&=\text{tr}(\rho_{0}A^{0}e^{-i\mathcal{H}_{0}t_{1}/\hbar}A^{1}e^{i\mathcal{H}_{0}t_{1}/\hbar}).\end{split}$
(211)

Writing the time evolution factors as

$\begin{split}e^{\pm i\mathcal{H}_{0}t_{1}/\hbar}=&\int
d\varepsilon\delta(\varepsilon-\mathcal{H}_{0})e^{\pm i\varepsilon
t_{1}/\hbar}\\\ =&\frac{i}{2\pi}\int
d\varepsilon(G^{r}(\varepsilon)-G^{a}(\varepsilon))e^{\pm i\varepsilon
t_{1}/\hbar}\\\ \equiv&\frac{i}{2\pi}\int d\varepsilon
G^{r-a}_{\varepsilon}e^{\pm i\varepsilon t_{1}/\hbar},\end{split}$ (212)

where we used (206), we have for the Fourier transform

$\begin{split}&\tilde{C}_{A^{0}A^{1}}(\omega_{1})=\int
dt_{1}\tilde{C}_{A^{0}A^{1}}(t_{1})e^{i\omega_{1}t_{1}}\\\ &=\int
dt_{1}\text{tr}(\rho_{0}A^{0}e^{-i\mathcal{H}_{0}t_{1}/\hbar}A^{1}e^{i\mathcal{H}_{0}t_{1}/\hbar})e^{i\omega_{1}t_{1}}\\\
&=\int d\varepsilon\int
d\varepsilon^{\prime}\text{tr}(\rho_{0}A^{0}\delta(\varepsilon^{\prime}-\mathcal{H}_{0})A^{1}\delta(\varepsilon-\mathcal{H}_{0}))\\\
&\qquad\times\int
dt_{1}e^{i\left(\omega_{1}-\frac{\varepsilon^{\prime}-\varepsilon}{\hbar}\right)t_{1}}\\\
&=2\pi\hbar\int
d\varepsilon\text{tr}(\rho_{0}A^{0}\delta(\varepsilon+\hbar\omega_{1}-\mathcal{H}_{0})A^{1}\delta(\varepsilon-\mathcal{H}_{0})),\end{split}$
(213)

where we used the identity

$\int
dt_{1}e^{i(\omega_{1}-(\varepsilon-\varepsilon^{\prime})/\hbar)t_{1}}=2\pi\hbar\delta(\varepsilon-\varepsilon^{\prime}-\hbar\omega_{1}).$
(214)

We can thus write the spectral representation of the 2-point stripped
correlator as

$\begin{split}\tilde{C}_{[A^{0}A^{1}]}(\omega_{1})=&\tilde{C}_{A^{0}A^{1}}(\omega_{1})-\tilde{C}_{A^{0}A^{1}}(-\omega_{1})\\\
=&2\pi\hbar\int
d\varepsilon\text{tr}(\rho_{0}(A^{0}\delta(\varepsilon+\hbar\omega_{1}-\mathcal{H}_{0})A^{1}\\\
&-A^{1}\delta(\varepsilon-\hbar\omega_{1}-\mathcal{H}_{0})A^{0})\delta(\varepsilon-\mathcal{H}_{0})).\end{split}$
(215)

Using the fact that the equilibrium density matrix is a function of
$\mathcal{H}_{0}$, i.e., $\rho_{0}\equiv\rho_{0}(\mathcal{H}_{0})$, we can use
the cyclic property of the trace and the standard trick of writing
$\rho_{0}(\mathcal{H}_{0})\delta(\varepsilon-\mathcal{H}_{0})=\rho_{0}(\varepsilon)\delta(\varepsilon-\mathcal{H}_{0})$
to ‘pull out’ the scalar-valued function $\rho_{0}(\varepsilon)$ from the
trace

$\begin{split}\tilde{C}_{[A^{0}A^{1}]}(\omega_{1})=&2\pi\hbar\int
d\varepsilon\rho_{0}(\varepsilon)\text{tr}((A^{0}\delta(\varepsilon+\hbar\omega_{1}-\mathcal{H}_{0})A^{1}\\\
&-A^{1}\delta(\varepsilon-\hbar\omega_{1}-\mathcal{H}_{0})A^{0})\delta(\varepsilon-\mathcal{H}_{0})).\end{split}$
(216)

At this point, we shift the integration variable in the second term within the
trace as $\varepsilon\to\varepsilon+\hbar\omega_{1}$ and once again use the
cyclic property to arrive at

$\begin{split}\tilde{C}_{[A^{0}A^{1}]}(\omega_{1})=-&2\pi\hbar\int
d\varepsilon(\rho_{0}(\varepsilon+\omega_{1})-\rho_{0}(\varepsilon))\\\
&\times\text{tr}((A^{0}\delta(\varepsilon+\hbar\omega_{1}-\mathcal{H}_{0})A^{1}\delta(\varepsilon-\mathcal{H}_{0}))\\\
\equiv-&2\pi\hbar\int
d\varepsilon(\rho_{0}(\varepsilon+\omega_{1})-\rho_{0}(\varepsilon))\\\
&\times\text{tr}(A^{0}G^{r-a}_{\omega_{1}}A^{1}G^{r-a}).\end{split}$ (217)

###  E.3 Spectral representation of the 3-point stripped correlator

The 3-point stripped correlator (the $n=2$ case of (182)) in terms of plain
correlators is

$\begin{split}&C_{[A^{0}A^{1}A^{2}]}(t_{0},t_{1},t_{2})\\\
&=C_{A^{0}A^{1}A^{2}}(t_{0},t_{1},t_{2})+C_{A^{2}A^{1}A^{0}}(t_{2},t_{1},t_{0})\\\
&\quad-
C_{A^{2}A^{0}A^{1}}(t_{0},t_{1},t_{2})-C_{A^{1}A^{0}A^{2}}(t_{2},t_{1},t_{0}).\end{split}$
(218)

Time-translation invariance allows us to write

$\begin{split}&\tilde{C}_{[A^{0}A^{1}A^{2}]}(t_{1},t_{2})\\\
&=\tilde{C}_{A^{0}A^{1}A^{2}}(t_{1},t_{2})+\tilde{C}_{A^{2}A^{1}A^{0}}(t_{1}-t_{2},-t_{2})\\\
&\quad-\tilde{C}_{A^{2}A^{0}A^{1}}(-t_{2},t_{1}-t_{2})-\tilde{C}_{A^{1}A^{0}A^{2}}(-t_{1},t_{2}-t_{1}).\end{split}$
(219)

The Fourier transform is

$\begin{split}&\tilde{C}_{[A^{0}A^{1}A^{2}]}(\omega_{1},\omega_{2})\\\
&=\tilde{C}_{A^{0}A^{1}A^{2}}(\omega_{1},\omega_{2})+\tilde{C}_{A^{2}A^{1}A^{0}}(\omega_{1},-\omega_{1}-\omega_{2})\\\
&\quad-\tilde{C}_{A^{2}A^{0}A^{1}}(-\omega_{1}-\omega_{2},\omega_{2})-\tilde{C}_{A^{1}A^{0}A^{2}}(-\omega_{1}-\omega_{2},\omega_{1}).\end{split}$
(220)

The explicit expression for the 3-point plain correlator
$\tilde{C}_{A^{0}A^{1}A^{2}}(t_{1},t_{2})\equiv
C_{A^{0}A^{1}A^{2}}(0,-t_{1},-t_{2})$ is given by the $n=2$ case of (183) as

$\begin{split}&\tilde{C}_{A^{0}A^{1}A^{2}}(t_{1},t_{2})\\\
&=\text{tr}(\rho_{0}A^{0}A^{1}_{\mathcal{H}_{0}}(-t_{1})A^{2}_{\mathcal{H}_{0}}(-t_{2}))\\\
&=\text{tr}(\rho_{0}A^{0}e^{-i\mathcal{H}_{0}t_{1}/\hbar}A^{1}e^{i\mathcal{H}_{0}(t_{1}-t_{2})/\hbar}A^{2}e^{i\mathcal{H}_{0}t_{2}/\hbar}).\end{split}$
(221)

Writing the time-evolution factors using delta functions, or, equivalently,
$G^{r-a}_{\varepsilon}$ terms, according to (212), we obtain the Fourier
transform

$\begin{split}&\tilde{C}_{A^{0}A^{1}A^{2}}(\omega_{1},\omega_{2})=\int
dt_{1}\int
dt_{2}\tilde{C}_{A^{0}A^{1}A^{2}}(t_{1},t_{2})e^{i\omega_{1}t_{1}}e^{i\omega_{2}t_{2}}\\\
&=\int d\varepsilon\int d\varepsilon^{\prime}\int
d\varepsilon^{\prime\prime}\text{tr}(\rho_{0}A^{0}\delta(\varepsilon^{\prime\prime}-\mathcal{H}_{0})A^{1}\delta(\varepsilon^{\prime}-\mathcal{H}_{0}))A^{2}\delta(\varepsilon-\mathcal{H}_{0}))\int
dt_{1}e^{i\left(\omega_{1}-\frac{\varepsilon^{\prime\prime}-\varepsilon^{\prime}}{\hbar}\right)t_{1}}\int
dt_{2}e^{i\left(\omega_{2}-\frac{\varepsilon^{\prime}-\varepsilon}{\hbar}\right)t_{2}}\\\
&=4\pi^{2}\hbar^{2}\int
d\varepsilon\text{tr}(\rho_{0}A^{0}\delta(\varepsilon+\hbar\omega_{1}+\hbar\omega_{2}-\mathcal{H}_{0})A^{1}\delta(\varepsilon+\hbar\omega_{2}-\mathcal{H}_{0})A^{2}\delta(\varepsilon-\mathcal{H}_{0}))\\\
&\equiv-\frac{i\hbar^{2}}{2\pi}\int
d\varepsilon\text{tr}(\rho_{0}A^{0}G^{r-a}_{\omega_{1}+\omega_{2}}A^{1}G^{r-a}_{\omega_{2}}A^{2}G^{r-a}).\end{split}$
(222)

We can then combine the corresponding terms according (220), and, just as for
the 2-point case above, use the delta function to pull out the density matrix
from the trace, and shift arguments to find

$\begin{split}\tilde{C}_{[A^{0}A^{1}A^{2}]}(\omega_{1},\omega_{2})=&-4\pi^{2}\hbar^{2}\int
d\varepsilon(\rho_{0}(\varepsilon+\hbar\omega_{2})-\rho_{0}(\varepsilon))\text{tr}(A^{0}\delta(\varepsilon+\hbar\omega_{1}+\hbar\omega_{2}-\mathcal{H}_{0})A^{1}\delta(\varepsilon+\hbar\omega_{2}-\mathcal{H}_{0})A^{2}\delta(\varepsilon-\mathcal{H}_{0}))\\\
&+4\pi^{2}\hbar^{2}\int
d\varepsilon(\rho_{0}(\varepsilon)-\rho_{0}(\varepsilon-\hbar\omega_{2}))\text{tr}(A^{1}\delta(\varepsilon-\hbar\omega_{1}-\hbar\omega_{2}-\mathcal{H}_{0})A^{0}\delta(\varepsilon-\mathcal{H}_{0})A^{2}\delta(\varepsilon-\hbar\omega_{2}-\mathcal{H}_{0}))\\\
\equiv&\frac{i\hbar^{2}}{2\pi}\int
d\varepsilon(\rho_{0}(\varepsilon+\hbar\omega_{2})-\rho_{0}(\varepsilon))\text{tr}(A^{0}G^{r-a}_{\omega_{1}+\omega_{2}}A^{1}G^{r-a}_{\omega_{2}}A^{2}G^{r-a})\\\
&-\frac{i\hbar^{2}}{2\pi}\int
d\varepsilon(\rho_{0}(\varepsilon)-\rho_{0}(\varepsilon-\hbar\omega_{2}))\text{tr}(A^{1}G^{r-a}_{-\omega_{1}-\omega_{2}}A^{0}G^{r-a}A^{2}G^{r-a}_{-\omega_{2}}).\end{split}$
(223)

###  E.4 Spectral representation of the 2-point retarded correlator

The 2-point plain correlator needed in (196) gives

$\begin{split}&-\frac{i}{\hbar}\theta_{t_{1}}\tilde{C}_{A^{0}A^{1}}(t_{1})\\\
&=\text{tr}\left(\rho_{0}A^{0}\left(\frac{-i}{\hbar}\theta_{t_{1}}e^{-i\mathcal{H}_{0}t_{1}/\hbar}\right)A^{1}e^{i\mathcal{H}_{0}t_{1}/\hbar}\right)\\\
&=i\hbar\text{tr}\left(\rho_{0}A^{0}G^{r}_{t_{1}}A^{1}(G^{r}_{-t_{1}}-G^{a}_{-t_{1}})\right)\\\
&=i\hbar\int\frac{d\varepsilon}{2\pi\hbar}\int\frac{d\varepsilon^{\prime}}{2\pi\hbar}\text{tr}(\rho_{0}A^{0}G^{r}_{\varepsilon}A^{1}(G^{r}_{\varepsilon^{\prime}}-G^{a}_{\varepsilon^{\prime}}))\\\
&\qquad\qquad\qquad\qquad\qquad\times
e^{-i(\varepsilon-\varepsilon^{\prime})t_{1}/\hbar},\end{split}$ (224)

where we used (205). Taking the Fourier transform results in

$\begin{split}&\int
dt_{1}\left(-\frac{i}{\hbar}\theta_{t_{1}}\tilde{C}_{A^{0}A^{1}}(t_{1})\right)e^{i\omega_{1}t_{1}}\\\
&=\frac{i}{2\pi}\int
d\varepsilon\text{tr}(\rho_{0}A^{0}G^{r}_{\varepsilon+\hbar\omega_{1}}A^{1}(G^{r}_{\varepsilon}-G^{a}_{\varepsilon})),\\\
&\equiv\frac{i}{2\pi}\int
d\varepsilon\text{tr}(\rho_{0}A^{0}G^{r}_{\omega_{1}}A^{1}G^{r-a}).\end{split}$
(225)

With the projector notation introduced in (200), the spectral representation
of the 2-point retarded correlator (196) in the frequency domain becomes

$\tilde{C}^{r}_{A^{0}A^{1}}(\omega_{1})=\frac{1}{\pi}\hat{\mathcal{P}}^{(+)}_{\mathcal{K}^{*}_{\omega}}\,i\int
d\varepsilon\,\text{tr}\left(\rho_{0}A^{0}G^{r}_{\omega_{1}}A^{1}G^{r-a}\right).$
(226)

###  E.5 Spectral representation of the 3-point retarded correlator

The 3-point retarded correlator (197) contains two different kinds of plain
correlators and is expressed in two different ways depending on what
combinations of step functions are attached to the second plain correlator.

The first correlator becomes

$\begin{split}-\frac{1}{\hbar^{2}}\theta_{t_{1}}\theta_{t_{2}t_{1}}\tilde{C}_{A^{0}A^{1}A^{2}}(t_{1},t_{2})&=\text{tr}\left(\rho_{0}A^{0}\left(\frac{-i}{\hbar}\theta_{t_{1}}e^{-i\mathcal{H}_{0}t_{1}/\hbar}\right)A^{1}\left(\frac{-i}{\hbar}\theta_{t_{2}t_{1}}e^{-i\mathcal{H}_{0}(t_{2}-t_{1})/\hbar}\right)A^{2}e^{i\mathcal{H}_{0}t_{2}/\hbar}\right)\\\
&=i\hbar\text{tr}(\rho_{0}A^{0}G^{r}_{t_{1}}A^{1}G^{r}_{t_{2}-t_{1}}A^{2}(G^{r}_{-t_{2}}-G^{a}_{-t_{2}}))\\\
&=i\hbar\int\frac{d\varepsilon}{2\pi\hbar}\int\frac{d\varepsilon^{\prime}}{2\pi\hbar}\int\frac{d\varepsilon^{\prime\prime}}{2\pi\hbar}\text{tr}(\rho_{0}A^{0}G^{r}_{\varepsilon}A^{1}G^{r}_{\varepsilon^{\prime}}A^{2}(G^{r}_{\varepsilon^{\prime\prime}}-G^{a}_{\varepsilon^{\prime\prime}}))e^{-i(\varepsilon-\varepsilon^{\prime})t_{1}/\hbar}e^{-i(\varepsilon^{\prime}-\varepsilon^{\prime\prime})t_{2}/\hbar}.\end{split}$
(227)

Moving to the frequency domain, we obtain

$\begin{split}&\int dt_{1}\int
dt_{2}\left(-\frac{1}{\hbar^{2}}\theta_{t_{1}}\theta_{t_{2}t_{1}}\tilde{C}_{A^{0}A^{1}A^{2}}(t_{1},t_{2})\right)e^{i\omega_{1}t_{1}}e^{i\omega_{2}t_{2}}\\\
&=\frac{i}{2\pi}\int
d\varepsilon\text{tr}(\rho_{0}A^{0}G^{r}_{\omega_{1}+\omega_{2}}A^{1}G^{r}_{\omega_{2}}A^{2}G^{r-a}).\end{split}$
(228)

The other plain correlator in (197) has two different combinations of step
functions attached to it. We have for the first type

$\begin{split}\frac{1}{\hbar^{2}}\theta_{t_{1}}\theta_{t_{2}t_{1}}\tilde{C}_{A^{1}A^{0}A^{2}}(-t_{1},t_{2}-t_{1})&=-\text{tr}\left(\rho_{0}A^{1}\left(\frac{i}{\hbar}\theta_{t_{1}}e^{i\mathcal{H}_{0}t_{1}/\hbar}\right)A^{0}e^{-i\mathcal{H}_{0}t_{2}/\hbar}A^{2}\left(\frac{i}{\hbar}\theta_{t_{2}t_{1}}e^{i\mathcal{H}_{0}(t_{2}-t_{1})/\hbar}\right)\right)\\\
&=-i\hbar\text{tr}(\rho_{0}A^{1}G^{a}_{-t_{1}}A^{0}(G^{r}_{t_{2}}-G^{a}_{t_{2}})A^{2}G^{a}_{t_{1}-t_{2}}).\end{split}$
(229)

Similarly,

$\frac{1}{\hbar^{2}}\theta_{t_{2}}\theta_{t_{1}t_{2}}\tilde{C}_{A^{1}A^{0}A^{2}}(-t_{1},t_{2}-t_{1})=-i\hbar\text{tr}(\rho_{0}A^{1}(G^{r}_{-t_{1}}-G^{a}_{-t_{1}})A^{0}G^{r}_{t_{2}}A^{2}G^{r}_{t_{1}-t_{2}}).$
(230)

The frequency-domain expression of the spectral representation then becomes

$\begin{split}&\int dt_{1}\int
dt_{2}\left(\frac{1}{\hbar^{2}}(\theta_{t_{1}}\theta_{t_{2}t_{1}}+\theta_{t_{2}}\theta_{t_{1}t_{2}})\tilde{C}_{A^{1}A^{0}A^{2}}(-t_{1},t_{2}-t_{1})\right)e^{i\omega_{1}t_{1}}e^{i\omega_{2}t_{2}}=\\\
&=-\frac{i}{2\pi}\int
d\varepsilon(\text{tr}(\rho_{0}A^{1}G^{r-a}A^{0}G^{r}_{\omega_{1}+\omega_{2}}A^{2}G^{r}_{\omega_{1}})+\text{tr}(\rho_{0}A^{1}G^{a}_{-\omega_{1}-\omega_{2}}A^{0}G^{r-a}A^{2}G^{a}_{-\omega_{2}})).\end{split}$
(231)

We also have the second type

$\frac{1}{\hbar^{2}}\theta_{t_{1}}\theta_{t_{2}}\tilde{C}_{A^{1}A^{0}A^{2}}(-t_{1},t_{2}-t_{1})=i\hbar\text{tr}(\rho_{0}A^{1}G^{a}_{-t_{1}}A^{0}G^{r}_{t_{2}}A^{2}(G^{r}_{t_{1}-t_{2}}-G^{a}_{t_{1}-t_{2}}),$
(232)

which yields

$\begin{split}&\int dt_{1}\int
dt_{2}\left(\frac{1}{\hbar^{2}}\theta_{t_{1}}\theta_{t_{2}}\tilde{C}_{A^{1}A^{0}A^{2}}(-t_{1},t_{2}-t_{1})\right)e^{i\omega_{1}t_{1}}e^{i\omega_{2}t_{2}}\\\
&=\frac{i}{2\pi}\int
d\varepsilon\text{tr}(\rho_{0}A^{1}G^{a}_{-\omega_{1}}A^{0}G^{r}_{\omega_{2}}A^{2}G^{r-a}).\end{split}$
(233)

Combining (228) with first (231) and then (233) and attaching the projectors,
the two expressions for the full 3-point retarded correlator (197) become

$\displaystyle C^{r}_{A^{0}A^{1}A^{2}}(\omega_{1},\omega_{2})=$
$\displaystyle\frac{1}{\pi}\hat{\mathcal{P}}^{(+)}_{\mathcal{K}^{*}_{\omega}}\hat{\mathcal{P}}^{(\Gamma_{1}^{+})}_{A^{1}_{\omega_{1}}A^{2}_{\omega_{2}}}i\int
d\varepsilon\,\text{tr}\bigg{(}\rho_{0}\bigg{[}A^{0}G^{r}_{\omega_{1}+\omega_{2}}A^{1}G^{r}_{\omega_{2}},A^{2}G^{r-a}\bigg{]}\bigg{)}$
(234) $\displaystyle=$
$\displaystyle\frac{1}{\pi}\hat{\mathcal{P}}^{(+)}_{\mathcal{K}^{*}_{\omega}}\hat{\mathcal{P}}^{(\Gamma_{1}^{+})}_{A^{1}_{\omega_{1}}A^{2}_{\omega_{2}}}i\int
d\varepsilon\,\text{tr}\bigg{(}\rho_{0}\bigg{(}A^{0}G^{r}_{\omega_{1}+\omega_{2}}A^{1}G^{r}_{\omega_{2}}A^{2}+\frac{1}{2}A^{1}G^{a}_{-\omega_{1}}A^{0}G^{r}_{\omega_{2}}A^{2}\bigg{)}G^{r-a}\bigg{)},$

where $[\dots,\dots]$ is the usual commutator and we defined the projector
$\hat{\mathcal{P}}^{(\Gamma_{1}^{+})}_{A^{1}_{\omega_{1}}\dots
A^{n}_{\omega_{n}}}$ onto the $\Gamma_{1}^{+}$ irreducible representation of
the permutation group $P^{A^{1}_{\omega_{1}}\dots A^{n}_{\omega_{n}}}(n)$ that
acts by permuting the observables and frequency labels simultaneously.

###  E.6 Spectral representation of the 4-point retarded correlator

Pairing the step functions with the interaction picture evolution operators,
recognizing Green’s operators and taking their Fourier transforms followed by
moving to the frequency domain in the same manner as we did for the 2 and
3-point correlators above, enables us to write (198) as

$\begin{split}C^{r}_{A^{0}A^{1}A^{2}A^{3}}(\omega_{1},\omega_{2},\omega_{3})=\frac{1}{\pi}\hat{\mathcal{P}}^{(+)}_{\mathcal{K}^{*}_{\omega}}\hat{\mathcal{P}}^{(\Gamma_{1}^{+})}_{A^{1}_{\omega_{1}}A^{2}_{\omega_{2}}A^{3}_{\omega_{3}}}i\int
d\varepsilon\,\text{tr}\bigg{(}\rho_{0}\bigg{(}&A^{0}G^{r}_{\omega_{1}+\omega_{2}+\omega_{3}}A^{1}G^{r}_{\omega_{2}+\omega_{3}}A^{2}G^{r}_{\omega_{3}}\\\
&+A^{1}G^{a}_{-\omega_{1}}A^{0}G^{r}_{\omega_{2}+\omega_{3}}A^{2}G^{r}_{\omega_{3}}\bigg{)}A^{3}G^{r-a}\bigg{)}.\end{split}$
(235)

###  E.7 Spectral representation of advanced correlators

Up to this point, the center stage has been given to the retarded correlator
$C^{r}$ since it expresses the total physical response of a system. However,
based on our discussion in section III, the advanced correlator $C^{a}$ plays
an equally significant role when we are concerned with discerning the
$C^{r}$’s behaviour under the action of the time-reversal group. Indeed,
formula (38) describes the reactive(dissipative) part of the response as
proportional to $C^{r+a}$($C^{r-a}$). Even when we are not interested in the
reactive/dissipative distinction, only the behaviour under magnetic-inversion,
we are left with formula (40) which contains both $C^{r}$ and a time-inverted
$C^{a}$. Consequently, it is important to obtain the spectral representation
of $C^{a}$ in addition to $C^{r}$ arrived at above. The recipe for this is
very simple.
Consider the 2-point advanced correlator arrived at by taking $n=1$ in (17)
and making use of time-translation invariance

$\begin{split}&C^{a}_{A^{0}A^{1}}(t_{1})=2\mathfrak{Re}(\frac{i}{\hbar}\theta_{-t_{1}}C_{A^{0}A^{1}}(0,-t_{1}))\\\
&=2\mathfrak{Re}\left(\text{tr}\left(\rho_{0}A^{0}\left(\frac{i}{\hbar}\theta_{-t_{1}}e^{-i\mathcal{H}_{0}t_{1}/\hbar}\right)A^{1}e^{i\mathcal{H}_{0}t_{1}/\hbar}\right)\right)\\\
&=2\mathfrak{Re}(i\hbar\text{tr}\left(\rho_{0}A^{0}G^{a}(t_{1})A^{1}(G^{r}(-t_{1})-G^{a}(-t_{1}))\right)).\end{split}$
(236)

Contrasting this with the retarded case (224), we note that the only
difference is that in the latter, we had $G^{r}(t_{1})$ as a result of the
step-function attachment, whereas in the present case it is $G^{a}(t_{1})$
that takes the stage. This means that we can obtain the spectral
representation of $C^{a}_{A^{0}A^{1}}(\omega_{1})$ by simply exchanging
$G^{r}_{\omega_{1}}$ for $G^{a}_{\omega_{1}}$ in (226) and vice-versa. Note,
however, that $G^{r}(\varepsilon)-G^{a}(\varepsilon)$ should be left as is
since this factor is not a result of step-function attachments. In a similar
manner, the spectral representations of the advanced 3 and 4-point correlators
can be found from the retarded expressions (234) and (235) by performing the
exchange $G^{r}\leftrightarrow G^{a}$ while leaving the $G^{r-a}\equiv
G^{r}(\varepsilon)-G^{a}(\varepsilon)$ factor unchanged.

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# PWR-Align: Leveraging Part-Whole Relationships for Part-wise Rigid Point
Cloud Registration in Mixed Reality Applications

Manorama Jha
GridRaster Inc.
<EMAIL_ADDRESS>Bhaskar Banerjee
GridRaster Inc.
<EMAIL_ADDRESS>
###### Abstract

We present an efficient and robust point cloud registration (PCR) workflow for
part-wise rigid point cloud alignment using the Microsoft HoloLens 2. Point
Cloud Registration (PCR) is an important problem in Augmented and Mixed
Reality use cases, and we present a study for a special class of non-rigid
transformations. Many commonly encountered objects are composed of rigid parts
that move relative to one another about joints resulting in non-rigid
deformation of the whole object such as, robots with manipulators, and
machines with hinges. The workflow presented allows us to register the point
cloud with various configurations of the point cloud.

##  1 Introduction

Mixed Reality (MR) [1] is used to create immersive experiences by blending
digital entities with the real-world environment. Point Cloud Registration
(PCR) [2], also called alignment, is the task of finding a spatial
transformation (rotation and translation) which when applied to a source point
cloud results in point-wise superposition with a destination point cloud of
the same object and/or scene. One of the primary use cases of PCR in MR is to
overlay a virtual reference model of an object on top of the corresponding
real object. For example, in manufacturing processes a virtual reference model
of a machine part can be overlaid on a damaged instance for inspection [3],
repair [4, 5] and training [6, 7, 8]. Real world applications of PCR face
several challenges like lack of order or structure [9, 10, 11, 12, 13],
sparsity [14], noise and outliers, [15, 16, 17, 18] and partial overlap [13,
19, 20, 21] of the source and target point clouds. ††Accepted for presentation
at the Women in Computer Vision (WiCV) workshop at CVPR-2023.

###  1.1 Background and Related Work

Traditionally, PCR involves two steps - a) feature extraction [22, 23, 13]
from the source and target point clouds and b) feature matching [17, 18, 24,
25, 26] to detect corresponding pairs of points in the two clouds. The
correspondence matches are then used to derive a transformation function that
maps the source to the target such that the corresponding points overlap. For
rigid objects, this transformation function is an affine transformation
consisting of rotation and translation. For non-rigid objects, this
transformation function is more complex and it involves warping the surface of
the source point cloud on to the target. Non-rigid Iterative Closest Point
[27, 28] is a state-of-the-art algorithm that applies a locally affine
regularization that assigns an affine transformation to each vertex and
minimizes the difference in the transformation of neighboring vertices. With
this regularization, the optimal deformation for fixed correspondences and
fixed stiffness can be determined precisely and efficiently. The algorithm
loops over a series of decreasing stiffness weights that results in
incremental deformation of the template surface towards the target.

Figure 1: Sample registration results for (a) NASA Viking Lander [29] and (b)
Cubebot [30] models. The target point cloud is scanned using Microsoft
Hololens 2 MR headset under real-world conditions. Our workflow achieves
robust and accurate performance even in the presence of sparse, noisy, and
partial data.

The scope of this paper is limited to finding a non-rigid transformation using
the set of predicted correspondences. In the general case, this involves
finding a transformation for each point on the source mesh and the
computational requirement can blow up for large meshes. Additional challenges
are posed by noise, holes and missing parts in real world 3D scans.
Computational efficiency also becomes crucial in MR applications to produce a
lag-free immersive experience.

###  1.2 Contributions

In this paper, we study PCR for a special class of non-rigid transformations
that happen in objects composed of rigid parts that can move relative to one
another (e.g., about a hinge). These are commonly encountered in enterprise MR
applications, e.g. machine parts, doors, wheels, antennas, etc. For such
objects, the non-rigid adjustment step after rigid fitting requires finding a
rigid transformation for each part that can move relative to one another. We
begin by constructing a graph to represent the part-whole relationships in the
source object. We present a novel approach to derive a rigid transformation
for each part, leveraging this graph to – a) make sure that the object model
does not disintegrate after applying part-wise registration; b) search for a
given part in the correct part of the target point cloud; and c) handle
outliers in the correspondence map.

##  2 Proposed Method - PWR-Align

In this section, we present PWR-Align, our proposed workflow that leverages
Part-Whole Relationships for Alignment. We start with constructing a
bidirectional graph of the source point cloud (from a CAD model or digital
twin). Each node of this graph represents one of the rigid parts. Each pair of
adjacent parts is connected with an edge. The next step is finding
corresponding points in the source and target point clouds for which we use
the Lepard [31] non-rigid correspondence matching algorithm. The third step is
the whole body rigid fitting, where we derive the rigid body transformation
for the whole source point cloud. This aligns the source and target point
clouds with possible misalignments of movable parts. The final step involves
part-wise tuning where each part is tuned in two steps - first using RANSAC
[17, 18] and then using the Iterative Closest Point (ICP) [32] algorithms. The
output of this step is a rotation and translation matrix for the part.

To prevent the object model from disintegrating upon part-wise transformation
we take the following steps:

  1. 1.

We adjust the parts in decreasing order of sizes.

  2. 2.

When using RANSAC, we add extra correspondences in the form of points that are
at the junction of the current part and its larger neighbor that must remain
at the same position where they were before the adjustment.

  3. 3.

If application of a transformation to a part causes a joint to be broken, we
skip the adjustment.

  4. 4.

If the number of detected correspondences is too few for a part, we skip the
RANSAC step for that part.

To make sure that a part is searched for in the correct part of the target
point cloud, we take the following measures:

  1. 1.

We perform whole body rigid fitting as the first step to roughly align the two
point clouds.

  2. 2.

We search for a given part only within a Region-of-Interest (RoI) of the
target point cloud. The RoI is calculated as the smallest axis-aligned
bounding box that contains the part in the source point cloud after whole body
rigid fitting and all the corresponding points in the target point cloud for
the feature points of the part.

Finally, in order to improve robustness against outliers, we take the
following measures:

  1. 1.

KPConv applies a uniform downsampling to prevent the compute load (quadratic
in the number of samples) from blowing up. If the density of a point cloud is
low, this leads to serious loss in detail and significant degradation in
performance. We make sure there are a minimum of $50$ points in each part of
the input point clouds.

  2. 2.

We perform ICP after RANSAC to correct for the effect of spurious
correspondences.

##  3 Results and Discussion

We evaluated our workflow on data collected using Microsoft Hololens 2 MR
headset [33] under real world conditions. Fig.1 shows two sample outcomes for
NASA Viking Lander [29] and Cubebot [30]. We observe that our workflow is able
to perform accurately even in the presence of noise, sparsity and missing
parts in the target point cloud scan. Performance degrades gracefully as
sparsity increases. Fig.1 (b) presents an extreme case where the target point
cloud is a single-view scan. As expected, we can observe some misalignment
near the deformed hand of the Cubebot. Please refer to the full thesis [34]
for a detailed discussion of the experiments and results.

##  4 Conclusion

In this paper, we study the problem of point cloud registration for a special
class of non-rigid deformations that are encountered in objects composed of
rigid parts that move about joints. We propose an efficient and robust
workflow called PWR-Align that leverages part-whole relationships. We evaluate
our method on real world MR data and demonstrate its robustness and viability
as a faster alternative to non-rigid ICP.

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