures of the algorithm uses the following task :

Bounded-sampling : A state ${Z_{rand}}$is placed randomly from obstacle free
region $\mathbb{Z}_{free}$.

Connectivity Region : A connectivity region ${\mathbb{C}_{\text{Region }}}$ is
identified in search space near to $\mathbb{z}_{src}$ using the area with
length.

Nearest : The nearest node from $T=(V,E)$ to ${z_{rand}}$ is identified by
$Nearest(T,z_{rand})$ based on the cost computed by the distance function as
given in equation (1).

$\Delta d=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}$
(1)

Steer : The procedure $\operatorname{Steer}\left(z_{\text{near
}},z_{\text{rand }}\right)$ provides a control input $u[0,T]$ that drives the
system from $x(0)=z_{\text{rand}}$ to $x(T)=z_{\text{near}}$ along the path
$x:[0,T]\rightarrow X$ giving $z_{\text{new}}$ at a distance $\Delta q$ from
$z_{\text{near}}$ towards $z_{\text{rand}}$, where $\Delta q$ is the
incremental distance.

Collision check : This procedure determines if a path $u[0,T]$ exists in the
obstacle free region $\mathbb{Z}_{free}$ for all $t=0$ to $t=T$.

Near : This procedure returns the nearby neighbouring nodes as given by
equation (2):

$k=\gamma(\log(n)/n)^{(1/d)}$ (2)

Choose parent : It selects the best parent $\mathbb{z}_{min}$ which gives the
minimum cost from the source point to $\mathbb{z}_{new}$.

Insert node : It adds a node $\mathbb{z}_{min}$ to $V$ in the tree $T=(V,E)$
and connects it to $\mathbb{z}_{min}$ as its parent.

Rewire : Rewiring is performed after the parent is selected. It computes the
cost of all nodes through $\mathbb{z}_{new}$ and compared with their previous
costs. If the cost is minimum for any node, then $\mathbb{z}_{new}$ will be
made as its new parent.

Prunepath : It returns the nodes that are connectable from source to
destination point in the obstacle free region. A tree $T$ is initialized with
$\mathbb{z}_{src}$ as the root node. Bounded sampling randomly selects nodes
restricted by an angle from ${\mathbb{C}_{\text{Region }}}$ and the tree is
populated with these nodes. ${\mathbb{C}_{\text{Region }}}$ is defined by
expansion distance scale in a search space between $\mathbb{z}_{src}$ and
$\mathbb{z}_{destn}$ using equation (3):

$D_{\text{scale }}=E/m$ (3)

where, E – size of the environment map
m – expansion factor
When a single scan of ${\mathbb{C}_{\text{Region }}}$ is completed using
bounded sampling and no path is found in the first scan and the time exceeds
the limit, then the angle of ${\mathbb{C}_{\text{Region }}}$ is increased as
mentioned in Steps 15-19 of the Algorithm 1.

##  IV Experimental results and Analysis

This section demonstrates the simulation results of proposed Angle-Based
Directed RRT* with CARLA simulator [59]. The experiments were carried out with
different scenario based on the density (cluster) of the obstacle nodes. The
proposed approach is implemented using Python in an Intel i7, NVIDIA 1080 GTX
GPU.

The proposed algorithm is experimented on six different scenarios i.e. ( S1,
S2, S3, S4, S5 and S6) which represents the distribution of obstacle nodes
(N). The scenarios are explained as follows :

  1. 1.

S1 – Obstacles densed near starting point with N=50

  2. 2.

S2 - Sparse distribution of obstacles throughout the path from starting to
destination point with N=50

  3. 3.

S3 - Sparse distribution of obstacles throughout the path from starting to
destination point with N=68

  4. 4.

S4 – Entire region equally densed with obstacles N=70

  5. 5.

S5 - Obstacles densed in middle region with N=80

  6. 6.

S6 - Obstacles densed near starting, middle and destination region with N=85

Fig 5 shows the path planning results of base RRT* and the proposed Angle-
Based Directed RRT* under different scenarios. The global path planning A*
algorithm finds the path between $Z_{src}$ and $Z_{destn}$. The local path
planning is performed by the proposed Angle-Based Directed RRT*. From the
Figure, it is clear that the base RRT* expands the tree in entire search space
and thus results in generating dense tree with more number of unnecessary
nodes. On the contrary, the proposed AD-RRT* restricts the search space to an
angle and finds the optimal path by generating less number of nodes i.e the
convergence rate is high with less time.

Table I shows the comparison between base RRT* and proposed AD-RRT* for
varying number of obstacles. The proposed method finds optimal path in less
time by generating less number of nodes. Also, the total path cost and average
path cost are less compared to base RRT*. Fig 6 shows the output of CARLA
simulator.

##  V Conclusion

In this paper, a two-stage path planning algorithm called Angle-based Directed
RRT* is proposed to find the optimal path for autonomous vehicles in complex
driving environments. The proposed method deployed A* algorithm to find global
path and compute the local path in restricted search space by modifying RRT*.
Analysis of the proposed algorithm is carried out in six different scenarios
using CARLA simulator and the results show faster convergence with less number
of nodes than the base RRT* algorithm. In future, the proposed algorithm is
extended to dynamic environment and the performance is analysed.

##  VI Acknowledgement

This publication is an outcome of the R&D work undertaken in the project under
TiHAN Faculty Fellowship of NMICPS Technology Innovation Hub on Autonomous
Navigation Foundation being implemented by Department of Science & Technology
National Mission on Interdisciplinary Cyber-Physical Systems (DST NMICPS) at
IIT Hyderabad.

## References

  * [1] M. Klingner, S. Borse, V. R. Kumar, B. Rezaei, V. Narayanan, S. Yogamani, F. Porikli, X3kd: Knowledge distillation across modalities, tasks and stages for multi-camera 3d object detection, in: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2023, pp. 13343–13353.
  * [2] S. Mohapatra, S. Yogamani, H. Gotzig, S. Milz, P. Mader, Bevdetnet: bird’s eye view lidar point cloud based real-time 3d object detection for autonomous driving, in: 2021 IEEE International Intelligent Transportation Systems Conference (ITSC), IEEE, 2021, pp. 2809–2815.
  * [3] H. Rashed, E. Mohamed, G. Sistu, V. R. Kumar, C. Eising, A. El-Sallab, S. Yogamani, Fisheyeyolo: Object detection on fisheye cameras for autonomous driving, in: Proceedings of the Machine Learning for Autonomous Driving NeurIPS 2020 Virtual Workshop, Virtual, Vol. 11, 2020.
  * [4] S. Thayalan, S. Muthukumarasamy, K. Santhakumar, K. B. Ravi, H. Liu, T. Gauthier, S. Yogamani, Monocular three-dimensional object detection using data augmentation and self-supervised learning in autonomous driving, Journal of Electronic Imaging 32 (1) (2022) 011004.  doi:10.1117/1.JEI.32.1.011004.
  * [5] A. Briot, P. Viswanath, S. Yogamani, Analysis of efficient cnn design techniques for semantic segmentation, in: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition Workshops, 2018, pp. 663–672.
  * [6] S. Chennupati, G. Sistu, S. Yogamani, S. Rawashdeh, Auxnet: Auxiliary tasks enhanced semantic segmentation for automated driving, arXiv preprint arXiv:1901.05808.
  * [7] G. Sistu, I. Leang, S. Yogamani, Real-time joint object detection and semantic segmentation network for automated driving, arXiv preprint arXiv:1901.03912.
  * [8] M. Siam, S. Valipour, M. Jagersand, N. Ray, S. Yogamani, Convolutional gated recurrent networks for video semantic segmentation in automated driving, in: 2017 IEEE 20th International Conference on Intelligent Transportation Systems (ITSC), IEEE, 2017, pp. 1–7.
  * [9] V. R. Kumar, S. Milz, C. Witt, M. Simon, K. Amende, J. Petzold, S. Yogamani, T. Pech, Near-field depth estimation using monocular fisheye camera: A semi-supervised learning approach using sparse lidar data, in: CVPR Workshop, Vol. 7, 2018, p. 2.
  * [10] V. R. Kumar, M. Klingner, S. Yogamani, M. Bach, S. Milz, T. Fingscheidt, P. Mäder, Svdistnet: Self-supervised near-field distance estimation on surround view fisheye cameras, IEEE Transactions on Intelligent Transportation Systems 23 (8) (2021) 10252–10261.
  * [11] A. R. Sekkat, Y. Dupuis, V. R. Kumar, H. Rashed, S. Yogamani, P. Vasseur, P. Honeine, Synwoodscape: Synthetic surround-view fisheye camera dataset for autonomous driving, IEEE Robotics and Automation Letters 7 (3) (2022) 8502–8509.
  * [12] V. R. Kumar, S. Yogamani, M. Bach, C. Witt, S. Milz, P. Mäder, Unrectdepthnet: Self-supervised monocular depth estimation using a generic framework for handling common camera distortion models, in: 2020 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), IEEE, 2020, pp. 8177–8183.
  * [13] M. M. Dhananjaya, V. R. Kumar, S. Yogamani, Weather and light level classification for autonomous driving: Dataset, baseline and active learning, in: 2021 IEEE International Intelligent Transportation Systems Conference (ITSC), IEEE, 2021, pp. 2816–2821.
  * [14] M. Uricár, J. Ulicny, G. Sistu, H. Rashed, P. Krizek, D. Hurych, A. Vobecky, S. Yogamani, Desoiling dataset: Restoring soiled areas on automotive fisheye cameras, in: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops, 2019, pp. 0–0.
  * [15] M. Uricár, D. Hurych, P. Krizek, et al., Challenges in designing datasets and validation for autonomous driving, in: Proceedings of the International Conference on Computer Vision Theory and Applications, 2019.
  * [16] S. Ramachandran, G. Sistu, J. McDonald, S. Yogamani, Woodscape fisheye semantic segmentation for autonomous driving–cvpr 2021 omnicv workshop challenge, arXiv preprint arXiv:2107.08246.
  * [17] M. Siam, H. Mahgoub, M. Zahran, S. Yogamani, M. Jagersand, A. El-Sallab, Modnet: Moving object detection network with motion and appearance for autonomous driving, arXiv preprint arXiv:1709.04821.
  * [18] H. Rashed, A. El Sallab, S. Yogamani, M. ElHelw, Motion and depth augmented semantic segmentation for autonomous navigation, in: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops, 2019, pp. 0–0.
  * [19] E. Mohamed, M. Ewaisha, M. Siam, H. Rashed, S. Yogamani, W. Hamdy, M. El-Dakdouky, A. El-Sallab, Monocular instance motion segmentation for autonomous driving: Kitti instancemotseg dataset and multi-task baseline, in: 2021 IEEE Intelligent Vehicles Symposium (IV), IEEE, 2021, pp. 114–121.
  * [20] B. R. Kiran, K. Ramakrishnan, Y. S. Kumar, K. Anoop, An improved connected component labeling by recursive label propagation, in: 2011 National Conference on Communications (NCC), IEEE, 2011, pp. 1–5.
  * [21] N. Tripathi, S. Yogamani, Trained trajectory based automated parking system using visual slam on surround view cameras, arXiv preprint arXiv:2001.02161.
  * [22] L. Yahiaoui, J. Horgan, B. Deegan, S. Yogamani, C. Hughes, P. Denny, Overview and empirical analysis of isp parameter tuning for visual perception in autonomous driving, Journal of Imaging 5 (10) (2019) 78.
  * [23] L. Yahiaoui, J. Horgan, S. Yogamani, C. Hughes, B. Deegan, Impact analysis and tuning strategies for camera image signal processing parameters in computer vision, in: Irish Machine Vision and Image Processing conference (IMVIP), Vol. 2, 2011.
  * [24] I. Leang, G. Sistu, F. Bürger, A. Bursuc, S. Yogamani, Dynamic task weighting methods for multi-task networks in autonomous driving systems, in: 2020 IEEE 23rd International Conference on Intelligent Transportation Systems (ITSC), IEEE, 2020, pp. 1–8.
  * [25] G. Sistu, I. Leang, S. Chennupati, S. Yogamani, C. Hughes, S. Milz, S. Rawashdeh, Neurall: Towards a unified visual perception model for automated driving, in: 2019 IEEE Intelligent Transportation Systems Conference (ITSC), IEEE, 2019, pp. 796–803.
  * [26] H. Rashed, S. Yogamani, A. El-Sallab, P. Krizek, M. El-Helw, Optical flow augmented semantic segmentation networks for automated driving, arXiv preprint arXiv:1901.07355.
  * [27] V. R. Kumar, C. Eising, C. Witt, S. Yogamani, Surround-view fisheye camera perception for automated driving: Overview, survey & challenges, IEEE Transactions on Intelligent Transportation Systems.
  * [28] M. Pöpperli, R. Gulagundi, S. Yogamani, S. Milz, Capsule neural network based height classification using low-cost automotive ultrasonic sensors, in: 2019 IEEE Intelligent Vehicles Symposium (IV), IEEE, 2019, pp. 661–666.
  * [29] K. Dasgupta, A. Das, S. Das, U. Bhattacharya, S. Yogamani, Spatio-contextual deep network-based multimodal pedestrian detection for autonomous driving, IEEE transactions on intelligent transportation systems 23 (9) (2022) 15940–15950.
  * [30] C. Eising, J. Horgan, S. Yogamani, Near-field perception for low-speed vehicle automation using surround-view fisheye cameras, IEEE Transactions on Intelligent Transportation Systems 23 (9) (2021) 13976–13993.
  * [31] X. Lan, S. D. Cairano, Continuous curvature path planning for semi-autonomous vehicle maneuvers using rrt, 2015 European Control Conference (ECC) (2015) 2360–2365.
  * [32] G. Lau, H. H.-T. Liu, Real-time path planning algorithm for autonomous border patrol: Design, simulation, and experimentation, Journal of Intelligent & Robotic Systems 75 (2014) 517–539.
  * [33] X. Zhang, J. Chen, B. Xin, Path planning for unmanned aerial vehicles in surveillance tasks under wind fields, Journal of Central South University 21 (2014) 3079–3091.  doi:10.1007/s11771-014-2279-7.
  * [34] N. Ahmidi, G. D. Hager, L. Ishii, G. L. Gallia, M. Ishii, Robotic path planning for surgeon skill evaluation in minimally-invasive sinus surgery, in: N. Ayache, H. Delingette, P. Golland, K. Mori (Eds.), Medical Image Computing and Computer-Assisted Intervention – MICCAI 2012, Springer Berlin Heidelberg, Berlin, Heidelberg, 2012, pp. 471–478.
  * [35] K. Naderi, J. Rajamäki, P. Hämäläinen, Rt-rrt* a real-time path planning algorithm based on rrt, in: Proceedings of the 8th ACM SIGGRAPH Conference on Motion in Games, 2015, pp. 113–118.
  * [36] J. Reif, Z. Sun, Nano-robotics motion planning and its applications in nanotechnology and biomolecular computing, in: NSF Design and Manufacturing Grantees Conference, 1999, pp. 5–8.
  * [37] P. E. Hart, N. J. Nilsson, B. Raphael, A formal basis for the heuristic determination of minimum cost paths, IEEE Transactions on Systems Science and Cybernetics 4 (2) (1968) 100–107.  doi:10.1109/TSSC.1968.300136.
  * [38] K. Daniel, A. Nash, S. Koenig, A. Felner, Theta*: Any-angle path planning on grids, J. Artif. Intell. Res. (JAIR) 39.  doi:10.1613/jair.2994.
  * [39] M. Elbanhawi, M. Simic, Sampling-based robot motion planning: A review, IEEE Access 2 (2014) 56–77.  doi:10.1109/ACCESS.2014.2302442.
  * [40] Y. K. Hwang, N. Ahuja, Gross motion planning—a survey, ACM Comput. Surv. 24 (1992) 219–291.
  * [41] C. L. Goerzen, Z. Kong, B. Mettler, A survey of motion planning algorithms from the perspective of autonomous uav guidance, Journal of Intelligent and Robotic Systems 57 (2010) 65–100.
  * [42] J.-H. Liang, C.-H. Lee, Efficient collision-free path-planning of multiple mobile robots system using efficient artificial bee colony algorithm, Adv. Eng. Softw. 79 (C) (2015) 47–56.  doi:10.1016/j.advengsoft.2014.09.006.
  * [43] X. Zhang, Y. Zhao, N. Deng, K. Guo, Dynamic path planning algorithm for a mobile robot based on visible space and an improved genetic algorithm, International Journal of Advanced Robotic Systems 13 (3) (2016) 91.  doi:10.5772/63484.
  * [44] R. Kala, Multi-robot path planning using co-evolutionary genetic programming, Expert Systems with Applications 39 (3) (2012) 3817–3831.  doi:https://doi.org/10.1016/j.eswa.2011.09.090.
  * [45] W. Zhu, H. Duan, Chaotic predator–prey biogeography-based optimization approach for ucav path planning, Aerospace Science and Technology 32 (1) (2014) 153–161.
  * [46] X. Wang, Z.-G. Hou, F. Lv, M. Tan, Y. Wang, Mobile robots’ modular navigation controller using spiking neural networks, Neurocomput. 134 (2014) 230–238.  doi:10.1016/j.neucom.2013.07.055.
URL https://doi.org/10.1016/j.neucom.2013.07.055

  * [47] D. Zhu, W. Li, M. Yan, S. X. Yang, The path planning of auv based on d-s information fusion map building and bio-inspired neural network in unknown dynamic environment, International Journal of Advanced Robotic Systems 11.
  * [48] M. S. Nosrati, R. Karimi, H. A. Hasanvand, Investigation of the * (star) search algorithms: Characteristics, methods and approaches - ti journals, World Applied Programming.
  * [49] M. Costa, M. Silva, A survey on path planning algorithms for mobile robots, 2019, pp. 7–18.  doi:10.1109/ICARSC.2019.8733623.
  * [50] I. Noreen, A. Khan, Z. Habib, Optimal path planning using rrt* based approaches: A survey and future directions, International Journal of Advanced Computer Science and Applications 7.
  * [51] S. Karaman, E. Frazzoli, Sampling-based algorithms for optimal motion planning, The International Journal of Robotics Research 30 (2011) 846 – 894.
  * [52] M. A. P. Castañeda, J. Savage, A. Hernández, F. A. Cosío, Local autonomous robot navigation using potential fields, in: X.-J. Jing (Ed.), Motion Planning, IntechOpen, Rijeka, 2008, Ch. 1.  doi:10.5772/6022.
URL https://doi.org/10.5772/6022

  * [53] F. Bounini, D. Gingras, H. Pollart, D. Gruyer, Modified artificial potential field method for online path planning applications, in: 2017 IEEE Intelligent Vehicles Symposium (IV), 2017, pp. 180–185.  doi:10.1109/IVS.2017.7995717.
  * [54] J. Cho, D. S. Pae, M.-T. Lim, T.-K. Kang, A real-time obstacle avoidance method for autonomous vehicles using an obstacle-dependent gaussian potential field, Journal of Advanced Transportation.
  * [55] B. Mashadi, M. Majidi, Global optimal path planning of an autonomous vehicle for overtaking a moving obstacle (2014).
  * [56] L. Kavraki, P. Svestka, J.-C. Latombe, M. Overmars, Probabilistic roadmaps for path planning in high-dimensional configuration spaces, IEEE Transactions on Robotics and Automation 12 (4) (1996) 566–580.  doi:10.1109/70.508439.
  * [57] J. Nasir, F. Islam, U. Malik, Y. Ayaz, O. Hasan, M. Khan, M. S. Muhammad, Rrt*-smart: A rapid convergence implementation of rrt*, International Journal of Advanced Robotic Systems 10 (7) (2013) 299.  doi:10.5772/56718.
  * [58] S. Sedighi, D.-V. Nguyen, K.-D. Kuhnert, Guided hybrid a-star path planning algorithm for valet parking applications, in: 2019 5th International Conference on Control, Automation and Robotics (ICCAR), 2019, pp. 570–575.  doi:10.1109/ICCAR.2019.8813752.
  * [59] A. Dosovitskiy, G. Ros, F. Codevilla, A. Lopez, V. Koltun, CARLA: An open urban driving simulator, in: Proceedings of the 1st Annual Conference on Robot Learning, 2017, pp. 1–16.


# Soft $g$-frames in soft Hilbert spaces

Sayyed Mehrab Ramezani

###### Abstract

In this paper, we define the concept of soft $g$-frame in soft Hilbert spaces
by extending the concept of soft from frame to $g$-frame. We then show some
properties of the soft $g$-frames in soft Hilbert spaces. Among other results,
we show that the $g$-frame operator is associated with a soft, finite, self-
adjoining, reversible, and finite inverse $g$-frame and get the dual g-frames.
In addition, we prove that every element in Hilbert’s soft space satisfies the
theorem of $g$-frame decomposition.

##  1 Introduction

Frames for Hilbert space were formally defined by Duffin and Schaeffer [7] in
1952 while studying some problems in non-harmonic Fourier series. Recall that
for a Hilbert space ${H}$ and a countable index set $J$, a collection
$\\{f_{j}\\}_{j\in J}\subset{H}$ is called a frame for the Hilbert space ${H}$
if there exist two positive constants $a$, $b$ such that for all $f\in H$

$a\|f\|^{2}\leq\sum_{j\in J}|\langle f,f_{j}\rangle|^{2}\leq b\|f\|^{2};$ (1)

$a$ and $b$ are called the lower and upper frame bounds, respectively. If only
the right-hand inequality in (3) is satisfied, we call $\\{f_{j}\\}_{j\in J}$
a Bessel sequence for ${H}$ with Bessel bound $b$. Wenchang Sun [13] developed
a generalization of frame ($g$-frame) including more other concepts and proved
that many basic properties can be derived from this more general context and
Ramezani [12] presented a generalization of orthonormal bases or simply
$g$-orthonormal bases.

A sequence $\\{\Lambda_{j}\in\mathcal{B}(\mathcal{H},\mathcal{K}_{j}):j\in
J\\}$ is called a $g$-frame for $\mathcal{H}$ with respect to
$\\{\mathcal{K}_{j}\\}_{j\in J}$ if there exist two positive constants $A$ and
$B$ such that:

$A\|f\|^{2}\leq\sum_{j\in J}\|\Lambda_{j}f\|^{2}\leq B\|f\|^{2},\hskip
85.35826pt(f\in\mathcal{H}).$ (2)

The bounded linear operator $T_{\Lambda}$ is defined by:

$T_{\Lambda}:\bigoplus_{j\in
J}\mathcal{K}_{j}\longrightarrow{\mathcal{H}},\hskip
28.45274ptT_{\Lambda}(\\{g_{j}\\}_{j\in J})=\sum_{j\in
J}\Lambda^{*}_{j}g_{j},$

is called the pre $g$-frame operator of $\\{\Lambda_{j}\\}_{j\in J}$. Also,
the bounded linear operator $S_{\Lambda}$ is defined by:

$S_{\Lambda}:\mathcal{H}\longrightarrow\mathcal{H},\hskip
28.45274ptS_{\Lambda}(f)=\sum_{j\in J}\Lambda^{*}_{j}\Lambda_{j}f,$

is called the $g$-frame operator of $\\{\Lambda_{j}\\}_{j\in J}$.

######  Example 1.

If $\mathcal{K}_{j}$=$\mathbb{C}$ for any $j\in J$ and $\Lambda_{j}f=\langle
f,f_{j}\rangle$ for any $f\in\mathcal{H}$, in this case, the $g$-frame is just
a frame for $\mathcal{H}$.

G-frames are generalized frames which include ordinary frames, bounded
invertible linear operators, as well as many recent generalizations of frames.

Soft set theory was introduced by Molodtsov [11] in 1999 as a new mathematical
tool for dealing with uncertainties while modeling problems in engineering,
physics, computer science, economics, social science, and medical sciences.
This theory began to receive special attention in 2002 when Maji et al. [9]
applied the soft sets to decision-making problems, using rough mathematics,
and later in 2003, with the definitions of various operations of soft sets
[10].

In [8], Osmin Ferrer , Arley Sierra and José Sanabria used soft linear
operators to introduce the notion of discrete frames on soft Hilbert spaces,
which extends the classical notion of frames on Hilbert spaces to the context
of algebraic structures on soft sets. Also, they showed that the frame
operator associated to a soft discrete frame is bounded, self-adjoint,
invertible and with a bounded inverse. Furthermore, they proved that every
element in a soft Hilbert space satisfies the frame decomposition theorem.

In this paper, we extend the concept of discrete frames in soft Hilbert spaces
to generalized frames, and define the $g$-frame concept in terms of soft
Hilbert spaces. Among other results, we show that the $g$-frame operator is
associated with a soft, finite, self-adjoining, reversible, and finite inverse
$g$-frame. In addition, we prove that every element in Hilbert’s soft space
satisfies the theorem of $g$-frame decomposition.

##  2 Preliminaries

Throughout this paper, $\mathcal{X}$ denotes a non-empty set (possibly without
algebraic structure), $\mathcal{P}(\mathcal{X})$ the power set of
$\mathcal{X}$, and $A$ a non-empty set of parameters.

######  Definition 1 ([11]).

A soft set on $\mathcal{X}$ is a pair $(F,A)$ where $F$ is a mapping given by
$F:A\longrightarrow\mathcal{P}(\mathcal{X})$. In this way, we can see a soft
set as the following:

$(F,A)=:\\{(\lambda,F(\lambda))\ \ :\ \ \lambda\in A,\
F(\lambda)\in\mathcal{P}(\mathcal{X})\\}.$

######  Definition 2 ([3]).

A soft set $(F,A)$ on $\mathcal{X}$ is said to be a null soft set if
$F(\lambda)=\emptyset$ for all $\lambda\in A$, and in this case we write
$(F,A)=\Phi$. If $F(\lambda)\neq\emptyset$ for some $\lambda\in A$, then the
soft set $(F,A)$ is said to be a non-null soft set on $\mathcal{X}$.

######  Definition 3 ([3]).

A soft set $(F,A)$ on $\mathcal{X}$ is said to be an absolute soft set if
$F(\lambda)=\mathcal{X}$ for all $\lambda\in A$; in this case, we write
$(F,A)=\check{\mathcal{X}}$.

######  Definition 4 ([3]).

A soft element on $\mathcal{X}$ is a function
$\varepsilon:A\longrightarrow\mathcal{X}$. Now if $\varepsilon(\lambda)\in
F(\lambda)$ for all $\lambda\in A$, the soft element $\varepsilon$ is said to
belong to the soft set $(F,A)$ on $\mathcal{X}$, which we denote by
$\varepsilon\tilde{\in}F$.

We denote the collection of all the soft elements of a soft set $(F,A)$ by
$SE\big{(}(F,A)\big{)}$; this is

$SE\big{(}(F,A)\big{)}:=\\{\varepsilon\ :\
\varepsilon\tilde{\in}F\\}=\\{\varepsilon\ :\ \varepsilon(\lambda)\in
F(\lambda),\ \ \forall\lambda\in A\\},$

and equivalently,

$F(\lambda)=\\{\varepsilon(\lambda)\ :\ \varepsilon\tilde{\in}F\\}.$

######  Definition 5 ([6]).

Let $\mathbb{R}$ be the set of real numbers and $\mathcal{P}(\mathbb{R})$ be
the collection of all non-empty bounded subsets of $\mathbb{R}$ and A be a set
of parameters. Then a mapping $F:A\longrightarrow\mathcal{P}(\mathbb{R})$ is
called a soft real set. It is denoted by $\mathbb{R}(A)$.

######  Definition 6 ([1]).

Let $\mathbb{C}$ be the set of complex numbers and $\mathcal{P}(\mathbb{C})$
be the collection of all non-empty bounded subsets of $\mathbb{C}$. A be a set
of parameters. Then a mapping $F:A\longrightarrow\mathcal{P}(\mathbb{C})$ is
called a soft complex set. It is denoted by $\mathbb{C}(A)$.

######  Definition 7 ([3]).

For two soft real numbers $\alpha\\!:A\\!\longrightarrow\\!\mathbb{R}$ and
$\beta\\!:A\\!\longrightarrow\\!\mathbb{R}$ we define the following:

  1. 1.

$\alpha\tilde{\leq}\beta,\ \text{if}\ \alpha(\lambda)\leq\beta(\lambda),\
\text{for all}\ \lambda\in A$,

  2. 2.

$\alpha\tilde{\geq}\beta,\ \text{if}\ \alpha(\lambda)\geq\beta(\lambda),\
\text{for all}\ \lambda\in A$,

  3. 3.

$\alpha\tilde{<}\beta,\ \text{if}\ \alpha(\lambda)<\beta(\lambda),\ \text{for
all}\ \lambda\in A$,

  4. 4.

$\alpha\tilde{>}\beta,\ \text{if}\ \alpha(\lambda)>\beta(\lambda),\ \text{for
all}\ \lambda\in A$.

######  Definition 8 ([4, 5]).

Let $\mathcal{X}$ be a $\mathbb{K}$-vector space (typically
$\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{C}$), $A$ be a non-empty set of
parameters and $(F,A)$ be a soft set on $\mathcal{X}$. The soft set $(F,A)$ is
said to be a soft $\mathbb{K}$-vector space on $\mathcal{X}$ if $F(\lambda)$
is a vector subspace of $\mathcal{X}$ for all $\lambda\in A$.

######  Definition 9 ([4, 5]).

Let $(F,A)$ be a soft $\mathbb{K}$-vector space on $\mathcal{X}$. A soft
vector $x$ of $(F,A)$ is said to be a null soft vector if $x(\lambda)=\theta$,
$\text{for all}\ \lambda\in A$, where $\theta$ is the zero element of
$\mathcal{X}$. This is denoted by $\Theta$.

######  Definition 10 ([14]).

Let $\check{\mathcal{X}}$ be an absolute soft $\mathbb{K}$-vector space, then
a mapping $\|\cdot\|:SE\big{(}(F,A)\big{)}\longrightarrow\mathbb{R}(A)$ is
said to be a soft norm on $\check{\mathcal{X}}$ if $\|\cdot\|$ satisfies the
following conditions:

  1. 1.

$\|x\|\tilde{\geq}\overline{0},\ \text{for all}\
x\tilde{\in}\check{\mathcal{X}}$,

  2. 2.

$\|x\|=\overline{0},\ \text{if and only if}\ x=\Theta$,

  3. 3.

$\|\alpha\cdot x\|=|\alpha|\|x\|,\ \text{for all}\
x\tilde{\in}\check{\mathcal{X}}\ \text{and for all soft scalar}\
\alpha,\vspace*{0.25cm}$

  4. 4.

$\|x+y\|\tilde{\leq}\|x\|+\|y\|,\ \text{for all}\
x,y\tilde{\in}\check{\mathcal{X}}$.

The soft $\mathbb{K}$-vector space $\check{\mathcal{X}}$ with a soft norm
$\|\cdot\|$ on $\check{\mathcal{X}}$ is said to be a soft $\mathbb{K}$-normed
space and is denoted by $(\check{\mathcal{X}},\|\cdot\|,A)$ or
$(\check{\mathcal{X}},\|\cdot\|)$.

######  Theorem 1 ([14]Decomposition theorem).

If $(\check{\mathcal{X}},\|\cdot\|,A)$ is a soft normed space, then for each
$\lambda\in A$, the mapping
$\|\cdot\|:{\mathcal{X}}\longrightarrow\mathbb{R}^{+}$ defined by
$\|\xi\|_{\lambda}=\|x\|(\lambda)$ where $x$ is such that $x(\lambda)=\xi$ is
a norm on $\mathcal{X}$ for all $\lambda\in A$.

######  Definition 11 ([4]).

If $\check{\mathcal{X}}$ is an absolute soft vector space, then a binary
operation $\langle\cdot,\cdot\rangle:SE\big{(}\check{\mathcal{X}}\big{)}\times
SE\big{(}\check{\mathcal{X}}\big{)}\longrightarrow\mathbb{C}(A)$ is said to be
a soft inner product on $\check{\mathcal{X}}$ if it satisfies the following
conditions:

  1. 1.

$\langle x,x\rangle\tilde{\geq}\overline{0},\ \text{for all}\
x\tilde{\in}\check{\mathcal{X}}\ \text{ and }\ \langle
x,x\rangle=\overline{0},\ \text{if and only if}\ x=\Theta$,

  2. 2.

$\langle x,y\rangle=\overline{\langle y,x\rangle},\ \text{for all}\
x,y\tilde{\in}\check{\mathcal{X}}$,

  3. 3.

$\langle\alpha\cdot x,y\rangle=\alpha\cdot{\langle x,y\rangle},\ \text{for
all}\ x,y\tilde{\in}\check{\mathcal{X}}\ \text{and for all soft scalar}\
\alpha,$

  4. 4.

$\langle x+y,z\rangle=\langle x,z\rangle+\langle y,z\rangle,\ \text{for all}\
x,y,z\tilde{\in}\check{\mathcal{X}}$.

The soft vector space $\check{\mathcal{X}}$ with a soft inner product
$\langle\cdot,\cdot\rangle$ on $\check{\mathcal{X}}$ is said to be a soft
inner product space and is denoted by
$(\check{\mathcal{X}},\langle\cdot,\cdot\rangle,A)$ or
$(\check{\mathcal{X}},\langle\cdot,\cdot\rangle)$.

######  Definition 12 ([2]).

A soft inner product space is said to be complete if it is complete with
respect to the soft metric defined by the soft inner product. A complete soft
inner product space is said to be a soft Hilbert space.

######  Example 2.

Let $\mathcal{X}\\!:=\\!\bigoplus_{j\in
J}\\!\mathcal{V}_{j}\\!=\\!\\{\\{g_{j}\\}_{j\in J}:\
g_{j}\\!\\!\in\\!\\!\mathcal{V}_{j},\ \sum_{j\in J}\\!|\langle
g_{j},g_{j}\rangle|\\!\\!<\\!\infty\\}$. It is well known that $\mathcal{X}$
is a Hilbert space with respect to the inner product $\langle
f,g\rangle=\sum_{j\in J}\langle f_{j},g_{j}\rangle_{\mathcal{V}_{j}}$ for
$f=\\{f_{j}\\}_{j\in J}$, $g=\\{g_{j}\\}_{j\in J}$ in $\bigoplus_{j\in
J}\mathcal{V}_{j}$. Now, let $\tilde{f}$, $\tilde{g}$ be soft elements of the
absolute soft vector space $\check{\mathcal{X}}$. Then,
$\tilde{f}(\lambda)=\\{f_{j}^{\lambda}\\}_{j\in J}$,
$\tilde{g}(\lambda)=\\{g_{j}^{\lambda}\\}_{j\in J}$ are elements of
$\bigoplus_{j\in J}\mathcal{V}_{j}$. Thus, the mapping
$\langle\cdot,\cdot\rangle:SE(\check{\mathcal{X}})\times
SE(\check{\mathcal{X}})\longrightarrow\mathbb{C}(A)$, defined by,
$\langle\tilde{f},\tilde{g}\rangle(\lambda)=\sum_{j\in J}\langle
f_{j}^{\lambda},g_{j}^{\lambda}\rangle_{\mathcal{V}_{j}}$ for all $\lambda\in
A$, is a soft inner product on $\check{\mathcal{X}}$. Therefore,
$(\check{\mathcal{X}},\langle\cdot,\cdot\rangle,A)$, with $A$ being a finite
set of parameters, is a soft Hilbert space.

##  3 Soft $g$-frames in soft Hilbert spaces

In this section, we define the concept of soft $g$-frame in soft Hilbert
spaces by extending the concept of soft from frame to $g$-frame. We then show
some properties of the soft $g$-frames in soft Hilbert spaces.

######  Definition 13.

Let $\\{\Lambda_{j}\\}_{j\in J}$ be a sequence of soft operator of a soft
Hilbert space $\check{\mathcal{U}}$ to soft Hilbert spaces
${\check{\mathcal{V}}}_{j}$ having a finite sets of parameters. We call a
sequence
$\\{\Lambda_{j}\in\mathcal{B}(\check{\mathcal{U}},{\check{\mathcal{V}}}_{j}):j\in
J\\}$ a soft $g$-frame for $\check{\mathcal{U}}$ with respect to
$\\{\check{\mathcal{V}}_{j}\\}_{j\in J}$ if there exist positive soft real
numbers $\overline{D}\tilde{\geq}\overline{C}\tilde{>}\overline{0}$ such that:

$\overline{C}\|f\|^{2}\tilde{\leq}\sum_{j\in
J}\|\Lambda_{j}f\|^{2}\tilde{\leq}\overline{D}\|f\|^{2},\hskip
85.35826pt(f\in\check{\mathcal{U}}).$ (3)

The soft real numbers $\overline{C}$ and $\overline{D}$ are called bounds of
the soft $g$-frame. These are not unique, as the optimal bounds are the
largest possible value of a and the smallest possible value of $\overline{D}$
that satisfy (3). In the case that $\overline{C}=\overline{D}$, the soft
$g$-frame is called tight. We call
$\\{\Lambda_{j}\in\mathcal{B}(\check{\mathcal{U}},\check{\mathcal{V}}_{j}):j\in
J\\}$ an exact soft $g$-frame if it ceases to be a soft $g$-frame whenever any
one of its elements is removed.

######  Example 3.

Let $\\{f_{j}\\}_{j\in J}$ be a soft frame for a soft Hilbert space
$\check{\mathcal{U}}$ Let $\Lambda_{f_{j}}$ be the functional induced by
$f_{j}$, i.e.,

$\Lambda_{f_{j}}=\langle f,f_{j}\rangle\hskip
42.67912pt(f\in\check{\mathcal{U}})$

It is easy to check that $\\{\Lambda_{f_{j}}\\}_{j\in J}$ is a soft $g$-frame
for $\check{\mathcal{U}}$ with respect to $\mathbb{C}(A)$.

##  4 soft $g$-frame operators and dual g-frames

Next, we introduce the notion of the soft pre $g$-frame operator and the soft
$g$-frame operator of a soft $g$-frame. In addition, we establish the most
important result, called the decomposition theorem of soft $g$-frames.

######  Definition 14.

Given a soft $g$-frame $\\{\Lambda_{j}\\}_{j\in J}$ for soft Hilbert space
$\check{\mathcal{U}}$ with respect to soft Hilbert space
$\\{\check{\mathcal{V}}_{j}\\}_{j\in J}$ having a finite set of parameters, we
define the soft pre $g$-frame operator by the following:

$\displaystyle T:SE(\check{\bigoplus_{j\in J}\mathcal{V}_{j}})\longrightarrow
SE(\check{\mathcal{U}}),$ $\displaystyle T(\\{{g_{j}}\\}_{j\in J})=\sum_{j\in
J}\Lambda^{*}_{j}g_{j}.$

######  Proposition 1.

The soft pre $g$-frame operator associated to soft $g$-frame
$\\{\Lambda_{j}\\}_{j\in J}$ is well defined and bounded.

  * Proof.

Let $g=\\{g_{j}\\}_{j\in J}$ be a soft element of $\check{\bigoplus_{j\in
J}\mathcal{V}_{j}}$ then

$\displaystyle\|T(g)\|$ $\displaystyle=\sup_{\|f\|=\overline{1}}|\langle
T(\\{{g_{j}}\\}_{j\in J}),f\rangle|$
$\displaystyle=\sup_{\|f\|=\overline{1}}|\langle\sum_{j\in
J}\Lambda_{j}^{*}g_{j},f\rangle|$
$\displaystyle=\sup_{\|f\|=\overline{1}}|\sum_{j\in J}\langle
g_{j},\Lambda_{j}f\rangle|$
$\displaystyle\tilde{\leq}\sup_{\|f\|=\overline{1}}\sum_{j\in J}|\langle
g_{j},\Lambda_{j}f\rangle|$ $\displaystyle\tilde{\leq}\left(\sum_{j\in
J}|\langle
g_{j},g_{j}\rangle|^{2}\right)^{\frac{1}{2}}\cdot\sup_{\|f\|=\overline{1}}\left(\sum_{j\in
J}|\langle\Lambda_{j}f,\Lambda_{j}f\rangle|^{2}\right)^{\frac{1}{2}}$
$\displaystyle=\|g\|\sup_{\|f\|=\overline{1}}\left(\sum_{j\in
J}\|\Lambda_{j}f\|^{2}\right)^{\frac{1}{2}}$
$\displaystyle\tilde{\leq}\|g\|\sqrt{\overline{D}}\sup_{\|f\|=\overline{1}}\|f\|$
$\displaystyle\tilde{\leq}\|g\|\sqrt{\overline{D}}\|f\|,$

so

$\displaystyle\|T\|$ $\displaystyle\tilde{\leq}\|g\|\sqrt{\overline{D}},$

and therefore the soft $g$-frame operator $T$ is well defined and bounded. ∎

######  Proposition 2.

The adjoint soft operator of $T$ is given by

$\displaystyle T^{*}:SE(\check{\mathcal{U}})\longrightarrow
SE(\check{\bigoplus_{j\in J}\mathcal{V}_{j}}),$ $\displaystyle
T^{*}(f)=\\{\Lambda_{j}f\\}_{j\in J}.$

  * Proof.

Let $f\in SE(\check{\mathcal{U}})$ and $g\in SE(\check{\bigoplus_{j\in
J}\mathcal{V}_{j}})$, then $g=\\{g_{j}\\}_{j\in J}$ with
$g(\lambda)=\\{g^{\lambda}_{j}\\}_{j\in J}$. Additionally, for all $\lambda\in
A$, the following holds:

$\displaystyle\langle T^{*}(f),\\{g_{j}\\}_{j\in
J}\rangle_{\bigoplus}(\lambda)$ $\displaystyle=\langle f,T(\\{g_{j}\\}_{j\in
J})\rangle_{\check{\mathcal{U}}}(\lambda)$ $\displaystyle=\langle f,\sum_{j\in
J}\Lambda^{*}_{j}g_{j}\rangle_{\check{\mathcal{U}}}(\lambda)$
$\displaystyle=\sum_{j\in
J}\langle\Lambda_{j}f,g_{j}\rangle_{\bigoplus}(\lambda)$
$\displaystyle=\langle\\{\Lambda_{j}f\\}_{j\in J},\\{g_{j}\\}_{j\in
J}\rangle_{\bigoplus}(\lambda).$

∎

######  Definition 15.

We define the associated soft $g$-frame operator to the soft $g$-frame as
follow

$\displaystyle S:SE(\check{\mathcal{U}})\longrightarrow
SE(\check{\mathcal{U}})$ $\displaystyle S(f)=TT^{*}(f),$

where $\Lambda^{*}_{j}$ is the adjoint operator of $\Lambda_{j}$. First of
all, $S$ is well defined on $SE(\check{\mathcal{U}})$. To see this, let
$n_{1}<n_{2}$ be integers. Then we have

$\displaystyle\|\sum_{j=n_{1}}^{j=n_{2}}\Lambda^{*}_{j}\Lambda_{j}f\|$
$\displaystyle=\sup_{g\in
SE(\check{\mathcal{U}}),\|g\|=\overline{1}}\left|\left\langle\sum_{j=n_{1}}^{j=n_{2}}\Lambda^{*}_{j}\Lambda_{j}f,g\right\rangle\right|$
$\displaystyle=\sup_{g\in
SE(\check{\mathcal{U}}),\|g\|=\overline{1}}\left|\left\langle\sum_{j=n_{1}}^{j=n_{2}}\Lambda_{j}f,\Lambda_{j}g\right\rangle\right|$
$\displaystyle\tilde{\leq}\sup_{g\in
SE(\check{\mathcal{U}}),\|g\|=\overline{1}}\left(\sum_{j\in
J}\|\Lambda_{j}f\|^{2}\right)^{\frac{1}{2}}\left(\sum_{j\in
J}\|\Lambda_{j}g\|^{2}\right)^{\frac{1}{2}}$
$\displaystyle\tilde{\leq}D^{\frac{1}{2}}\left(\sum_{j\in
J}\|\Lambda_{j}f\|^{2}\right)^{\frac{1}{2}}.$

Therefore, $Sf$ is well defined for any $f\in SE(\check{\mathcal{U}})$ and,

$\displaystyle\|S\|$ $\displaystyle=\sup_{f\in
SE(\check{\mathcal{U}}),\|f\|=\overline{1}}\left\langle
Sf,f\right\rangle=\sup_{f\in
SE(\check{\mathcal{U}}),\|f\|=\overline{1}}\sum_{j\in
J}\|\Lambda_{j}f\|^{2}\tilde{\leq}D.$

Hence $S$ is a bounded self-adjoint operator.

######  Corollary 1.

Let $S:SE(\check{\mathcal{U}})\longrightarrow SE(\check{\mathcal{U}})$ be the
associated soft $g$-frame operator to the soft $g$-frame
$\\{\Lambda_{j}\\}_{j\in J}$ for $\check{\mathcal{U}}$ with respect to
$\\{\check{\mathcal{V}}_{j}\\}_{j\in J}$. Then

$\displaystyle S(f)=\sum_{j\in J}\Lambda^{*}_{j}\Lambda_{j}f.$

  * Proof.

Let $f\in SE(\check{\mathcal{U}})$ then

$\displaystyle S(f)=TT^{*}(f)=T(\\{\Lambda_{j}f\\}_{j\in J})=\sum_{j\in
J}\Lambda^{*}_{j}\Lambda_{j}f.$

∎

######  Remark 1.

Note that for all $f\in SE(\check{\mathcal{U}})$ we have the following:

$\displaystyle\left\langle Sf,f\right\rangle=\left\langle\sum_{j\in
J}\Lambda^{*}_{j}\Lambda_{j}f,f\right\rangle=\sum_{j\in
J}\left\langle\Lambda^{*}_{j}\Lambda_{j}f,f\right\rangle=\sum_{j\in
J}\left\langle\Lambda_{j}f,\Lambda_{j}f\right\rangle=\sum_{j\in
J}\|\Lambda_{j}f\|^{2}.$

Hence, the soft $g$-frame condition can be written in the form

$\displaystyle\overline{C}\|f\|^{2}\tilde{\leq}\left\langle
Sf,f\right\rangle\tilde{\leq}\overline{D}\|f\|^{2},\hskip
85.35826pt(f\in\check{\mathcal{U}}).$

The above mentioned cases can be summarized in the following proposition:

######  Proposition 3.

Given a sequence $\\{\Lambda_{j}\\}_{j\in J}$ of soft operator of a soft
Hilbert space $\check{\mathcal{U}}$ to soft Hilbert spaces
${\check{\mathcal{V}}}_{j}$ having a finite sets of parameters, the following
statements are equivalent:

  1. 1.

$\\{\Lambda_{j}\in\mathcal{B}(\check{\mathcal{U}},{\check{\mathcal{V}}}_{j}):j\in
J\\}$ is a soft $g$-frame for $\check{\mathcal{U}}$ with respect to
$\\{\check{\mathcal{V}}_{j}\\}_{j\in J}$ with bounds
$\overline{D}\tilde{\geq}\overline{C}\tilde{>}\overline{0}$.

  2. 2.

$S(f)=\sum_{j\in J}\Lambda^{*}_{j}\Lambda_{j}f$, is a positive and bounded
soft linear operator of $SE(\check{\mathcal{U}})$ to
$SE(\check{\mathcal{U}})$, which satisfies
$\overline{C}Id\tilde{\leq}S\tilde{\leq}\overline{D}Id$.

Here we give an application of soft $g$-frame. Let $\\{\Lambda_{j}\\}_{j\in
J}$ be a soft $g$-frame for soft Hilbert space $\check{\mathcal{U}}$ with
respect to soft Hilbert spaces $\\{\check{\mathcal{V}}_{j}\\}_{j\in J}$.
Suppose that $\\{\tilde{\Lambda}_{j}\\}_{j\in J}$ is the canonical dual soft
$g$-frame. Then for any $f\in\check{\mathcal{U}}$, we have

$\displaystyle f=\sum_{j\in J}\Lambda^{*}_{j}\tilde{\Lambda}_{j}f=\sum_{j\in
J}\tilde{\Lambda}^{*}_{j}\Lambda_{j}f.$

Let $T$ be a bounded soft linear operator on $\check{\mathcal{U}}$. We see
that

$\displaystyle T(f)=\sum_{j\in
J}\Lambda^{*}_{j}\tilde{\Lambda}_{j}T(f)=\sum_{j\in
J}\tilde{\Lambda}^{*}_{j}\Lambda_{j}T(f).$ (4)

We call (4) soft atomic resolutions of a soft operator T.

######  Theorem 2.

Let $\\{\Lambda_{j}\\}_{j\in J}$ and $\\{\Gamma_{j}\\}_{j\in J}$ be a pair of
dual soft $g$-frame for for soft Hilbert space $\check{\mathcal{U}}$ with
respect to soft Hilbert spaces $\\{\check{\mathcal{V}}_{j}\\}_{j\in J}$.
Suppose $\\{f_{j,k}:k\in K_{j}\\}$ and $\\{{g}_{j,k}:k\in K_{j}\\}$ be a pair
of dual soft frames for $\check{\mathcal{V}}_{j}$, respectively. Then
$\\{\Lambda^{*}_{j}(f_{j,k}):j\in J\ k\in K_{j}\\}$ and
$\\{\Gamma^{*}_{j}({g}_{j,k}):j\in J,\ k\in K_{j}\\}$ are a pair of dual soft
frames for $\check{\mathcal{U}}$ provided the soft frame bounds for
$\\{f_{j,k}:k\in K_{j}\\}$ satisfying
$\overline{C}\tilde{\leq}\overline{A_{j}}\tilde{\leq}\overline{B_{j}}\tilde{\leq}\overline{D}$
for some constants $\overline{C},\overline{D}\tilde{>}\overline{0}$.

  * Proof.

Assume that $\\{f_{j,k}:k\in K_{j}\\}$ be a soft frame for
$\check{\mathcal{V}}_{j}$ then for soft element $\Lambda_{j}f$ in
$\check{\mathcal{V}}_{j}$ we have

$\overline{A_{j}}\|\Lambda_{j}f\|^{2}\tilde{\leq}\sum_{k\in
K_{j}}|\langle\Lambda_{j}f,(f_{j,k})\rangle|^{2}\tilde{\leq}\overline{B_{j}}\|\Lambda_{j}f\|^{2},\hskip
122.34692pt(f\in\check{\mathcal{U}}),$

so

$\overline{A}\|\Lambda_{j}f\|^{2}\tilde{\leq}\overline{A_{j}}\|\Lambda_{j}f\|^{2}\tilde{\leq}\sum_{k\in
K_{j}}|\langle\Lambda_{j}f,(f_{j,k})\rangle|^{2}\tilde{\leq}\overline{B_{j}}\|\Lambda_{j}f\|^{2}\tilde{\leq}\overline{B}\|\Lambda_{j}f\|^{2},\hskip
14.22636pt(f\in\check{\mathcal{U}}).$

therefore

$\sum_{j\in J}\overline{A}\|\Lambda_{j}f\|^{2}\tilde{\leq}\sum_{j\in
J}\sum_{k\in
K_{j}}|\langle\Lambda_{j}f,(f_{j,k})\rangle|^{2}\tilde{\leq}\sum_{j\in
J}\overline{B}\|\Lambda_{j}f\|^{2},\hskip
68.28644pt(f\in\check{\mathcal{U}}).$

$\\{\Lambda_{j}\\}_{j\in J}$ is a soft $g$-frame for for soft Hilbert space
$\check{\mathcal{U}}$ with respect to soft Hilbert spaces
$\\{\check{\mathcal{V}}_{j}\\}_{j\in J}$, then for all soft element $f$ in
$\check{\mathcal{U}}$ we have

$\overline{A}\ \overline{C}\|f\|^{2}\tilde{\leq}\sum_{j\in
J}\overline{A}\|\Lambda_{j}f\|^{2}\tilde{\leq}\sum_{j\in J}\sum_{k\in
K_{j}}|\langle\Lambda_{j}f,(f_{j,k})\rangle|^{2}\tilde{\leq}\sum_{j\in
J}\overline{B}\|\Lambda_{j}f\|^{2}\tilde{\leq}\overline{B}\
\overline{D}\|f\|^{2},$

consequently

$\overline{A}\ \overline{C}\|f\|^{2}\tilde{\leq}\sum_{j\in J}\sum_{k\in
K_{j}}|\langle\Lambda_{j}f,(f_{j,k})\rangle|^{2}\tilde{\leq}\overline{B}\
\overline{D}\|f\|^{2},\hskip 85.35826pt(f\in\check{\mathcal{U}}).$

Therefore, $\\{\Lambda^{*}_{j}(f_{j,k}):j\in J\ k\in K_{j}\\}$ is a soft frame
for $\check{\mathcal{U}}$ and similarly we can get that
$\\{\Gamma^{*}_{j}({g}_{j,k}):j\in J,\ k\in K_{j}\\}$ is a soft frame for
$\check{\mathcal{U}}$. On the other hand for any $f\in\check{\mathcal{U}}$, we
have

$\displaystyle\sum_{j\in J}\\!\sum_{k\in
K_{j}}\langle\Lambda^{*}_{j}(f_{j,k}),f\rangle\Gamma^{*}_{j}({g}_{j,k})\\!=\\!\sum_{j\in
J}\Gamma^{*}_{j}\\!\left(\sum_{k\in
K_{j}}\langle(f_{j,k}),\Lambda_{j}f\rangle({g}_{j,k})\right)\\!=\\!\sum_{j\in
J}\Gamma^{*}_{j}\Lambda_{j}f=f.$

Similarly we can get that

$\displaystyle\sum_{j\in J}\sum_{k\in
K_{j}}\langle\Gamma^{*}_{j}({g}_{j,k}),f\rangle\Lambda^{*}_{j}(f_{j,k})=f.$

Hence $\\{\Lambda^{*}_{j}(f_{j,k}):j\in J\ k\in K_{j}\\}$ and
$\\{\Gamma^{*}_{j}({g}_{j,k}):j\in J,\ k\in K_{j}\\}$ are dual soft frames for
$\check{\mathcal{U}}$. ∎

######  Corollary 2.

Suppose that $\\{\Lambda_{j}\\}_{j\in J}$ and $\\{\tilde{\Lambda}_{j}\\}_{j\in
J}$ are canonical dual soft g-frames, $\\{f_{j,k}:k\in K_{j}\\}$ and
$\\{\tilde{f}_{j,k}:k\in K_{j}\\}$ are canonical dual soft frames, and that
$\\{f_{j,k}:k\in K_{j}\\}$ is a tight soft frame with soft frame bounds
$\overline{A_{j}}=\overline{B_{j}}=\overline{A}$. Then
$\\{\Lambda^{*}_{j}(f_{j,k}):j\in J\ k\in K_{j}\\}$ and
$\\{\tilde{\Lambda}^{*}_{j}(\tilde{f}_{j,k}):j\in J,\ k\in K_{j}\\}$ are
canonical dual soft frames.

  * Proof.

Assume that $\\{\Lambda_{j}\\}_{j\in J}$ and $\\{\tilde{\Lambda}_{j}\\}_{j\in
J}$ are canonical dual soft g-frames and $\\{f_{j,k}:k\in K_{j}\\}$ is a tight
frame with frame bounds $\overline{A}_{j}=\overline{B}_{j}=\overline{A}$. Then
$\tilde{f}_{j,k}=\dfrac{1}{\overline{A}}f_{j,k}$. Let $S_{\Lambda}$ and
$S_{\Lambda^{*}}$ be the soft frame operators associated with
$\\{\Lambda_{j}\\}_{j\in J}$ and $\\{\Lambda^{*}_{j}(f_{j,k}):j\in J\ k\in
K_{j}\\}$, respectively. Then we have

$\displaystyle S_{\Lambda^{*}}f$ $\displaystyle=\sum_{j\in J}\sum_{k\in
K_{j}}\langle\Lambda^{*}_{j}({f}_{j,k}),f\rangle\Lambda^{*}_{j}(f_{j,k})$
$\displaystyle=\sum_{j\in J}\Lambda^{*}_{j}\left(\sum_{k\in
K_{j}}\langle{f}_{j,k},\Lambda_{j}f\rangle f_{j,k}\right)$
$\displaystyle=\overline{A}\sum_{j\in J}\Lambda^{*}_{j}\Lambda_{j}f$
$\displaystyle=\overline{A}S_{\Lambda}f.$

Hence

$\displaystyle
S_{\Lambda^{*}}^{-1}\Lambda^{*}_{j}(f_{j,k})=\dfrac{1}{\overline{A}}S_{\Lambda}\Lambda^{*}_{j}(f_{j,k})=\tilde{\Lambda}^{*}_{j}(\tilde{f}_{j,k}).$

This completes the proof. ∎

######  Theorem 3.

If
$\\{\Lambda_{j}\in\mathcal{B}(\check{\mathcal{U}},{\check{\mathcal{V}}}_{j}):j\in
J\\}$ is a soft $g$-frame for $\check{\mathcal{U}}$ with respect to
$\\{\check{\mathcal{V}}_{j}\\}_{j\in J}$ with bounds
$\overline{D}\tilde{\geq}\overline{C}\tilde{>}\overline{0}$ , then the
following statements are satisfied:

  1. 1.

The soft linear operator $S:SE(\check{\mathcal{U}})\longrightarrow
SE(\check{\mathcal{U}})$ is invertible and
$\overline{D}\tilde{\geq}\overline{C}\tilde{>}\overline{0}$

$\displaystyle\overline{D}^{-1}I\tilde{<}S^{-1}\tilde{<}\overline{C}^{-1}I,$

where $\overline{C}^{-1}(\lambda)=\dfrac{1}{\overline{C}}$ , for all
$\lambda\in A$.

  2. 2.

$\\{\Lambda_{j}S^{-1}\\}_{j\in J}$ is a soft $g$-frame for
$\check{\mathcal{U}}$ with respect to $\\{\check{\mathcal{V}}_{j}\\}_{j\in J}$
with bounds
$\overline{C}^{-1}\tilde{\geq}\overline{D}^{-1}\tilde{>}\overline{0}$, called
the dual soft $g$-frame of $\\{\Lambda_{j}S^{-1}\\}_{j\in J}$.

  * Proof.

    1. 1.

The proof is the same as the proof of Theorem $18$ in reference [8].

    2. 2.

Since $S^{-1}$ is positive and self-adjoint, we have the following:

$\displaystyle\sum_{j\in
J}(\Lambda_{j}S^{-1})^{*}\Lambda_{j}S^{-1}(f)\\!=\\!S^{-1}(\sum_{j\in
J}\Lambda^{*}_{j}\Lambda_{j}(S^{-1}f))\\!=\\!S^{-1}(S(S^{-1}f))\\!=\\!S^{-1}f.$

As $\overline{D}^{-1}Id\tilde{\leq}S^{-1}{\leq}\overline{C}^{-1}Id$, by
Proposition 3, we obtain that $\\{\Lambda_{j}S^{-1}\\}_{j\in J}$ is a soft
$g$-frame for $\check{\mathcal{U}}$ with respect to
$\\{\check{\mathcal{V}}_{j}\\}_{j\in J}$ with bounds
$\overline{C}^{-1}\tilde{\geq}\overline{D}^{-1}\tilde{>}\overline{0}$.

∎

######  Theorem 4 (Decomposition Theorem for soft frames).

Let
$\\{\Lambda_{j}\in\mathcal{B}(\check{\mathcal{U}},{\check{\mathcal{V}}}_{j}):j\in
J\\}$ a soft $g$-frame for $\check{\mathcal{U}}$ with respect to
$\\{\check{\mathcal{V}}_{j}\\}_{j\in J}$ with bounds
$\overline{D}\tilde{\geq}\overline{C}\tilde{>}\overline{0}$ and $S$ be the
associated $g$-frame operator. Then, for all $f\in SE(\check{\mathcal{U}})$,
we have the following:

$\displaystyle f=\sum_{j\in J}\Lambda^{*}_{j}\Lambda_{j}S^{-1}f=\sum_{j\in
J}S^{-1}\Lambda^{*}_{j}\Lambda_{j}f.$

  * Proof.

If $f\in SE(\check{\mathcal{U}})$, then the following statements are
satisfied:

$\displaystyle f=S(S^{-1}f)=\sum_{j\in
J}\Lambda^{*}_{j}\Lambda_{j}(S^{-1}f)=\sum_{j\in
J}\Lambda^{*}_{j}\Lambda_{j}S^{-1}f,$

and

$\displaystyle f=S^{-1}(Sf)=S^{-1}(\sum_{j\in
J}\Lambda^{*}_{j}\Lambda_{j}f)=\sum_{j\in
J}S^{-1}\Lambda^{*}_{j}\Lambda_{j}f.$

∎

## R E F E R E N C E S

  * [1] Das, S.; Samanta, S.K. On soft complex sets and soft complex numbers. J. Fuzzy Math. 2013, 21, 195–216.
  * [2] Das, S.; Samanta, S.K. On soft inner product spaces. Ann. Fuzzy Math. Inform. 2013, 6, 151–170.
  * [3] Das, S.; Samanta, S.K. On soft metric spaces. J. Fuzzy Math. 2013, 21, 707–734.
  * [4] Das, S.; Samanta, S.K. Operators on soft inner product spaces. Fuzzy Inf. Eng. 2014, 6, 435–450.
  * [5] Das, S.; Samanta, S.K. Soft linear operators in soft normed linear spaces. Ann. Fuzzy Math. Inform. 2013, 6, 295–314.
  * [6] S. Das and S. K. Samanta, Soft real sets, soft real numbers and their properties, The journal of fuzzy math. 20(3) (2012), 551-576.
  * [7] R. Duffin, A. Schaeffer . A class of non-harmonic Fourier series. Trans. Amer. Math. Soc. 72, (1952), 341-366.
  * [8] O. Ferrer, A. Sierra, J. Sanabria. Soft Frames in Soft Hilbert Spaces. Mathematics. 2021; 9(18):2249. https://doi.org/10.3390/math9182249
  * [9] P.K. Maji, R. Biswas, R. Roy, An application of soft sets in a decision making problem. Comput. Math. Appl. 2002, 44, 1077–1083.
  * [10] P.K. Maji, R. Biswas, R. Roy, Soft set theory. Comput. Math. Appl. 2003, 45, 555–562.
  * [11] D. Molodtsov, Soft set theory-First results. Comput. Math. Appl. 1999, 37, 19–31.
  * [12] Ramezani S M, Nazari A. g-orthonormal bases, g-Riesz bases and $g$-dual of g-frames.U.P.B. Sci. Bull. 78, (2016), 91-98.
  * [13] W. Sun . Stability of g-frames.J. Math. Anal. Appl.326(2), (2007), 858-868.
  * [14] Thakur, R.; Samanta, S.K. Soft Banach Algebra. Ann. Fuzzy Math. Inform. 2015, 10, 397–412.