# Vision-aided Localization and Navigation Based on Trifocal Tensor

Qiang Fang Q. Fang is with the College of Mechatronics and Automation,
National University of Defense Technology, Hunan, Changsha, China. e-mail:
(qiangfang<EMAIL_ADDRESS>
###### Abstract

In this paper, a novel method for vision-aided navigation based on trifocal
tensor is presented. The main goal of the proposed method is to provide
position estimation in GPS-denied environments for vehicles equipped with a
standard inertial navigation systems(INS) and a single camera only. We treat
the trifocal tensor as the measurement model, being only concerned about the
vehicle state and do not estimate the the position of the tracked landmarks.
The performance of the proposed method is demonstrated using simulation and
experimental data.

##  I Introduction

Accurate navigation state estimation is essential in many fields, and Global
Positioning System(GPS) aided inertial measurement unit(IMU) navigation is the
primary method today. However, GPS might be unavailable or unreliable, such as
operating indoors, under water or on other planets. In these scenarios,
vision-based methods constitute an attractive alternative for navigation
aiding due to their relatively low cost and autonomous nature.

Vision-aided navigation[1-2] has indeed become an active research field over
the past decades. Many works have addressed visual-inertial motion estimation,
and several camera-IMU solutions have been proposed to track the state of a
system in real-time on computationally constrained platforms and in real
environments. Existing methods vary by the number of overlapping images and by
the techniques used for fusing the imagery data with the navigation system.
Considering two overlapping images, it is only possible to determine camera
rotation and up-to-scale translation[3](translation during the interval is
associated with an unknown scale). Therefore, two-view based methods for
navigation aiding[4-5] are incapable of eliminating the developing navigation
errors in all states.

Given multiple images($>$2), it is possible to determine the camera motion up
to a common scale[3](translations during the intervals are associated with a
common unknown scale). Considering the computational requirements, several
muti-view methods using the augmented state technique for navigation aiding
have been already proposed[6-9], an approach commonly referred to as
simultaneous localization and mapping (SLAM)[10-13]. For the ability to handle
loops(vehicle returns to the original region), [14] proposed to cope with
loops using bundle adjustment[15], however, this method might be hardly
possible in the real-time performance.

In contrast to SLAM, [16] presented a method based on three-view geometry.
Compared with bundle adjustment and SLAM, the proposed method reduces the
computational resources. Through linearizing the residual measurement and
calculating the relevant Jacobian matrix, the authors obtained good position
results by applying the implicit extended Kalman filter(IEKF). In order to
provide better position estimations, [17] proposed an unscented Kalman
filter(UKF) method based on the idea of three-view geometry. Given three
overlapping images, the familiar constraints are the trifocal tensor[3], and
in 1997, Hartley[18] applied the trifocal tensor to solve the main problem of
computing the motion of the camera (along with the scene structure). For
uncalibrated image sequences, Torr[19] focused on the problem of degeneracy in
structure and motion recovery using trifocal tensor. Guerrero[20] directly
solved the trifocal tensor using singular value decomposition (SVD) method for
location and mapping. Based on the augmented state technique[7], Hu[21] added
the trifocal tensor to the multi-view measurement model to estimate the
current vehicle state and the features’ state during some interval.

In this paper, we present an algorithm that is able to fuse the INS and vision
based on trifocal tensor. Our approach is motivated by the three-view geometry
constraints derived by Indelman[16], which is different from the augmented
state technique. We treat the trifocal tensor as the measurement model, being
only concerned about the vehicle state, which means the position of the
tracked landmarks does not have to be estimated. It is obvious that the
constraints of the trifocal tensor and three-view geometry are both derived
from three overlapping images, so there might be some connection between them.
After a brief discussion of the reference frames in the next section, the
trifocal tensor will be described and its relationship with the three-view
geometry analysed in section III; in section IV the details of the IEKF
estimator based on trifocal tensor will be presented; the performance of the
algorithm will be verified by simulation and experimental results in section
V; and finally, in section VI the conclusions of this work will be drawn.

##  II Reference frames

In this section the reference frames used to derive the Vision/INS equations
are summarized as follows:

1) $n$–North-East-Down(NED) coordinate system. Its origin is set at the
location of the navigation system. It points North, East, and completes a
Cartesian right hand system.

2) $b$–Body-fixed reference frame. Its origin is set at center-of-mass of the
vehicle. It points towards front of the vehicle, or right when viewed from
above, and completes the setup to yield a Cartesian right hand system.

3) $c$–Camera-fixed reference frame. Its origin is set at the center-of-
projection of the camera. It points toward the field-of-view(FOV) center, or
toward the right half of the FOV when viewed from the center-of-projection of
the camera, and completes the setup to yield a Cartesian right hand system.

##  III Trifocal Tensor

We first briefly describe the trifocal tensor and then analyze its
relationship with the three-view geometry developed by Indelman ${[16]}$.

Figure 1 shows a 3-view point correspondence, where $\bm{X}$ is a space point,
$\bm{\widehat{x},\widehat{x}^{{}^{\prime}},\widehat{x}^{{}^{\prime\prime}}}$
are the space point $\bm{X}$ projected in the first, second and third camera
image respectively, $c_{1},c_{2},c_{3}$ are the camera centres of the first,
second and third camera respectively, and
$\pi,\pi^{{}^{\prime}},\pi^{{}^{\prime\prime}}$ are the image planes.

Figure 1: 3-view point correspondence among three camera images

Assuming the camera calibration matrix is known, then the camera projection
matrices can be defined as[3]

$\bm{U}=[\bm{I}|\bm{0}],\bm{U}^{{}^{\prime}}=[\bm{A}|\bm{a}_{4}],\bm{U}^{{}^{\prime\prime}}=[\bm{B}|\bm{b}_{4}]$
(1)

where $\bm{A}=[\bm{a}_{1},\bm{a}_{2},\bm{a}_{3}]=\bm{C}_{c_{1}}^{c_{2}}$ is
the rotation matrix from camera $c_{1}$ to camera $c_{2}$,
$\bm{B}=[\bm{b}_{1},\bm{b}_{2},\bm{b}_{3}]=\bm{C}_{c_{1}}^{c_{3}}$ is the
rotation matrix from camera $c_{1}$ to camera
$c_{3}$,$\bm{a}_{4}=\bm{T}_{12}^{c_{2}}$ is the translation expressed in
camera $c_{2}$ from $c_{1}$ camera to camera $c_{2}$, and
$\bm{b}_{4}=\bm{T}_{13}^{c_{3}}$ is the translation expressed in camera
$c_{3}$ from camera $c_{1}$ to camera $c_{3}$.

According to [3], the trifocal tensor incidence relations using matrix
notation can be given as

$[\widehat{\bm{x}}^{{}^{\prime}}\times](\sum_{i=1}^{3}\widehat{x}_{i}\bm{T}_{i})[\widehat{\bm{x}}^{{}^{\prime\prime}}\times]=\bm{0}_{3\times
3}$ (2)

where $\bm{T}_{i}=\bm{a}_{i}\bm{b}_{4}^{T}-\bm{a}_{4}\bm{b}_{i}^{T}(i=1,2,3)$
and image points are normalized as the LOS vectors expressed in the camera
systems, that is,
$\widehat{\bm{x}}=[\widehat{x}_{1},\widehat{x}_{2},\widehat{x}_{3}]^{T}=[x_{1},y_{1},f_{1}]^{T},\widehat{\bm{x}}^{{}^{\prime}}=[\widehat{x}^{{}^{\prime}}_{1},\widehat{x}^{{}^{\prime}}_{2},\widehat{x}^{{}^{\prime}}_{3}]^{T}=[x_{2},y_{2},f_{2}]^{T},\widehat{\bm{x}}^{{}^{\prime\prime}}=[\widehat{x}^{{}^{\prime\prime}}_{1},\widehat{x}^{{}^{\prime\prime}}_{2},\widehat{x}^{{}^{\prime\prime}}_{3}]^{T}=[x_{3},y_{3},f_{3}]^{T}$,
$x_{i},y_{i}$ is the point expressed in the camera system $c_{i}$ and $f_{i}$
is the camera focal length.

The variables used in eq.(2) are expressed in camera systems. Rearranging
eq.(2), we can yield the trifocal tensor expression in the navigation system
as(the proof is in the Appendix A):

$[\bm{\widehat{x}}^{{}^{\prime}n}\times]\bm{\widehat{x}}^{n}\bm{T}_{23}^{nT}[\bm{\widehat{x}}^{{}^{\prime\prime}n}\times]-[\bm{\widehat{x}}^{{}^{\prime}n}\times][(\bm{\widehat{x}}^{n}\times\bm{T}_{12}^{n})\times][\bm{\widehat{x}}^{{}^{\prime\prime}n}\times]=\bm{0}_{3\times
3}$ (3)

where
$\bm{\widehat{x}}^{n}=\bm{C}_{c_{2}}^{n}\bm{\widehat{x}}^{{}^{\prime}},\bm{\widehat{x}}^{{}^{\prime}n}=\bm{C}_{c_{2}}^{n}\bm{\widehat{x}}^{{}^{\prime}},\bm{\widehat{x}}^{{}^{\prime\prime}n}=\bm{C}_{c_{2}}^{n}\bm{\widehat{x}}^{{}^{\prime\prime}}$,$\bm{T}_{23}^{n}=\bm{Pos}_{3}^{n}-\bm{Pos}_{2}^{n}$
and $\bm{T}_{12}^{n}=\bm{Pos}_{2}^{n}-\bm{Pos}_{1}^{n}$. Here
$\bm{C}_{c_{i}}^{n}=\bm{C}_{b_{i}}^{n}(\bm{\theta})\bm{C}_{c_{i}}^{b_{i}}(i=1,2,3)$
is the rotation matrix from camera $c_{i}$ to the navigation system(
$\bm{C}_{b_{i}}^{n}$ is the rotation matrix from the body system at time
$t_{i}$ to the navigation system which is changed with the vehicle’s rotation
angle $\bm{\theta}$ and $\bm{C}_{c_{i}}^{b_{i}}$ is the rotation matrix from
camera $c_{i}$ to body system $b_{i}$, which is calibrated in the beginning
and maintained unchanged), and $\bm{Pos}_{i}^{n}(i=1,2,3)$ is the position
expressed in the navigation system at time $t_{i}$.

As we introduced in section I, the constraints of the trifocal tensor and the
three-view geometry are both derived from three overlapping images. There
should be some connection between them. Through the analysis, we have obtained
the following Lemma(the detailed proof is listed in the Appendix B):

_Lemma 1_ : The trifocal tensor constraints are the sufficient conditions of
the three-view geometry constraints.

According to [16] , given multiple matching features, one can determine the
translation vectors $\bm{T}_{12}$ and $\bm{T}_{23}$, respectively, up to
scale. In general, these two scale unknowns are different. The two scales can
be connected through three-view geometry, which relates between the magnitudes
of $\bm{T}_{12}$ and $\bm{T}_{23}$. Consequently, if the magnitude of
$\bm{T}_{12}$ is known, it is possible to calculate both the direction and the
magnitude of $\bm{T}_{23}$, and in [16], the authors have obtained good
results. Then from Lemma 1, we might conclude that the performance of the
trifocal tensor constraints is better than that of the three-view geometry
used in a navigation system such as [16]. And this conclusion will be verified
in the following sections.

##  IV Estimator description

###  IV-A Structure of the EKF state vector

The evolving INS state is described by the vector[7]

$\bm{X}=[\bm{q}_{n}^{bT},\bm{b}_{g}^{T},\bm{V}_{n}^{T},\bm{b}_{a}^{T},\bm{Pos}_{n}^{T}]^{T}$
(4)

where $\bm{q}_{n}^{b}$ is the unit quaternion describing the rotation from
frame n to b, $\bm{Pos}_{n}$ and $\bm{V}_{n}$ are the vehicle position and
velocity with respect to frame n, and finally $\bm{b}_{g}$ and $\bm{b}_{a}$
are $3\times 1$ vectors that describe the biases affecting the gyroscope and
accelerometer measurements, respectively. The IMU biases are modeled as random
walk processes, driven by the white Gaussian noise vectors $\bm{n}_{wg}$ and
$\bm{n}_{wa}$, respectively. Following eq.(4), the IMU error-state is defined
as:

$\delta\bm{X}=[\delta\bm{\theta}^{T},\delta\bm{b}_{g}^{T},\delta\bm{V}_{n}^{T},\delta\bm{b}_{a}^{T},\delta\bm{Pos}_{n}^{T}]^{T}$
(5)

For the position, velocity, and biases, the error is defined as the standard
additive error, i.e., the estimate $\widehat{x}$ of a quantity $x$ is defined
as $\delta{x}=x-\widehat{x}$. However,for the quaternion,the error quaternion
$\delta{q}$ describes the small error rotation, and it is defined as

$\delta{\bm{q}}=[0.5\delta\bm{\theta}^{T}1]^{T}$ (6)

where $\delta\bm{\theta}$ is the attitude error.

###  IV-B System model

The linearized continuous-time model for the IMU error-state is [7]

$\delta\dot{\bm{X}}=\bm{F}\delta\bm{X}+\bm{G}\bm{n}_{IMU}$ (7)

where
$\bm{n}_{IMU}=[\bm{n}_{g}^{T},\bm{n}_{wg}^{T},\bm{n}_{a}^{T},\bm{n}_{wa}^{T}]$
is the system noise. The matrices $\bm{F}$ and $\bm{G}$ are

$\bm{F}=\begin{bmatrix}-[\widehat{\bm{w}}\times]&-\bm{I_{3}}&\bm{0}_{3\times
3}&\bm{0}_{3\times 3}&\bm{0}_{3\times 3}\\\ \bm{0}_{3\times 3}&\bm{0}_{3\times
3}&\bm{0}_{3\times 3}&\bm{0}_{3\times 3}&\bm{0}_{3\times 3}\\\
-\bm{C}_{b}^{n}[\widehat{\bm{a}}\times]&\bm{0}_{3\times
3}&-2[\bm{w}_{G}\times]&-\bm{C}_{b}^{n}&-[\bm{w}_{G}\times]^{2}\\\
\bm{0}_{3\times 3}&\bm{0}_{3\times 3}&\bm{0}_{3\times 3}&\bm{0}_{3\times
3}&\bm{0}_{3\times 3}\\\ \bm{0}_{3\times 3}&\bm{0}_{3\times
3}&\bm{I_{3}}&\bm{0}_{3\times 3}&\bm{0}_{3\times 3}\\\ \end{bmatrix}$

where $\bm{I_{3}}$ is the $3\times 3$ identity matrix,$\bm{w}_{G}$ is the
planet’s rotation,
$\widehat{\bm{w}}=\bm{w}_{m}-\widehat{\bm{b}}_{g}-\bm{C}_{n}^{b}\bm{w}_{G}$
and $\widehat{\bm{a}}=\bm{a}_{m}-\widehat{\bm{b}}_{a}$,$\bm{w}_{m}$ is the
gyroscope measurements and $\bm{a}_{m}$ is the accelerometer measurements,and

$\bm{G}=\begin{bmatrix}-\bm{I}_{3}&\bm{0}_{3\times 3}&\bm{0}_{3\times
3}&\bm{0}_{3\times 3}\\\ \bm{0}_{3\times 3}&\bm{I}_{3}&\bm{0}_{3\times
3}&\bm{0}_{3\times 3}\\\ \bm{0}_{3\times 3}&\bm{0}_{3\times
3}&-\bm{C}_{b}^{n}&\bm{0}_{3\times 3}\\\ \bm{0}_{3\times 3}&\bm{0}_{3\times
3}&\bm{0}_{3\times 3}&\bm{I}_{3}\\\ \bm{0}_{3\times 3}&\bm{0}_{3\times
3}&\bm{0}_{3\times 3}&\bm{0}_{3\times 3}\end{bmatrix}$

###  IV-C Mesurement model

Define matrix
$\bm{M}=[\bm{\widehat{x}}^{{}^{\prime}n}\times]\bm{\widehat{x}}^{n}\bm{T}_{23}^{nT}[\bm{\widehat{x}}^{{}^{\prime\prime}n}\times]-[\bm{\widehat{x}}^{{}^{\prime}n}\times][(\bm{\widehat{x}}^{n}\times\bm{T}_{12}^{n})\times][\bm{\widehat{x}}^{{}^{\prime\prime}n}\times]$,and
the elements of matrix $\bm{M}$ are

$\bm{M}=\begin{bmatrix}m_{11}&m_{12}&m_{13}\\\ m_{21}&m_{22}&m_{23}\\\
m_{31}&m_{32}&m_{33}\\\ \end{bmatrix}\\\ $ (8)

where $m_{ij}$ represents the row $i$ and column $j$ element of $\bm{M}$, and

$\displaystyle m_{ij}=$
$\displaystyle-(\bm{\widehat{x}}^{{}^{\prime}n}\times\bm{\widehat{x}}^{nT})\bm{E}_{ij}(\bm{\widehat{x}}^{{}^{\prime\prime}n}\times\bm{T}_{23}^{n})-(\bm{\widehat{x}}^{{}^{\prime}n}\times\bm{\widehat{x}}^{n})^{T}\bm{E}_{ij}(\bm{\widehat{x}}^{{}^{\prime\prime}n}$
(9)
$\displaystyle\times\bm{T}_{12}^{n})+(\bm{\widehat{x}}^{{}^{\prime}n}\times\bm{T}_{12}^{n})^{nT}\bm{E}_{ij}(\bm{\widehat{x}}^{{}^{\prime\prime}n}\times\bm{\widehat{x}}^{n})$

with $\bm{E}_{ij}$ representing a $3\times 3$ dimension matrix, and the row
$i$ and column $j$ element being one and others zeros.

Noting that the trifocal tensor constraints in eq.(3) contain nine
trilinearites, only four are linearly independent[3]. Geometrically these four
trilinearites arise from special choices of the lines in the second and third
images for the point-line-line relation.

Here we choose $m_{11},m_{13},m_{31},m_{33}$ as the basis, which means we
choose a line parallel to the image x-axis and a line through the image
coordinate origin in the second and third images respectively. Then the
trifocal tensor constraints in eq.(3) can be simplied as four linearly
independent constraints.

First consider a single feature, which is observed in three images captured at
time instances $t_{1},t_{2},t_{3}$($t_{1}<t_{2}<t_{3}$), and then the trifocal
tensor constraints can be given as follows:

$\displaystyle\bm{z}_{1}$ $\displaystyle=[m_{11},m_{13},m_{31},m_{33}]^{T}$
(10)
$\displaystyle=h_{1}(\bm{Pos}_{t_{1}},\bm{Pos}_{t_{2}},\bm{Pos}_{t_{3}},\bm{\theta}_{t_{1}},\bm{\theta}_{t_{2}},\bm{\theta}_{t_{3}},\bm{\widehat{x}},\bm{\widehat{x}^{{}^{\prime}}},\bm{\widehat{x}}^{{}^{\prime\prime}})$

While in typical scenarios, the number of the matching pairs of features is
usually not only one, assuming there are $N$ matching features from all the
three views. Then the measurement model can be written as

$\bm{z}=h(\bm{Pos}_{t_{1}},\bm{Pos}_{t_{2}},\bm{Pos}_{t_{3}},\bm{\theta}_{t_{1}},\bm{\theta}_{t_{2}},\bm{\theta}_{t_{3}},\bm{\widehat{x}},\bm{\widehat{x}^{{}^{\prime}}},\bm{\widehat{x}}^{{}^{\prime\prime}})=\begin{pmatrix}\bm{z}_{1}\\\
\vdots\\\ \bm{z}_{N}\end{pmatrix}$ (11)

###  IV-D Implicit Extended Kalman Filtering

In this section, we present the implicit Extended Kalman Filter(IEKF) used to
analyze the performance of the fusion with a navigation system.

Equation (11) shows that the measurement model $\bm{z}$ implicitly contains
the system states(position $\bm{Pos}$, attitude $\bm{\theta}$)and the
measurement noise of the images. Linearizing $\bm{z}$ about
$\bm{Pos}_{t_{1}},\bm{\theta}_{t_{1}},\bm{\theta}_{t_{2}},\bm{Pos}_{t_{2}},\bm{Pos}_{t_{3}},\bm{\theta}_{t_{3}}$,
and
$\bm{\widehat{x}},\bm{\widehat{x}^{{}^{\prime}}},\bm{\widehat{x}}^{{}^{\prime\prime}}$,
and keeping the first-order yields

$\bm{z}\approx\bm{H_{3}}\delta\bm{X_{t_{3}}}+\bm{H_{2}}\delta\bm{X_{t_{2}}}+\bm{H_{1}}\delta\bm{X_{t_{1}}}+\bm{Dv}$
(12)

where $\bm{H_{3}},\bm{H_{2}},\bm{H_{3}}$ are defined as

$\displaystyle\bm{H_{3}}=[\frac{\partial
z}{\partial\bm{\theta}_{t_{3}{}}},\bm{0}_{3\times
12},\frac{\partial\bm{z}}{\partial\bm{Pos}_{t_{3}{}}}]$ (13)
$\displaystyle\bm{H_{2}}=[\frac{\partial
z}{\partial\bm{\theta}_{t_{2}{}}},\bm{0}_{3\times
12},\frac{\partial\bm{z}}{\partial\bm{Pos}_{t_{2}{}}}]$
$\displaystyle\bm{H_{1}}=[\frac{\partial
z}{\partial\bm{\theta}_{t_{1}{}}},\bm{0}_{3\times
12},\frac{\partial\bm{z}}{\partial\bm{Pos}_{t_{1}{}}}]$

$\bm{v}$ is the image noise associated with the LOS vectors
$\bm{\widehat{x},\widehat{x}^{{}^{\prime}},\widehat{x}^{{}^{\prime\prime}}}$,
and its covariance matrix is R. The matrix $D$ in (12) is the gradient of $h$
with respect to the LOS vectors and

$\bm{D}=[\frac{\partial\bm{z}}{\partial\bm{\widehat{x}}},\frac{\partial\bm{z}}{\partial\bm{\widehat{x}^{{}^{\prime}}}},\frac{\partial\bm{z}}{\partial\bm{\widehat{x}^{{}^{\prime\prime}}}}]\\\
$ (14)

During the time interval $(t_{k},t_{k+T})$, the IMU state estimate
$\bm{X}^{-}$is propagated using Runge-Kutta numerical integration of the basic
INS equations in [7]. Moreover, the propagation step of the filter is carried
out using state transition matrix $\bm{\Phi}$ computed by numerical
integration of the differential equation

$\dot{\bm{\Phi}}(t_{k+\tau},t_{k})=\bm{F}\bm{\Phi}(t_{k+\tau},t_{k}),\tau\in(0,T)$
(15)

The propagation covariance is similarly calculated by numerical integration of
the Lyapunov equation:

$\dot{\bm{P}}=\bm{F}\bm{P}+\bm{P}\bm{F}+\bm{G}\bm{Q}_{IMU}\bm{G}^{T}$ (16)

where $\bm{Q}_{IMU}$ is the covariance matrix of system noise $\bm{n}_{IMU}$.

Noting that we are only interested in estimating the navigation errors at the
current time instant $\bm{X}_{t_{3}}$, the navigation errors at the first two
time instances are considered as random parameters in the measurement
equation. Therefore, since $\bm{X}_{t_{2}}$ and $\bm{X}_{t_{1}}$ are not
estimated, these errors are transmitted in the filter covariance matrices
$\bm{P}_{t_{2}},\bm{P}_{t_{1}}$, respectively, which are attached to the first
two images.

Then the Kalman gain matrix can be given by[16]

$\bm{K}=\bm{P}_{\bm{X}_{t_{3}}\bm{z}}\bm{P}_{\bm{z}}^{-1}$ (17)

Since $\bm{z}^{-}=\bm{H_{3}}\delta{\bm{X}}^{-}_{t_{3}}$

$\displaystyle\delta{\bm{z}}$ $\displaystyle=\bm{z}-\bm{z}^{-}$ (18)
$\displaystyle=\bm{H_{3}}\delta{\bm{X}}^{-}_{t_{3}}+\bm{H_{2}}\delta\bm{X_{t_{2}}}+\bm{H_{1}}\delta\bm{X_{t_{1}}}+\bm{Dv}$

Hence

$\displaystyle\bm{P}_{\bm{X}_{t_{3}}}$
$\displaystyle=\bm{P}_{3}^{-}\bm{H}_{3}^{T}+\bm{P}_{32}^{-}\bm{H}_{2}^{T}+\bm{P}_{31}^{-}\bm{H}_{1}^{T}$
(19) $\displaystyle\bm{P}_{\bm{z}}$
$\displaystyle=\bm{H}_{3}\bm{P}_{3}^{-}\bm{H}_{3}^{T}+\begin{bmatrix}\bm{H}_{2}&\bm{H}_{1}\end{bmatrix}\begin{bmatrix}\bm{P}_{2}&\bm{P}_{21}\\\
\bm{P}_{21}^{T}&\bm{P}_{1}\end{bmatrix}\begin{bmatrix}\bm{H}_{2}&\bm{H}_{1}\end{bmatrix}^{T}$
$\displaystyle+\bm{D}\bm{R}\bm{D}^{T}$

Then the corrected state and covariance are given as

$\displaystyle\delta{\bm{X}}_{t_{3}}$ $\displaystyle=\bm{K}\delta{\bm{z}}$
(20) $\displaystyle\bm{P}_{3}^{+}$
$\displaystyle=\bm{P}_{3}^{-}-\bm{K}\bm{P}_{\bm{z}}\bm{K}^{T}$

The estimated state $\delta{\bm{X}}_{t_{3}}$ is then used for correcting the
navigation solution and IMU bias paremetrization $\bm{X}_{t_{3}}$.

Referring to (20), the matrices $\bm{P}_{1},\bm{P}_{2},\bm{P}_{3}^{-}$ are
known, and one of the challenges during IEKF is how to calculate the cross-
correlation matrices $\bm{P}_{21},\bm{P}_{31}^{-},\bm{P}_{32}^{-}$ . Here the
term $\bm{P}_{21}$ may be calculated as

$\bm{P}_{21}=\bm{\Phi}(t_{2},t_{1})\bm{P}_{1}$ (21)

The other two cross-correlation terms, $\bm{P}_{31}^{-},\bm{P}_{32}^{-}$, may
be neglected (e.g. loops). In case the above assumptions regarding
$\bm{P}_{21},\bm{P}_{31}^{-},\bm{P}_{32}^{-}$ are not satisfied, these terms
may be calculated using the methods developed in [22-24].

###  IV-E Computation Requirements

It is interesting to examine the computational complexity of the operations
needed during the IEKF, and analyze its relationship with other methods, such
as the ”classical” MSCKF[7].

Assume that at some step $k$ there are $N$ features included in the three
overlapping images. The computational complexity of carrying out the algorithm
IEKF at step k involves the computation of the predicted state $\bm{X}^{-}$
and propagation covariance $\bm{P}_{3}^{-}$, which also requires the
computation of the corresponding jacobians $\bm{F}_{k},\bm{G}_{k}$ and the
updated $\bm{X}_{k},\bm{P}_{k}$, which involves the computation of the
corresponding jacobian $\bm{H}_{i},\bm{D}$, the Kalman gain matrix $K$, the
innovation $\delta{\bm{z}}$ , and its covariance $\bm{P}_{z}$.

Note that during the IEKF process we do not estimate the state of the
features, and the state dimension r remains unchanged. Here the state
dimension $r=15$, so the computation of the predicted state $\bm{X}^{-}$ and
propagation covariance $\bm{P}_{3}^{-}$ is $O(1)$. The computation of the
Jacobian matrices $\bm{H}_{1},\bm{H}_{2},\bm{H}_{3},\bm{D}$ is linear in $N$,
that is, $O(N)$ operations. Consider as an example the innovation covariance
matrix $\bm{P}_{z}$ in eq.(19). Normally, the computation of this matrix would
require multiplications $(4Nr^{2}+16rN^{2})+(16Nr^{2}+32rN^{2})+(36N+48N^{2})$
and $(4Nr(r-1)+16(r-1)N^{2})+(8Nr(2r-1)+16(2r-1)N^{2})+(24N+32N^{2})$ sums,
that is, $O(N^{2})$ operations. Similar analysis leads to the conclusion that
the cost of computing the innovation $\delta{\bm{z}}$ is $O(N)$, the
covariance matrix $\bm{P}_{k}$ is $O(N^{2})$, and that the greatest cost in an
IEKF update is the computation of the Kalman gain matrix $K$, which is
$O(N^{3})$. Thus, the computational cost per step of IEKF is third power of
the size of the features.

To the ”classical” MSCKF, the IMU poses at the times the last 2 images were
recorded, so the state dimension is $r=27$, and the dimension of the
measurement state is $2N$. Similarly considering the innovation covariance
matrix, the computation of this matrix would require multiplications
$2Nr^{2}+4rN^{2}$ and $(2Nr(r-1)+4(r-1)N^{2})+(4N^{2})$ sums, that is,
$O(N^{2})$ operations. Similar analysis leads to the conclusion that the
greatest cost in an MSCKF update is the computation of the Kalman gain matrix
$K$, which is $O(N^{3})$ . It is the same as that in an IEKF update.

##  V Simulation and experimental Results

In this section, we use the IEKF to analyze the performance of the fusion with
a navigation system through simulation and experimental results.

###  V-A Simulation results

TABLE I: Initial navigation errors and IMU errors States | Description | Value | Units
---|---|---|---
$\delta\bm{Pos}$ | position(1$\delta$) | $[0.1,0.1,0.1]^{T}$ | m
$\delta\bm{V}$ | velocity(1$\delta$) | $[0.1,0.1,0.1]^{T}$ | m/s
$\delta\bm{\theta}$ | attitude(1$\delta$) | $[1,1,1]^{T}$ | deg
$\delta\bm{b}_{g}$ | drift(1$\delta$) | $[0.2,0.2,0.2]^{T}$ | deg/s
$\delta\bm{b}_{a}$ | drift(1$\delta$) | $[10,10,10]^{T}$ | mg

We present statistical results obtained by applying the trifocal tensor
constraints to a trajectory containing a loop based on a simulated navigation
system and synthetic imagery features. The assumed initial navigation errors
and IMU errors are summarized in Table I. The IMU is sampled at 100Hz and the
synthetic imagery features are sampled at 1Hz, which are obtained by assuming
camera calibration matrix $\bm{K}_{cam}$

$\bm{K}_{cam}=\begin{bmatrix}260&0&376\\\ 0&260&240\\\ 0&0&1\\\
\end{bmatrix}\\\ $

and the image noise covariance $\bm{R}$ is

$\bm{R}=\begin{bmatrix}1&0&0\\\ 0&1&0\\\ 0&0&0\\\ \end{bmatrix}\\\ $

Assume the simulated trajectory is a loop repeated once as shown in Fig.2. In
order to demonstrate the performance of the algorithm in loop scenarios, two
different update modes are demonstrated: 1) “sequential update”, in which all
the three images are acquired closely to each other, and 2) “loop update”, in
which the first two images are captured while the platform passes a given
region for the first time, whereas the third image is obtained at the second
passing of the same region. The total running time is about 215s and at
$t=210s$ the vehicle returns to its original area, which makes it possible to
deal with the loop update mode using the method developed in this paper.

Figure 2: loop path

Figure 3: loop path and features

Figure 4: Navigation performance associated with features’ number

During the simulation, the image features are generated randomly(Fig.3), which
remain in the field of the camera view among the three overlapping images
according to the flight data using the pinhole camera model, so it is
necessary to analyze the impact of the features’ number on the navigation
performance, especially the position estimations.

Figure 4 shows the Monte-Carlo results(200 runs) of the RMSE(root-mean-square
error) of the position at time $t_{3}=210s$ using the developed trifocal
tensor and the three-view geometry constraints. It can be seen from Fig.4
that, as the number of the features is greater than 120, the performance of
the change is not very obvious, so in our simulation, we randomly select 120
matching triplets for the three images and use the trifocal tensor and three-
view geometry constraints to analyse the system performance.

Figure 5 shows the Monte-Carlo results (200 runs) using the developed trifocal
tensor. The curve is the square root of the filter covariance, defined for the
$ith$ component in the state vector as $\sqrt{P(i,i)}$ , which is the
a-posteriori covariance matrix. In addition, the three-view geometry and INS-
only estimations are shown for comparison. For $t\leq 150s$, the sequential
upadate mode is used, the time instances $(t_{1},t_{2},t_{3})$ are chosen such
that $t_{1}=0s,t_{2}=1s$ and $t_{3}=10s$. The position and velocity errors in
the INS method grow unbounded during the flight. The position errors are reset
in all axes to a few decimeters(Fig.5(a)) in the trifocal tensor and three-
view geometry, and the position errors of the trifocal tensor are much closer
to the initial errors(0.1m). The velocity errors and accelerometer bias are
considerably reduced(nearly zeros) in all axes as a result of the algorithm
activation(see Fig.5(b-c)) both in the two methods. The attitude
errors(Fig.5(d)) decrease to about 0.2 degrees in all axes in the trifocal
tensor, while they are about 0.5 degrees in the three-view geometry. To the
gyro drift(Fig.5(e)), they decrease to about 0.002 degrees per second in all
axes in the trifocal tensor, however, they are about 0.007 degrees per second
in the three-view geometry. Once $t=210s$, the vehicle returns to its orignal
area, and it is possible to apply the algorithm in the loop update mode.

Figure 5: 200 Monte-Carlo results error estimations(covariance) using the
trifocal tensor,three-view geometry and INS. (a) Position errors. (b) Velocity
errors. (c)Accelerate bias (d)Attitude errors (e)Gyro bias.

The time instances $(t_{1},t_{2},t_{3})$ are chosen such that
$t_{1}=0s,t_{2}=1s$ and $t_{3}=210s$. Taking the position errors as an
example, they are reset in all axes to a few decimeters(Fig.5(a)) from 4
meters and the position errors of the trifocal tensor are much closer to the
initial errors(0.1m) than those of the three-view geometry. Similar analysis
leads to the conclusion that the performance of the trifocal tensor is better
than that of the three-view geometry and INS.

###  V-B Experimental results

An experiment is carried out in this section. During the experiment, the
dataset package collected by Lee[25] at ETHz is applied to validate the
proposed method. There are five synchronized datasets in the package. Here we
select the 1LoopDown dataset(the flight trajectory is a loop repeated once
using the downward looking camera). The inertial sensor measurements and
camera images are recorded for postprocessing at 200 Hz and 50 Hz,
respectively, and the ture position of the quadrotor is obtained by the vicon
system located in the room for comparison. The features are detected and
matched using the SIFT${{}^{[}26]}$ algorithm added RANSAC[27] method.

Figure 6: Experiment processing.(a) the quadrotor and flight environment (b)
flight trajectory .

Figure 6(a) shows the quadrotor collecting the dataset package in the indoor
flight environment and (b) shows the flight trajectory. The quadrotor flies
for about 32 seconds and returns to its original area at ${t=27s}$. In the
experiment, the time instances $(t_{1},t_{2},t_{3})$ are chosen such that
$t_{2}-t_{1}=0.1s$ and $t_{3}-t_{1}=1s$, and the first time instances are
$t_{1}=0s,t_{2}=0.1s,t_{3}=1s$. At first, we utilize the sequential update
mode, and once the quadrotor returns to its original area, e.g., at ${t=27s}$,
we begin to apply the loop update mode, and the time instances are
$t_{1}=0s,t_{2}=0.1s,t_{3}=27s$.

Figure 7 shows the detected matching features during the experiment. It can be
seen that sometimes there are no matching features, such as at
$t=2s,3s,14s,21s,22s$. In this case, we might use the INS-only update method
instead, that is, we just execute the propagation step of the filter using
eq.(16).

Figure 7: Number of detected features

Figure 8: Estimated position using different methods. (a)position errors in
three axes (b) the total position error

The position estimations are shown in Fig.8. It can be intuitively seen that
the position errors increase greatly in the INS-only method, but reset in all
axes to meters(Fig.8(a)) in the trifocal tensor and three-view geometry both.
At time ${t=27s}$, the position errors decrease most(Fig.8(b)), because at
time ${t=27s}$, the loop update mode is used. Moreover, the postion error
using the trifocal tensor is much closer to zero than that of the three-view
geometry.

As shown in Table II, we can also quantitatively analyze the performance of
the proposed method in the mean position error, max position error and end
position error. Taking the mean position error as an example, the error in the
trifocal tensor method is 2.85m, which is just a half of that in the three-
view geometry method and one-sixteenth of that in the INS-only method. Similar
analysis on max position error and end position error leads to the conclusion
that the trifocal tensor method in this paper outperforms the three-view
geometry and the INS-only methods.

TABLE II: The position errors Algorithm | mean position | Max position | End position
---|---|---|---
| error(m) | error(m) | error(m)
trifocal tensor | 2.8500 | 5.4342 | 0.5801
three-view | 4.7225 | 8.5094 | 0.5944
INS only | 44.9512 | 120.6157 | 120.6157

##  VI Conclusion and future work

This paper presented a new method for the purpose of Vision/INS applications
based on trifocal tensor. The trifocal tensor constraints derived here were
the sufficient conditions of the three-view geometry constraints. The proposed
method utilized three overlapping images to formulate constraints relating
between the platform motions at the time instances of the three images, and
the constraints were further fused with an INS using the IEKF. Simulation and
experimental results indicated that the performance of the trifocal tensor
constraints exceeded the three-view geometry constraints. This paper applied
the IEKF to analyse the navigation system. As we know, the UKF uses a selected
set of points to more accurately map the probability distribution of the
measurement model than the linearization of the IEKF, leading to faster
convergence from inaccurate initial conditions in estimation problems, so
using the UKF method to analyse the performance of the navigaiton might be our
future work.

##  VII Acknowledgment

The authors are grateful to the editors Zeynep Celik–Butler and Amy DiMaria
and anonymous reviewers for their useful suggestions and detailed comments.

##  Appendix A Proof of Equation (3)

Proof: From the notation definition of eq.(1) and eq.(2), we know that
$\widehat{\bm{x}}=[x^{1},x^{2},x^{3}]^{T},\bm{A}=[\bm{a}_{1},\bm{a}_{2},\bm{a}_{3}]=\bm{C}_{c_{1}}^{c_{2}},\bm{a}_{4}=\bm{T}_{12}^{c_{2}},\bm{B}=[\bm{b}_{1},\bm{b}_{2},\bm{b}_{3}]=\bm{C}_{c_{1}}^{c_{3}},\bm{b}_{4}=\bm{T}_{13}^{c_{3}}$,
and $\bm{T}_{i}=\bm{a}_{i}\bm{b}_{4}^{T}-\bm{a}_{4}\bm{b}_{i}^{T}$, then the
middle term of eq.(2) can be written as

$\displaystyle\sum_{i}x^{i}\bm{T}_{i}=x^{1}\bm{T}_{1}+x^{2}\bm{T}_{2}+x^{3}\bm{T}_{3}$
(22)
$\displaystyle=x^{1}(\bm{a}_{1}\bm{b}_{4}^{T}-\bm{a}_{4}\bm{b}_{1}^{T})+x^{2}(\bm{a}_{2}\bm{b}_{4}^{T}-\bm{a}_{4}\bm{b}_{2}^{T})+x^{3}(\bm{a}_{3}\bm{b}_{4}^{T}-\bm{a}_{4}\bm{b}_{3}^{T})$
$\displaystyle=(x^{1}\bm{a}_{1}+x^{2}\bm{a}_{2}+x^{3}\bm{a}_{3})\bm{b}_{4}^{T}-\bm{a}_{4}(x^{1}\bm{b}_{1}^{T}+x^{2}\bm{b}_{2}^{T}+x^{3}\bm{b}_{3}^{T})$
$\displaystyle=\bm{A}\widehat{\bm{x}}\bm{b}_{4}^{T}-\bm{a}_{4}\widehat{\bm{x}}\bm{B}^{T}=\bm{C}_{c_{1}}^{c_{2}}\widehat{\bm{x}}\bm{T}_{13}^{c_{3}T}-\bm{T}_{12}^{c_{2}}\widehat{\bm{x}}\bm{C}_{c_{3}}^{c_{1}}$
$\displaystyle=\bm{C}_{c_{1}}^{c_{2}}\widehat{\bm{x}}(\bm{T}_{12}^{c_{3}T}+\bm{T}_{23}^{c_{3}T})-\bm{T}_{12}^{c_{2}}\widehat{\bm{x}}\bm{C}_{c_{3}}^{c_{1}}$
$\displaystyle=\bm{C}_{c_{1}}^{c_{2}}\widehat{\bm{x}}\bm{T}_{23}^{c_{3}T}+(\bm{C}_{c_{1}}^{c_{2}}\widehat{\bm{x}}\bm{T}_{12}^{c_{3}T}-\bm{T}_{12}^{c_{2}}\widehat{\bm{x}}\bm{C}_{c_{3}}^{c_{1}})$

Substituting eq.(22) into eq.(2) yields

$\displaystyle\bm{0}=[\widehat{\bm{x}}^{{}^{\prime}}\times](\sum_{i=1}^{3}\widehat{x}^{i}\bm{T}_{i})[\widehat{\bm{x}}^{{}^{\prime\prime}}\times]$
(23)
$\displaystyle=[\widehat{\bm{x}}^{{}^{\prime}}\times](\bm{C}_{c_{1}}^{c_{2}}\widehat{x}\bm{T}_{23}^{c_{3}T}+(\bm{C}_{c_{1}}^{c_{2}}\widehat{x}\bm{T}_{12}^{c_{3}T}-\bm{T}_{12}^{c_{2}}\widehat{x}\bm{C}_{c_{3}}^{c_{1}}))[\widehat{\bm{x}}^{{}^{\prime\prime}}\times]$
$\displaystyle=[\widehat{\bm{x}}^{{}^{\prime}}\times](\bm{C}_{c_{1}}^{c_{2}}\widehat{x}\bm{T}_{23}^{c_{3}T})[\widehat{\bm{x}}^{{}^{\prime\prime}}\times]+[\widehat{\bm{x}}^{{}^{\prime}}\times](\bm{C}_{c_{1}}^{c_{2}}\widehat{x}\bm{T}_{12}^{c_{3}T}-\bm{T}_{12}^{c_{2}}\widehat{\bm{x}}^{T}\bm{C}_{c_{3}}^{c_{1}})[\widehat{\bm{x}}^{{}^{\prime\prime}}\times]$
$\displaystyle=(\bm{C}_{n_{2}}^{c_{2}}[\widehat{\bm{x}}^{{}^{\prime}n_{2}}\times]\bm{C}_{c_{2}}^{n_{2}})(\bm{C}_{c_{1}}^{c_{2}}(\bm{C}_{n_{2}}^{c_{1}}\widehat{\bm{x}}^{n_{2}})(\bm{C}_{n_{2}}^{c_{3}}\bm{T}_{23}^{n_{2}})^{T})(\bm{C}_{n_{2}}^{c_{3}}[\widehat{\bm{x}}^{{}^{\prime\prime}n_{2}}\times]\bm{C}_{c_{3}}^{n_{2}})$
$\displaystyle+(\bm{C}_{n_{2}}^{c_{2}}[\widehat{\bm{x}}^{{}^{\prime}n_{2}}\times]\bm{C}_{c_{2}}^{n_{2}})(\bm{C}_{c_{1}}^{c_{2}}(\bm{C}_{n_{2}}^{c_{1}}\widehat{\bm{x}}^{n_{2}})(\bm{C}_{n_{2}}^{c_{3}}\bm{T}_{12}^{n_{2}})^{T}$
$\displaystyle-(\bm{C}_{n_{2}}^{c_{2}}\bm{T}_{12}^{n_{2}})(\bm{C}_{n_{2}}^{c_{1}}\widehat{\bm{x}}^{n_{2}})^{T}\bm{C}_{c_{3}}^{c_{1}})(\bm{C}_{n_{2}}^{c_{3}}\bm{T}_{23}^{n_{2}})^{T})(\bm{C}_{n_{2}}^{c_{3}}[\widehat{\bm{x}}^{{}^{\prime\prime}n_{2}}\times]\bm{C}_{c_{3}}^{n_{2}})$
$\displaystyle=\bm{C}_{n_{2}}^{c_{2}}[\widehat{\bm{x}}^{{}^{\prime}n_{2}}\times]\widehat{\bm{x}}^{n_{2}}\bm{T}_{23}^{n_{2}T}[\widehat{\bm{x}}^{{}^{\prime\prime}n_{2}}\times]\bm{C}_{c_{3}}^{n_{2}}$
$\displaystyle+\bm{C}_{n_{2}}^{c_{2}}[\widehat{\bm{x}}^{{}^{\prime}n_{2}}\times](\widehat{\bm{x}}^{n_{2}}\bm{T}_{12}^{n_{2}T}-\bm{T}_{12}^{n_{2}}\widehat{\bm{x}}^{n_{2}T})[\widehat{\bm{x}}^{{}^{\prime\prime}n_{2}}\times]\bm{C}_{c_{3}}^{n_{2}}$
$\displaystyle=\bm{C}_{n_{2}}^{c_{2}}([\widehat{\bm{x}}^{{}^{\prime}n_{2}}\times]\widehat{\bm{x}}^{n_{2}}\bm{T}_{23}^{nT}[\widehat{\bm{x}}^{{}^{\prime\prime}n_{2}}\times]$
$\displaystyle+[\widehat{\bm{x}}^{{}^{\prime}n_{2}}\times](\widehat{\bm{x}}^{n_{2}}\bm{T}_{12}^{nT}-\bm{T}_{12}^{n}\bm{x}^{n_{2}T})[\widehat{\bm{x}}^{{}^{\prime\prime}n_{2}}\times])\bm{C}_{c_{3}}^{n_{2}}$
$\displaystyle=\bm{C}_{n_{2}}^{c_{2}}([\widehat{\bm{x}}^{{}^{\prime}n_{2}}\times]\widehat{\bm{x}}^{n_{2}}\bm{T}_{23}^{T}[\widehat{\bm{x}}^{{}^{\prime\prime}n_{2}}\times]-[\widehat{\bm{x}}^{{}^{\prime}n_{2}}\times][(\widehat{\bm{x}}^{n_{2}}\times\bm{T}_{12}^{n})\times][\widehat{\bm{x}}^{{}^{\prime\prime}n_{2}}\times])\bm{C}_{c_{3}}^{n_{2}}$

Since the rotation matrixs are both invertible, so pre-multiply
$\bm{C}_{n_{2}}^{c_{2}}$ and post-multiply $\bm{C}_{c_{3}}^{n_{2}}$. Here we
assume that the navigation system is rotated very slowly and can be neglected,
so we define the navigation system as $n$, then eq.(23) can be simplified as

$[\widehat{\bm{x}}^{{}^{\prime}n}\times]\widehat{\bm{x}}^{n}\bm{T}_{23}^{T}[\widehat{\bm{x}}^{{}^{\prime\prime}n}\times]-[\widehat{\bm{x}}^{{}^{\prime}n}\times][(\widehat{\bm{x}}^{n}\times\bm{T}_{12}^{n})\times][\widehat{\bm{x}}^{{}^{\prime\prime}n}\times]=\bm{0}_{3\times
3}$ (24)

Proved.

##  Appendix B Proof of Lemma 1

Define matrix
$\bm{M}=[\bm{\widehat{x}}^{{}^{\prime}n}\times]\bm{\widehat{x}}^{n}\bm{T}_{23}^{nT}[\bm{\widehat{x}}^{{}^{\prime\prime}n}\times]-[\bm{\widehat{x}}^{{}^{\prime}n}\times][(\bm{\widehat{x}}^{n}\times\bm{T}_{12}^{n})\times][\bm{\widehat{x}}^{{}^{\prime\prime}n}\times]$,
and the elements of matrix $\bm{M}$ are

$\bm{M}=\begin{bmatrix}m_{11}&m_{12}&m_{13}\\\ m_{21}&m_{22}&m_{23}\\\
m_{31}&m_{32}&m_{33}\\\ \end{bmatrix}\\\ $ (25)

where $m_{ij}$ represents the row $i$ and column $j$ element of $\bm{M}$, and

$\displaystyle m_{ij}=$
$\displaystyle-(\bm{\widehat{x}}^{{}^{\prime}n}\times\bm{\widehat{x}}^{nT})\bm{E}_{ij}(\bm{\widehat{x}}^{{}^{\prime\prime}n}\times\bm{T}_{23}^{n})-(\bm{\widehat{x}}^{{}^{\prime}n}\times\bm{\widehat{x}}^{n})^{T}\bm{E}_{ij}(\bm{\widehat{x}}^{{}^{\prime\prime}n}$
(26)
$\displaystyle\times\bm{T}_{12}^{n})+(\bm{\widehat{x}}^{{}^{\prime}n}\times\bm{T}_{12}^{n})^{T}\bm{E}_{ij}(\bm{\widehat{x}}^{{}^{\prime\prime}n}\times\bm{\widehat{x}}^{n})$

with $\bm{E}_{ij}$ represent a $3\times 3$ dimension matrix,with the $i$ row
and the $j$ column element as one and others as zeros.

Next we investigate the relationship between the trifocal tensor constraints
and the three-view geometry constraints proposed by Indelman[16]. Adding the
elements $m_{11},m_{22},m_{33}$ of $\bm{M}$ matrix together, we have

$\displaystyle m_{11}+m_{22}+m_{33}=$
$\displaystyle-(\widehat{\bm{x}}^{{}^{\prime}n}\times\widehat{\bm{x}}^{n})^{T}(\widehat{\bm{x}}^{{}^{\prime\prime}n}\times\bm{T}_{23}^{n})$
(27)
$\displaystyle+(\widehat{\bm{x}}^{n}\times\bm{T}_{12}^{n})^{T}(\widehat{\bm{x}}^{{}^{\prime\prime}n}\times\widehat{\bm{x}}^{{}^{\prime}n})=\bm{0}$

Rearranging (27) yields

$(\widehat{\bm{x}}^{{}^{\prime}n}\times\widehat{\bm{x}}^{n})^{T}(\widehat{\bm{x}}^{{}^{\prime\prime}n}\times\bm{T}_{23}^{n})=(\widehat{\bm{x}}^{n}\times\bm{T}_{12}^{n})^{T}(\widehat{\bm{x}}^{{}^{\prime\prime}n}\times\widehat{\bm{x}}^{{}^{\prime}n})$
(28)

Equation (28) is just the three-view geometry constraints proposed by
Indelman[16]. On the other hand, the trifocal tensor contains four linearly
independent constraints, while the three-view geometry constraints are less
than four, so the trifocal tensor constraints are the sufficient conditions of
the three-view geometry constraints.

Proved.

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Higgs factory, (i.e. a collider capable of producing statistically significant
numbers of Higgs bosons) could achieve. This type of machine would allow us to
explore the existence of such a symmetric point by directly heating the
electroweak sector to populate it, and to explore the properties of the EW
phase transition.

With the data in hand, the volume could be obtained by using Hanbury-Brown-
Twiss (HBT) or its particle physics analogue Goldhaber-Goldhaber-Lee-Pais
(GGLP) interferometry: the Higgs bosons exiting the collision would retain
memory (by interference) of the radius of the source that emitted them. The
technique is routinely used in astrophysics to establish the size of
astrophysical objects from the emitted photons, and in nuclear collisions by
analyzing pions. That future electroweak collider could likewise obtain the
radius of a hot electroweak ball from the Goldstone and Higgs bosons emitted.

Let us state schematically how this interferometry works (a comprehensive
review can be found in [83]). Suppose that a source has emission points
continuously distributed in a space-time volume $V_{4}$ with an emission
probability amplitude, $\Pi({r};\boldsymbol{k})$, of emitting a particle with
momentum $\boldsymbol{k}$ (on the mass-shell with plane-wave wave-function
$\psi_{\boldsymbol{k}}(\boldsymbol{r})\propto
e^{i\boldsymbol{k}\cdot\boldsymbol{r}}$) at the space-time point $r$. Hence,
the total probability of observing the emission of one particle with momentum
$\boldsymbol{k}$ from the source is
$P(\boldsymbol{k})=\int_{V_{4}}d^{4}{r}|\Pi({r};\boldsymbol{k})|^{2}$.
Likewise, the total probability of measuring two particles with momenta
$\boldsymbol{k}_{1}$ and $\boldsymbol{k}_{2}$, assuming the two emissions are
uncorrelated, i.e.
$\Pi({r}_{1},{r}_{2};\boldsymbol{k}_{1},\boldsymbol{k}_{2})=\Pi({r}_{1};\boldsymbol{k}_{1})\Pi({r}_{2};\boldsymbol{k}_{2})$,
amounts to

$\displaystyle P(\boldsymbol{k}_{1},\boldsymbol{k}_{2})$
$\displaystyle=\int_{V_{4}}d^{4}{r}_{1}d^{4}{r}_{2}\left|\frac{\psi_{\boldsymbol{k}_{1}}({\boldsymbol{r}}_{1})\psi_{{\boldsymbol{k}}_{2}}({\boldsymbol{r}}_{2})+\psi_{{\boldsymbol{k}}_{1}}({\boldsymbol{r}}_{2})\psi_{{\boldsymbol{k}}_{2}}({\boldsymbol{r}}_{1})}{\sqrt{2}}\right|^{2}|\Pi({r_{1}};\boldsymbol{k}_{1})|^{2}|\Pi({r_{2}};\boldsymbol{k}_{2})|^{2}$
$\displaystyle=P(\boldsymbol{k}_{1})P(\boldsymbol{k}_{2})+\int_{V_{4}}d^{4}{r}_{1}d^{4}{r}_{2}\cos{\big{[}(\boldsymbol{r}_{1}-\boldsymbol{r}_{2})\cdot(\boldsymbol{k}_{1}-\boldsymbol{k}_{2})\big{]}}|\Pi({r_{1}};\boldsymbol{k}_{1})|^{2}|\Pi({r_{2}};\boldsymbol{k}_{2})|^{2}\;.$
(121)

The GGLP experiment could be adapted to measuring, at two detection points
$\boldsymbol{d}_{1}$ and $\boldsymbol{d}_{2}$, two Higgs bosons with precise
momentum $\boldsymbol{k}_{1}$ and $\boldsymbol{k}_{2}$ respectively (see Fig.
5). The correlation function among the two momenta is

$C(\boldsymbol{k}_{1},\boldsymbol{k}_{2}):=\frac{P(\boldsymbol{k}_{1},\boldsymbol{k}_{2})}{P(\boldsymbol{k}_{1})P(\boldsymbol{k}_{2})}=1+\frac{\int_{V_{4}}d^{4}{r}_{1}d^{4}{r}_{2}\cos{\big{[}(\boldsymbol{r}_{1}-\boldsymbol{r}_{2})\cdot(\boldsymbol{k}_{1}-\boldsymbol{k}_{2})\big{]}}|\Pi({r_{1}};\boldsymbol{k}_{1})|^{2}|\Pi({r_{2}};\boldsymbol{k}_{2})|^{2}}{P(\boldsymbol{k}_{1})P(\boldsymbol{k}_{2})}.$
(122)

Under the assumptions explained thoroughly in [84] (neglection of higher order
symmetrization, smoothness and equal time approximations, useful for large
(RHIC-like) sources) the correlation function in Eq. (122) simplifies to

$\displaystyle C(\boldsymbol{k}_{1},\boldsymbol{k}_{2})-1$ $\displaystyle=\int
d^{3}\boldsymbol{r}^{\prime}\mathcal{S}_{\boldsymbol{K}}(\boldsymbol{r}^{\prime})\cos{\big{[}(\boldsymbol{r}_{1}-\boldsymbol{r}_{2})\cdot(\boldsymbol{k}_{1}-\boldsymbol{k}_{2})\big{]}}$
$\displaystyle\mathcal{S}_{\boldsymbol{K}}(\boldsymbol{r}^{\prime}):$
$\displaystyle=\frac{\int_{V_{4}}d^{4}{r}_{1}d^{4}{r}_{2}|\Pi({r_{1}};\boldsymbol{k}_{1})|^{2}|\Pi({r_{2}};\boldsymbol{k}_{2})|^{2}\delta(\boldsymbol{r}^{\prime}-\boldsymbol{r}_{1}+\boldsymbol{r}_{2})}{P(\boldsymbol{k}_{1})P(\boldsymbol{k}_{2})},$
(123)

where $\boldsymbol{K}=\boldsymbol{k}_{2}+\boldsymbol{k}_{1}$ is the total
momentum of the pair of outgoing particles. The function
$\mathcal{S}_{\boldsymbol{K}}(\boldsymbol{r}^{\prime})$ encodes “the
distribution of relative positions of particles with identical velocities and
total momentum $\boldsymbol{K}$” [84] and it gives information about the size
of the region of homogeneity of a source (i.e. the region where the
equilibrium assumptions can be taken). The curvature of
$C(\boldsymbol{k}_{1},\boldsymbol{k}_{2})$ at
$\boldsymbol{q}:=\boldsymbol{k}_{1}-\boldsymbol{k}_{2}=0$ is related to the
mean-square separation of the three-dimensional quadrupolar moments [84]

$-\frac{C(\boldsymbol{k}_{1},\boldsymbol{k}_{2})}{dq_{i}dq_{j}}\Bigg{|}_{\boldsymbol{q}=0}=\int
d^{3}\boldsymbol{r}S_{\boldsymbol{K}}(\boldsymbol{r})r_{i}r_{j}\;.$ (124)

In this way we can obtain the volume of the region where the source can be
considered homogeneous and the equilibrium conditions apply.

##  VIII Conclusions

In this work we have bridged between the SMEFT and HEFT formalisms following
the work of two other groups [21, 20, 19, 22, 23]. We have focused on the
Higgs-flare function $\mathcal{F}$ that controls the derivative couplings of
two Goldstone bosons $\omega_{i}$ to any number of Higgs bosons. We have
exhaustively studied this flare function $\mathcal{F}$ and particularly
addressed the existence of its key zero at a symmetric point in the
$(\omega_{i},h)$ field space.

We extend previous results concerning the expression of a few coefficients of
this function in terms of the $c_{H\Box}$ Wilson coefficient of SMEFT; in our
work we have addressed a larger number of such coefficients, we have employed
the Warsaw basis, and we have proceeded to the next order ($1/\Lambda^{4}$) in
the SMEFT expansion.

Further, we have completely eliminated the Wilson coefficient and obtained
correlations that are intrinsic to HEFT and can be used to falsify SMEFT
itself from the wrapping theory. This is the central result of a companion
letter submitted together with this manuscript that provides in extenso
discussion. We provide simple recipes that analysts following upcoming and
future experimental data can follow to test the framework of SMEFT.

With the latest ATLAS and CMS bounds on the $a_{1}$ (also known as
$\kappa_{2V}$) and $a_{2}$ coefficients we have explicitly given 95%
confidence intervals for a few $a_{i}$, $i>2$ ones, that if exceeded would
automatically rule out SMEFT, at least to the orders here considered, and
point out to the need of extending the SMEFT framework.

It may be useful for those analysis to have explicit expressions of the
$\omega\omega\to nh$ amplitudes in HEFT and we provide their leading order in
perturbation theory. In future work, we will additionally address cross-
sections with the physical experimental cuts so that the parameter space can
directly be compared with data. These processes allow to access $\mathcal{F}$
order by order; we also observe that if a future collider could substantially
increase the temperature of the collision environment, the entire function
could be accessed from the $p_{T}$-spectrum of created Higgs bosons.

In the Appendix C and also in the companion letter we have also given similar
correlations that we have extracted among the coefficients of the $V(h)$
nonderivative Higgs potential. This is attractive because it does not require
the Equivalence Theorem (the process $\omega\omega\to nh$ needs to be
extracted from $W_{L}W_{L}\to nh$ data and corrections are needed at low-
energy, while $V(h)$ does not involve the Goldstone bosons) and is already
accessible at LHC energies. Interestingly, it is affected by the properties of
$\mathcal{F}$ since it is this function which controls the change of variable
between HEFT and SMEFT, $h_{1}\to h$.

The same reasoning applies to the Yukawa fermion-Higgs couplings which is
another interesting alley of investigation for future works where activity is
ongoing [85].

## Acknowledgments

Supported by Spanish MICINN PID2019-108655GB-I00 grant, and Universidad
Complutense de Madrid under research group 910309 and the IPARCOS institute;
ERC Starting Grant REINVENT-714788; UCM CT42/18-CT43/18; the Fondazione
Cariplo and Regione Lombardia, grant 2017-2070. A.S.B. thanks Professor Juan
Ferrera-Cuesta for pointing out the relevance of using Schwarz’s lemma. We
also thank Carlos Quezada-Calonge for discussions on $WW\to hh$ scattering.

##  Appendix A Schwarz’s lemma in complex analysis and its corollaries

This brief appendix provides a short overview of Schwarz’s lemma, which we
quickly use to demonstrate the corollary of interest for subsection VI.2.
Again, the idea is whether the image set under $\mathcal{F}$ includes or not a
disk around $\mathcal{F}=1$ large enough to encompass the origin
$\mathcal{F}=0$. The second corollary below gives a sufficient condition for
this to be true. The point $h^{*}$ which is the preimage of $\mathcal{F}=0$ is
the symmetric point around which the SMEFT expansion can be constructed. If
the conditions of the second corollary are met, we know that $\mathcal{F}$
will be analytic in a region broad enough to guarantee the power-expansion.

To start, take a disk $D_{z}(0,1)$ around the origin in the preimage complex
space (in our application, the extension $h\to z$ of the singlet Higgs field
to be a complex variable). Second, $D_{f}(0,1)$ is a disk in the image complex
space (also extending $\mathcal{F}\to f\in\mathbb{C}$), both disks having
radius 1 and being centered around 0 as the notation indicates. We can then
state the lemma [86] as follows.

###  A.1 Schwarz’s Lemma

Let $f\;:\;D_{z}(0,1)\;\to\;D_{f}(0,1)$ be holomorphic with $f(0)=0$. Then
$|f(z)|\leq|z|$ and $|f^{\prime}(0)|\leq 1$. Furthermore, if
$|f(z_{0})|=|z_{0}|$ for some $z_{0}\in D_{z}(0,1)$, then $|f(z)|=1$ $\forall
z\in D(0,1).$

##### Proof

Given those $f(0)$ and $f^{\prime}(0)$, write $f(z)=zg(z)$: $g$ is also
holomorphic. Take $r<1$, if $|z|=r$ we have that $|g(z)|=\frac{|f(z)|}{r}$ and
hence $|g(z)|=\frac{|f(z)|}{r}\leq\frac{1}{r}$ (since the image of $f$ is
$D_{f}(0,1)$). The inequality $|g(z)|\leq\frac{1}{r}$ is satisfied for all
$z\in\bar{D}(0,r)$ thanks to the Maximum modulus principle (if $f$ is a
holomorphic function, then the modulus $|f|$ cannot exhibit a strict local
maximum that is in the interior of the domain of $f$). This means that if
$g(z_{0})=1/r$, that is, it reaches its maximum for some $z_{0}$ satisfying
$|z_{0}|<r$, then the function $g$ must be a constant (and the maximum is
reached at the boundary anyway). Now, taking the limit $r\to 1$ from the left
we obtain $|g(z)|\leq 1$ and consequently $|f(z)|\leq|z|$ for all $z\in
D_{z}(0,1)$.

Noticing that $f^{\prime}(z)=g(z)+zg^{\prime}(z)$ it is immediate to prove
that $|f^{\prime}(0)|\leq 1$.

###  A.2 Corollaries

##### First Corollary

Let $f\;:\;D_{z}(0,1)\;\to\;D_{f}(0,M)$ analytic such that $f(0)=0$ and
$|f^{\prime}(0)|=1$. Then we will have that $M\geq 1$ and
$D_{f}(0,\frac{\sqrt{2}}{\left(\sqrt{2}+1\right)\left(\sqrt{2}+2\right)M})\subset
f(D_{z}(0,1))$ ( the open disc of radius
$\frac{\sqrt{2}}{\left(\sqrt{2}+1\right)\left(\sqrt{2}+2\right)M}$ is
contained in the image through $f$ of the open unit disc).

##### Proof

Thanks to Schwarz’s lemma we know that $M\geq 1$ since otherwise
$|f^{\prime}(0)|<1$. We can then write $f$ as

$f(z):=z+\sum_{n=2}^{\infty}a_{n}z^{n}$ (125)

The triangular inequality of the complex norm yields

$\displaystyle|z|=|f(z)-\sum_{n=2}^{\infty}a_{n}z^{n}|\leq|f(z)|+|\sum_{n=2}^{\infty}a_{n}z^{n}|\
.$ (126)

Choosing to evaluate with $|z_{M}|:=\frac{1}{\alpha M}$ with $\alpha>1$, we
find

$\displaystyle|f(z_{M})|$
$\displaystyle\geq|z_{M}|-|\sum_{n=2}^{\infty}a_{n}z_{M}^{n}|\geq|z_{M}|-\sum_{n=2}^{\infty}|a_{n}||z_{M}|^{n}=\frac{1}{\alpha
M}-\sum_{n=2}^{\infty}\frac{|a_{n}|}{(\alpha M)^{n}}\ .$ (127)

Thanks to Cauchy’s estimates (cancelling factorials) we know that for all
$r<1$, $|a_{n}|\leq\frac{M}{r^{n}}$ and hence $|a_{n}|\leq M$; we may take the
worst bound with $r\to 1$. Then,

$\displaystyle|f(z_{M})|$ $\displaystyle\geq\frac{1}{\alpha
M}-M\sum_{n=2}^{\infty}\frac{1}{(\alpha M)^{n}}=\frac{1}{\alpha
M}-\frac{M}{(\alpha M)^{2}}\frac{1}{1-\frac{1}{\alpha M}}=\frac{1}{\alpha
M}\Big{(}1-\frac{M}{\alpha M-1}\Big{)}\;,$ (128)

(having reconstructed the geometric series). Now, since $M\geq 1$ and we have
taken $\alpha>1$ we have that

$\displaystyle\frac{1}{\alpha
M}\Big{(}1-\frac{1}{\alpha-1/M}\Big{)}\geq\frac{1}{\alpha
M}\Big{(}1-\frac{1}{\alpha-1}\Big{)}=\frac{1}{M}\Big{(}\frac{\alpha-2}{\alpha^{2}-\alpha}\Big{)},$
(129)

giving the highest lower bound for $\alpha=2+\sqrt{2}$. Hence, taking
$|z_{M}|=\frac{1}{(2+\sqrt{2})M}$, we have

$\displaystyle|f(z_{M})|$
$\displaystyle\geq\frac{\sqrt{2}}{\left(\sqrt{2}+1\right)\left(\sqrt{2}+2\right)M}\;.$
(130)

We now proceed to prove that the image of the disk in the Higgs field
$f(D_{z}(0,1))$ contains the disk of the $\mathcal{F}$ function, namely
$D_{f}(0,\frac{\sqrt{2}}{\left(\sqrt{2}+1\right)\left(\sqrt{2}+2\right)M})$.

Use for this an auxiliary $w_{f}\in
D_{f}(0,\frac{\sqrt{2}}{\left(\sqrt{2}+1\right)\left(\sqrt{2}+2\right)M})$,
that is, $|w_{f}|\leq|f(z_{0})|$; the function $g(z)=w_{f}-f(z)$ verifies then

$|f(z_{M})+g(z_{M})|=|w|<\frac{\sqrt{2}}{\left(\sqrt{2}+1\right)\left(\sqrt{2}+2\right)M}\leq|f(z_{M})|\;\;\;\text{for
}\;\;|z_{M}|=\frac{1}{(2+\sqrt{2})M}\;.$ (131)

Now we make use of Rouché’s theorem to state that $f$ and $g$ have the same
number of zeroes in $D(0,\frac{1}{(2+\sqrt{2})M})$, i.e. at least one by
definition (because by hypothesis $f(0)=0$). As a consequence there exists
$z_{0}\in D(0,1)$ such that $f(z_{0})=w_{f}$, in other words $f(D_{z}(0,1))$
contains
$D_{f}(0,\frac{\sqrt{2}}{\left(\sqrt{2}+1\right)\left(\sqrt{2}+2\right)M})$.

##### Second Corollary

Let $g\;:\;D(0,R)\;\to\;D_{g}(0,M)$ analytic, such that $g(0)=0$ and
$|g^{\prime}(0)|=\mu>0$. Then $g(D(0,R))$ contains another disk where $g$ is
analytic,

$g(D(0,R))\supset
D_{g}(0,\frac{\sqrt{2}R^{2}\mu^{2}}{\left(\sqrt{2}+1\right)\left(\sqrt{2}+2\right)M})$
(132)

The $R$ appearing there is what can be tested to guarantee that the image
includes a disk that in turn includes 0 (and therefore, $\exists
h_{*}|\mathcal{F}(h_{*})=0$ ), and the function is analytic between the vacuum
and that symmetric point.

##### Proof

It follows in a relatively straightforward manner by applying the first
corollary to the auxiliary function

$f(z):=\frac{1}{Rg^{\prime}(0)}g(Rz)\;\;\text{ for }\;\;|z|<1\;.$ (133)

##  Appendix B From SMEFT to HEFT

In this appendix we show the two alternative (complementary) procedures
employed to obtain the effective Lagrangian simplifications discussed along
the article: first by means of Higgs field redefinitions; in a second
subsection, we show how to get this same simplifications of the effective
action by means of partial integration and the use of the field equation of
motion for the classical fields.

These two approaches for the simplifications of the effective theory are but
two sides of the same coin: field redefinitions in the integration of the
generating functional modifies the classical action in the exponential
$e^{iS}$, introducing new operators that are proportional to the classical EoM
(this is, through an appropriate field redefinition one can remove operators
in $S$ that are proportional to the EoM, while keeping the same generating
functional); on the other hand, at tree-level the effective action $\Gamma$
coincides with the classical action $S$ evaluated in the (quantum) classical
field, where, by construction, any operator in $\Gamma$ proportional to the
EoM is deemed to vanish. Further details can be found, e.g., in App. B in Ref.
[28], App. A in Ref. [87], and Ref. [88].

###  B.1 Simplification of the action by field redefinitions

We will take the SMEFT Lagrangian as our starting point, which written in the
modulus-phase form has the structure:

$\displaystyle\mathcal{L}_{\rm SMEFT}$ $\displaystyle=$
$\displaystyle\frac{v^{2}}{4}\left(1+\frac{h}{v}\right)^{2}\langle
D_{\mu}U^{\dagger}D^{\mu}U\rangle$ (134)
$\displaystyle\qquad\,+\frac{1}{2}\left(1+(v+h)^{2}B(h)\right)(\partial_{\mu}h)^{2}\,\,\,-\,\,\,V_{SM}(h)-\Delta
V_{\text{dim-6}}(h)\,,$

which, up to $\mathcal{O}(\Lambda^{-2})$, is given by
$B(h)=\,-\,\frac{2c_{H\Box}}{\Lambda^{2}}$.

We want to make the Higgs kinetic term canonical, as it is customary in the
HEFT Lagrangian. Thus, one has the relation between the SMEFT Higgs modulus
field $h$ and the HEFT Higgs singlet $h_{1}$ is provided by,

$\displaystyle\left(1+(v+h)^{2}B(h)\right)^{1/2}\partial_{\mu}h\,=\
\partial_{\mu}h_{1}\quad\Longrightarrow\quad\left(1+(v+h)^{2}B(h)\right)^{1/2}\frac{dh}{dh_{1}}\,=\,1\,,$
(135)

that leads to

$\displaystyle
dh_{1}=\left(1+(v+h)^{2}B(h)\right)^{1/2}dh\quad\Longrightarrow\quad
h_{1}=\int_{0}^{h}\left(1+(v+h)^{2}B(h)\right)^{1/2}\,dh\,.$

With this integration range prescription, one recovers the LO result
$h_{1}=h$. 888Here we are not performing the necessary Higgs field shift
required to take into account that the potential minimum of $V_{\rm HEFT}(h)$
is not the same as the minimum for the tree-level SM potential ($\langle
h\rangle=0$). This can be always preformed in a second field redefinition
afterwards if needed.  If we go up to NLO in $1/\Lambda^{2}$ we can compute
this redefinition in a perturbative way: first, use the pertubative expansion
of the integrand in Eq. (B.1), e.g.,
$\left(1+(v+h)^{2}B(h)\right)^{1/2}=1-c_{H\Box}(v+h)^{2}/\Lambda^{2}$ up to
NLO in SMEFT; second, trivially integrate each term up to the considered
perturvative order. This two steps lead to the $h_{1}=h_{1}(h)$ relation up to
the desired perturbative order where, e.g., Eq. (33) in the text provides the
expression up to $\mathcal{O}(\Lambda^{-2}$ in the SMEFT expansion.

Finally, through an iterative perturbative procedure, it is possible to invert
this relation and extract the relation $h=h(h_{1})$ as an expansion in powers
of $1/\Lambda^{2}$, as it was shown in Eq. (33) for SMEFT at NLO.

Under this field redefinition the SMEFT Lagrangian then becomes:

$\displaystyle\mathcal{L}_{SMEFT}$ $\displaystyle=$
$\displaystyle\frac{v^{2}}{4}\left(1+\frac{h(h_{1})}{v}\right)^{2}\langle
D_{\mu}U^{\dagger}D^{\mu}U\rangle\,+\frac{1}{2}(\partial_{\mu}h_{1})^{2}\,\,\,-\,\,\,V_{SM}(h(h_{1}))-\Delta
V_{\text{dim-6}}(h(h_{1}))$ (136)
$\displaystyle=\frac{v^{2}}{4}\mathcal{F}(h_{1})\langle
D_{\mu}U^{\dagger}D^{\mu}U\rangle\,+\frac{1}{2}(\partial_{\mu}h_{1})^{2}\,\,-\,\,V_{\rm
HEFT}(h_{1})\,.$

with the flare function $\mathcal{F}(h_{1})$ provided in Eq. (4).

###  B.2 Alternative route: partial integration and equations of motion on the
field $h$.

The previous transformations to cast a SMEFT Lagrangian into a HEFT,
particularly Eq. (12) for the partial integration in terms of $H$ fields and
Eq. (30) changing the variable from $h$ (SMEFT’s) to $h_{1}$ (HEFT’s), lay a
path that can be traded for a different one. We can easily start with the
quantum effective action $\Gamma$ in terms of $h$, then apply partial
integration (Green’s theorem) and finally use the classical Higgs equation of
motion (EoM) of $h$. The resulting effective action is then valid only up to a
given order, and the field appearing therein should be interpreted as $h_{1}$.
This will allow us to transform and remove any operator with only Higgs fields
and two derivatives on the same field, for example to express
$\mathcal{O}^{m}_{2}:=h^{m}\partial^{2}h$ in terms of
$\mathcal{O}^{n}_{1}=h^{n}(\partial_{\mu}h)^{2}$. First, we will show that the
former ones can always be converted in the second ones up to a total
derivative. For this, we will once more use partial integration (Leibniz’s
rule) to rewrite a Lagrangian term of this form as

$\displaystyle\mathcal{O}^{n}_{1}$ $\displaystyle=$
$\displaystyle-\,n\,\mathcal{O}^{n}_{1}\,-\,\mathcal{O}^{n+1}_{2}\,\,\,+\,\,\,\partial^{\mu}(h^{n+1}\partial_{\mu}h)\,,$
(137)

which, as $n>0$ entails $n\neq-1$, and up to the now omitted total divergence,
can be recast in the form

$\displaystyle\mathcal{O}^{n}_{1}=h^{n}(\partial_{\mu}h)^{2}\,\,\,$
$\displaystyle=$
$\displaystyle\,\,\,-\,\frac{1}{n+1}\mathcal{O}^{n+1}_{2}=\,\,\,-\,\frac{1}{n+1}h^{n+1}\partial^{2}h.$
(138)

Thus, we find that all the possible $\mathcal{O}^{n}_{1}$ operators of our
Lagrangian ($n\geq 0$) can be always rewritten in the $\mathcal{O}^{m}_{2}$
form (up to a total derivative) and viceversa. Specifically, up to a total
derivative,

$\displaystyle(h+v)^{m}\,\partial_{\mu}h\,\partial^{\mu}h$ $\displaystyle=$
$\displaystyle-\,\frac{\left[(v+h)^{m+1}-v^{m+1}\right]}{m+1}\,\partial^{2}h\,,$
(139)

This is equivalent to Eq. (22) but written in terms of the singlet and now for
operators with arbitrarily large $n$ powers of $h$; we can use it to rewrite
more general SMEFT operators in the HEFT form. 999Another useful relation for
this type of partial integration simplifications is given by the identity:
$\Delta\mathcal{L}=A\,\partial_{\mu}B\,\partial^{\mu}C=\frac{1}{2}\left((\partial^{2}A)\,B\,C-A\,(\partial^{2}B)\,C-A\,B\,(\partial^{2}C)\right)+\partial_{\mu}\chi^{\mu}$,
with the irrelevant total derivative term given by
$\chi^{\mu}=((\partial^{\mu}A)\,B\,C-A\,(\partial^{\mu}B)\,C-A\,B\,(\partial^{\mu}C))/2$.
This relation is essentially the position-space representation of the relation
$p_{B}p_{C}=\frac{1}{2}(p_{A}^{2}-p_{B}^{2}-p_{C}^{2})$, which is the square
of the momentum conservation equation with all $p_{A,B,C}$ incoming,
$p_{B}+p_{C}=-P_{A}$. By means of it one also can rewrite the
$\mathcal{O}^{n}_{1}$ Lagrangian terms into $\mathcal{O}^{m}_{2}$ operators up
to a total derivative (although it is a little less efficient than the
simplifications in the main text).

Moving on, we transform and remove this second type of operators including
$\partial^{2}h$ that are generated in SMEFT; we will make use of the EoM of
the Higgs field, which causes a difference that is pushed to higher orders in
the expansion that are not included anyway, so that at fixed order they are
equivalent. The EoM (within the electroweak sector alone) reads [88, 89, 21,
19]

$\displaystyle\partial^{2}h$ $\displaystyle=$
$\displaystyle\frac{(v+h)}{2}\,\langle
D_{\mu}U^{\dagger}D^{\mu}U\rangle\,-\,V^{\prime}(h)\,,$ (140)

with Higgs potential (ignorable at high $s$) given by
$V(h)=V_{SM}(h)=m_{h}^{2}\left(-\frac{v^{2}}{8}+\frac{h^{2}}{2}+\frac{h^{3}}{2v}+\frac{h^{4}}{8v^{2}}\right)$
at lowest order in the $1/\Lambda^{2}$ SMEFT expansion, i.e., the tree-level
SM potential, and
$V^{\prime}_{0}(h)=m_{h}^{2}\left(h+\frac{3h^{2}}{2v}+\frac{h^{3}}{2v^{2}}\right)$.
For further detail, see Eqs. (3) and (10) in [88], where one can indeed see
that we also get additional fermion operators whose discussion is beyond the
scope of this article. Thus, we can make use of the EoM of the classical Higgs
field to simplify the operators in the quantum effective action $\Gamma$
(which amounts to appropriate field redefinitions in the generating
functional).

The construction derived from a $\mathcal{O}_{H\Box}$-type operator can then
be transformed as, up to a total derivative,

$\displaystyle(v+h)^{n}(\partial_{\mu}h)^{2}=\,-\,\frac{1}{n+1}\,[(v+h)^{n+1}-v^{n+1}]\,\partial^{2}h$
(141)
$\displaystyle\qquad=\,\,-\,\frac{(v+h_{1})\,[(v+h_{1})^{n+1}-v^{n+1}]}{2(n+1)}\,\langle
D_{\mu}U^{\dagger}D^{\mu}U\rangle\,+\,\frac{1}{n+1}\,[(v+h_{1})^{n+1}-v^{n+1}]\,\,V^{\prime}(h_{1})\,,$

with $h_{1}$ now understood as the HEFT field. In the simplest non-trivial
case, $n=2$, the term $\Delta\mathcal{L}=c_{H\Box}\mathcal{O}_{H\Box}$ yields,
up to a total derivative,

$\displaystyle\Delta\mathcal{L}=\frac{c_{H\Box}}{\Lambda^{2}}\mathcal{O}_{H\Box}=-\,\frac{c_{H\Box}}{\Lambda^{2}}(v+h)^{2}(\partial_{\mu}h)^{2}$
$\displaystyle\qquad=\frac{c_{H\Box}(v+h)}{6\Lambda^{2}}\,[(v+h)^{3}\,-\,v^{3}]\,\langle
D_{\mu}U^{\dagger}D^{\mu}U\rangle\,\,\,-\,\,\,\frac{c_{H\Box}}{3\Lambda^{2}}\,[(v+h)^{3}\,-\,v^{3}]\,V^{\prime}(h)\,.$
(142)

With this, the SMEFT Lagrangian up to dimension-6 can be rewritten in the form

$\displaystyle\mathcal{L}_{\rm
SMEFT}=\frac{v^{2}}{4}\left(1+\frac{h_{1}}{v}\right)^{2}\langle
D_{\mu}U^{\dagger}D^{\mu}U\rangle\,+\frac{1}{2}\left(1-\frac{2c_{H\Box}(h_{1}+v)^{2}}{\Lambda^{2}}\right)(\partial_{\mu}h_{1})^{2}\,\,\,-\,\,\,V(h_{1})$
$\displaystyle\qquad=\frac{v^{2}}{4}\mathcal{F}(h_{1})\langle
D_{\mu}U^{\dagger}D^{\mu}U\rangle+\frac{1}{2}(\partial_{\mu}h_{1})^{2}-V(h)-\frac{c_{H\Box}\,[(v+h_{1})^{3}\,-\,v^{3}]}{3\Lambda^{2}}V^{\prime}(h_{1})\,.$
(143)

with $\mathcal{F}$ the flare function shown in the main text, given at this
order by

$\displaystyle\mathcal{F}(h_{1})=1+\frac{2h_{1}}{v}\left(1+\frac{c_{H\Box}v^{2}}{\Lambda^{2}}\right)+\frac{h_{1}^{2}}{v^{2}}\left(1+\frac{4c_{H\Box}v^{2}}{\Lambda^{2}}\right)+\frac{h_{1}^{3}}{v^{3}}\left(\frac{8c_{H\Box}v^{2}}{3\Lambda^{2}}\right)+\frac{h_{1}^{4}}{v^{4}}\left(\frac{2c_{H\Box}v^{2}}{3\Lambda^{2}}\right)\,.$
(144)

Satisfactorily, this result is in agreement with earlier investigations [47,
64], although now we are extending the relation up to
$\mathcal{O}(h_{1}^{4})$. Terms of $\mathcal{O}(h_{1}^{5})$ in
$\mathcal{F}(h_{1})$ and higher start at $\mathcal{O}(\Lambda^{-4})$ or above,
and they are thus suppressed in the SMEFT counting.

##  Appendix C Restrictions on the coefficients of the HEFT potential
$V(h_{1})$ required for valid SMEFT

Though a full phenomenological analysis will be left for other work (see [90]
for a precis), we here want to call attention on restrictions over the
coefficients of the generic HEFT Higgs potential $V(h)$ that are required for
SMEFT to be applicable, and are thus of interest to falsify it too, beyond
those for $\mathcal{F}$ that we have concentrated over for most of the
manuscript. These relations may be interesting at lower $E<0.5$ TeV which
appears to make them more attractive; but overall, the physics is more
contrived due to the important corrections that the equivalence theorem takes,
so that distinguishing $\omega$ and $W_{L}$ becomes necessary. A full standard
model analysis of the relations is needed, which we will not address now. In
contrast, the equivalence theorem allows a much cleaner analysis of
$\mathcal{F}$ in the TeV region (at the prize of exacting experimental
conditions).

Again, as in subsection IV.1.2, a proper expansion of the SMEFT Lagrangian
that is analytic, and may thus be approximated by a polynomial of the
quadratic singlet $H^{\dagger}H$, requires to avoid any singularities in
passing to it from HEFT. After the transformation to SMEFT coordinates as in
Eq. (III.3), we must recover a SMEFT potential usable in Eq. (1),

$V_{\rm SMEFT}={\rm const.}\times(H^{\dagger}H)+{\rm
const^{\prime}}\times(H^{\dagger}H)^{2}+{\rm
const^{{}^{\prime\prime}}}\times(H^{\dagger}H)^{3}+\dots$ (145)

In a nutshell, when the change of variable from HEFT to SMEFT is attempted,
making the Higgs kinetic term canonical involves $\mathcal{F}$ intruding in
the relation between the potentials as per Eq. (48),

$V_{\rm SMEFT}(H^{\dagger}H)=V_{\rm HEFT}(h_{1}(z))=V_{\rm
HEFT}(\mathcal{F}^{-1}(z))\ ,$ (146)

with $z:=\sqrt{2H^{\dagger}H/v^{2}}$ as in Eq. (53).

Given a generic $V_{\rm HEFT}$ and any coordinate transformation $F$, odd
powers of $z$ appear in $V_{\rm SMEFT}$ upon expanding in this variable around
$z\simeq 0$. But, as such odd-power terms are nonanalytic (odd powers of the
squared root $\sqrt{H^{\dagger}H}$) and cannot appear in SMEFT, which is a
Taylor expansion, we need to demand that they vanish.

Thus, for SMEFT to be valid, the expansion of $V_{\rm HEFT}(h_{1})$ can only
contain even terms in its $(h_{1}-h_{\ast})^{n}$ expansion. This mimicks our
discussion of $\mathcal{F}$, and thus the nonderivative potential density
$V_{\rm HEFT}(h_{1})$ must be an even function when expanded around the
symmetric point $h_{\ast}$; ignoring the potential’s zero-point value, that
expansion starts at order two, namely

$V_{\rm
HEFT}(h_{1})=\frac{m_{h}^{2}v^{2}}{2}\left[\frac{v_{2}^{\ast}}{v^{2}}(h_{1}-h_{\ast})^{2}+\frac{v_{4}^{\ast}}{v^{4}}(h_{1}-h_{\ast})^{4}+\frac{v_{6}^{\ast}}{v^{6}}(h_{1}-h_{\ast})^{6}+\dots\right]$
(147)

This is completely analogous to the properties of $\mathcal{F}$ found in
subsection III.3, that upon expanding around the symmetric point where
$\mathcal{F}(h_{\ast})=0$ needed to have an expansion with only even powers of
$h_{1}-h_{\ast}$.

$\mathcal{F}(h_{1})=\frac{1}{v^{2}}(h_{1}-h_{\ast})^{2}+\frac{a^{*}_{4}}{v^{4}}(h_{1}-h_{\ast})^{4}+\frac{a^{*}_{6}}{v^{6}}(h_{1}-h_{\ast})^{6}+\dots$
(148)

The global factor $\frac{1}{2}m_{h}^{2}v^{2}$ that has been extracted in Eq.
(147) helps us make a direct comparison between the correlations constraining
the potential constants and the very same correlations that we find for the
$a_{i}$ coefficients of the flare function $\mathcal{F}$.

That is, the trilinear, quadrilinear, pentalinear $\dots$ coefficients of the
potential expanded around the physical vacuum,

$V_{\rm
HEFT}=\frac{m_{h}^{2}v^{2}}{2}\left[\left(\frac{h_{1}}{v}\right)^{2}+v_{3}\left(\frac{h_{1}}{v}\right)^{3}+v_{4}\left(\frac{h_{1}}{v}\right)^{4}+\dots\right]$
(149)

are determined by the $v_{4}^{\ast}$, $v_{6}^{\ast}\dots$, coefficients around
the symmetric vacuum, in analogy to Eq. (IV.1.2) where the $a_{i}$ were
determined by the $a_{i}^{\ast}$.

One difference among the two discussions is that, while $\mathcal{F}$ in the
Standard Model Eq. (44) (and hence, the zeroth order SMEFT) is only of order
$h^{2}$, the SM potential $V$ reaches order $h^{4}$. Likewise, SMEFT at one
more order takes $\mathcal{F}$ up to $\mathcal{O}(h^{4})$ while the potential
$V$ reaches $\mathcal{O}(h^{6})$ and so on. Translated in terms of the
expansion coefficients of $\mathcal{F}$, the SM limit implies
$a_{2}^{\ast}\neq 0$ while $a_{4}^{\ast}\ ,\ a_{4}^{\ast}\dots=0$; but those
of $V_{\rm HEFT}(h_{1})$ are one higher order, $v_{2}^{\ast}\ ,\
v_{4}^{\ast}\neq 0$ with $v_{6}^{\ast}\ ,\ v_{8}^{\ast}\dots=0$.

Truncating both series around the symmetric and the physical vacua to order
$h_{1}^{6}$ we can solve for the first three coefficients around the physical
vacuum, yielding (where $h_{\ast}/v\to h_{\ast}$ absorbs the normalization for
visibility)

$\begin{pmatrix}v_{2}^{\ast}\\\ v_{4}^{\ast}\\\
v_{6}^{\ast}\end{pmatrix}=-\frac{1}{8h_{\ast}^{4}}\left(\begin{array}[]{ccc}\frac{15}{2}h_{\ast}^{3}&7h_{\ast}^{4}&3h_{\ast}^{5}\\\
\frac{5}{2}h_{\ast}&5h_{\ast}^{2}&3h_{\ast}^{3}\\\
\frac{1}{2}h_{\ast}^{-1}&1&h_{\ast}\\\
\end{array}\right)\begin{pmatrix}v_{1}\\\ v_{2}\\\ v_{3}\end{pmatrix}\,,$
(150)

with $v_{1}=0$ and $v_{2}=1$ at the potential minimum. This can then be
employed to solve the relation, at the same order, for $v_{4}$, $v_{5}$,
$v_{6}$ that then yields

$\begin{pmatrix}v_{4}\\\ v_{5}\\\
v_{6}\end{pmatrix}=\frac{1}{4h_{\ast}^{4}}\left(\begin{array}[]{ccc}-\frac{5}{2}h_{\ast}&-5h_{\ast}^{2}&-6h_{\ast}^{3}\\\
\frac{3}{2}&3h_{\ast}&3h_{\ast}^{2}\\\
-\frac{1}{4}h_{\ast}^{-1}&-\frac{1}{2}&-\frac{h_{\ast}}{2}\\\
\end{array}\right)\begin{pmatrix}v_{1}\\\ v_{2}\\\ v_{3}\end{pmatrix}\,.$
(151)

These relations allow expressing $v_{4}$, $v_{5}$ and $v_{6}$ in terms of
$v_{1}=0$, $v_{2}=1$, the trilinear coupling $v_{3}$ and the symmetric point
position $h_{\ast}$.

In SMEFT the symmetric point is always set at $\sqrt{2}|H|=(h+v)=0$. In the
absence of derivative operators of the $c_{H\Box}^{(j)}$ type, the SMEFT
radial coordinate $\sqrt{2}|H|=(h+v)$ coincides with the HEFT field
combination $(h_{1}+v)$. Thus, $h=h_{1}=0$ at the potential minimum
$|H|=v/\sqrt{2}$ and $h=h_{1}=h_{\ast}=-v$ at the symmetric point $H=0$. Note
that $v$ is given by the minimum of the $V$ potential at the given SMEFT order
(see Eq. (90) and below). Setting $h_{\ast}/v=-1$ we have the three
correlations in Table 3:

Table 3:  Correlations among the HEFT Higgs potential $V(h)$ around the physical vacuum (thus, directly tree-level observables) derived from assuming that the flare function $\mathcal{F}(h)$ has a zero at $h^{\ast}=-v$; the field is normalized with the physical $v$ corresponding to the observation and not the bare Lagrangian parameter, additionally, $c_{H\Box}=0$. If these correlations are violated, SMEFT needs to be extended to a more general HEFT.  $\displaystyle{v_{4}=\frac{1}{4}(-5+6v_{3})}$ | $\displaystyle{\Delta v_{4}=\frac{3}{2}\Delta v_{3}}$
---|---
$\displaystyle{v_{5}=\frac{3}{4}(v_{3}-1)}$ | $\displaystyle{\Delta v_{5}=\frac{3}{4}\Delta v_{3}}$
$\displaystyle{v_{6}=\frac{1}{8}(v_{3}-1)}$ | $\displaystyle{\Delta v_{6}=\frac{1}{8}\Delta v_{3}}$

If we take the particular value $v_{3}\to 1$, we recover $v_{4}=\frac{1}{4}$
and $v_{5}=0=v_{6}$ that imply no new physics.

This result can be easily generalized to the case with SMEFT operators of the
form $c_{H\Box}^{(6)}$, $c_{H\Box}^{(8)}$, etc. One must simply substitute the
expression for $h_{\ast}$ at $\mathcal{O}(\Lambda^{-2})$ (at
$\mathcal{O}(\Lambda^{-4})$) in Eq. (36) (in Eq. (38)), instead of the value
$h_{\ast}=-v$ employed in Table 3.

##  Appendix D Zeroes of the Flare function in SMEFT

The flare function to order $1/\Lambda^{2}$ is:

$\displaystyle\mathcal{F}(h)=$ $\displaystyle
1+2\frac{h}{v}\underbrace{\left(1+\frac{c_{H\Box}v^{2}}{\Lambda^{2}}\right)}_{a}+\left(\frac{h}{v}\right)^{2}\underbrace{\left(1+\frac{4c_{H\Box}v^{2}}{\Lambda^{2}}\right)}_{b}+\left(\frac{h}{v}\right)^{3}\underbrace{\left(\frac{8c_{H\Box}v^{2}}{3\Lambda^{2}}\right)}_{c}+\left(\frac{h}{v}\right)^{4}\underbrace{\left(\frac{2c_{H\Box}v^{2}}{3\Lambda^{2}}\right)}_{d}$
$\displaystyle=\left(1+\frac{h}{v}\right)^{2}+\left(\frac{h}{v}\right)\left(\frac{2c_{H\Box}v^{2}}{\Lambda^{2}}\right)+\left(\frac{h}{v}\right)^{2}\left(\frac{4c_{H\Box}v^{2}}{\Lambda^{2}}\right)+\left(\frac{h}{v}\right)^{3}\left(\frac{8c_{H\Box}v^{2}}{3\Lambda^{2}}\right)+\left(\frac{h}{v}\right)^{4}\left(\frac{2c_{H\Box}v^{2}}{3\Lambda^{2}}\right)$
$\displaystyle=\left(1+\frac{h}{v}\right)^{2}+\left(\frac{c_{H\Box}v^{2}}{\Lambda^{2}}\right)\left[2\left(\frac{h}{v}\right)+4\left(\frac{h}{v}\right)^{2}+\frac{8}{3}\left(\frac{h}{v}\right)^{3}+\frac{2}{3}\left(\frac{h}{v}\right)^{4}\right]$
(152)

We can see that, at this order in $\Lambda$, both $h=-v$ and
$h=-v+x/\Lambda^{2}$ are possible zeroes:

$\displaystyle\mathcal{F}(-v+x/\Lambda^{2})=$
$\displaystyle=\left(\frac{x}{\Lambda^{2}}\right)^{2}+\left(\frac{c_{H\Box}v^{2}}{\Lambda^{2}}\right)\left[\underbrace{-2+4-8/3+2/3}_{=0}+\mathcal{O}\left(\frac{x}{\Lambda^{2}}\right)\right]$
(153)

This is, the coefficients in front of $c_{H\Box}$ conspire to make the fixed
point of $\mathcal{F}(h)$ be the one of the SM. I.e. this cancellation would
not hold if $a,b,c,d$ wouldn’t be related the way they are in Eq. (152), as
$(-2,4,-8/3,2/3)$. Using the SMEFT basis, it is only at quadratic level that a
shift is allowed. The flare function at order
$\mathcal{O}(\rm{dim-6},\Lambda^{-4})$ is given in Eq. (II.3.2). We can test
again the solution $h\to-v$:

$\displaystyle\mathcal{F}(-v)=$
$\displaystyle\left(-1\right)\left[2\left(\frac{c_{H\Box}v^{2}}{\Lambda^{2}}+\frac{3c_{H\Box}^{2}v^{4}}{2\Lambda^{4}}\right)-\left(\frac{4c_{H\Box}v^{2}}{\Lambda^{2}}+\frac{12c_{H\Box}^{2}v^{4})}{\Lambda^{4}}\right)\right.+\left(\frac{8c_{H\Box}v^{2}}{3\Lambda^{2}}+\frac{56c_{H\Box}^{2}v^{4}}{3\Lambda^{4}}\right)$
$\displaystyle-\left(\frac{2c_{H\Box}v^{2}}{3\Lambda^{2}}+\frac{44c_{H\Box}^{2}v^{4}}{3\Lambda^{4}}\right)+\left.\left(\frac{88c_{H\Box}^{2}v^{4}}{15\Lambda^{4}}\right)-\left(\frac{44c_{H\Box}^{2}v^{4}}{45\Lambda^{4}}\right)\right]=$
$\displaystyle=\left(-\frac{c_{H\Box}v^{2}}{\Lambda^{2}}\right)\underbrace{\left[2-4+\frac{8}{3}-\frac{2}{3}\right]}_{=0}-\left(\frac{c_{H\Box}v^{2}}{\Lambda^{2}}\right)^{2}\underbrace{\left[\frac{3}{2}-12+\frac{56}{3}-\frac{44}{3}+\frac{88}{15}-\frac{44}{45}\right]}_{=-1/9}$
(154)

in this case we do observe a shift of the solution
$\Delta=\left(\frac{c_{H\Box}v}{3\Lambda^{2}}\right)^{2}$. Coincidentally
similar to the solution $h_{*}=-v+\frac{c_{H\Box}v^{3}}{3\Lambda^{2}}$, given
in Eq. (38).

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