Endoscopic Equivalences: Imaging & Harmonic Analysis

• Endoscopic Equivalences are defined as frameworks that preserve an invariant substrate, uniting differing modalities in computational colonoscopy and endoscopic transfer.
• In computational colonoscopy, the model isolates 3D geometry from appearance using encoders, decoders, and adversarial frameworks to translate between OC and VC images.
• In harmonic analysis, equivalence is established by matching stable conjugacy classes through explicit κ‐orbital integrals, transfer factors, and Fourier-transform compatibility.

Searching arXiv for the cited papers to ground the article in the literature. arxiv_search query: (Mathew et al., 2021) Endoscopic equivalences designate two technically distinct constructions in recent arXiv literature. In computational colonoscopy, the term refers to a learned correspondence between optical colonoscopy (OC) and virtual colonoscopy (VC) in which both modalities are treated as different renderings of the same underlying 3D colon geometry, with color, texture, and specular reflections handled separately as appearance variables (Mathew et al., 2021). In representation theory and relative harmonic analysis, the closely related notion is endoscopic transfer: regular semisimple classes on unitary groups or spherical varieties are matched through transfer factors, stable conjugacy, and $\kappa$-orbital integrals, yielding precise correspondences between orbital integrals on a group and its endoscopic data (Lee, 2021, Xiao, 2018). This suggests a common structural pattern—preservation of an invariant substrate across changes of presentation—even though the two usages are mathematically and methodologically unrelated.

1. Terminological scope and disciplinary separation

In the colonoscopy literature, the relevant invariant is geometric. The core assumption is that OC and VC share the same underlying 3D colon geometry, while differing in color, texture, and specular reflections. Endoscopic equivalence in this sense is therefore a geometry-preserving correspondence between image domains.

In the automorphic and representation-theoretic literature, the invariant is not geometric appearance but stable conjugacy data. The operative objects are regular semisimple elements, their centralizers, endoscopic characters, transfer factors, and orbital integrals. Here equivalence is expressed through matching orbits and equality of weighted orbital integrals after transfer.

A common misconception is to read the phrase as denoting a single unified theory. In the cited literature it does not. The medical-imaging usage belongs to cross-domain representation learning for endoscopic video, whereas the mathematical usage belongs to Langlands-style endoscopy, relative trace formulae, and harmonic analysis on unitary groups and spherical varieties.

2. Shared-latent endoscopic equivalence in colonoscopy

The OC/VC framework is formulated on four spaces: the OC image domain $X$, the VC image domain $Y$, a shared latent space $Z$, and a noise space $N$. Samples are written $x\sim p_X(x)$, $y\sim p_Y(y)$, and $z\sim p_N(z)$. The latent space $Z$ is intended to capture pure geometry, while $N$ encodes appearance variations such as lighting, texture, and reflections.

The architecture consists of two encoders,

$X$0

two decoders,

$X$1

and two discriminators. $X$2 judges real versus fake OC images. $X$3 judges the correct direction of a translation pair. The design is asymmetric: $X$4 re-renders geometry in OC style using a noise vector, while $X$5 re-renders geometry in VC style without extra noise.

The paper characterizes this as a lossy unpaired image-to-image translation model with enforced shared latent space. The latent code $X$6 or $X$7 contains only geometry. Appearance is injected solely through the additional Gaussian noise input $X$8. This permits one-to-many mappings from VC to OC and from OC to OC. In the paper’s summary formulation, any OC or VC image is mapped to the same $X$9 if its colon geometry matches, and $Y$0 or $Y$1 then re-renders that geometry in either OC or VC style (Mathew et al., 2021).

3. Objective functions, identifiability constraints, and empirical behavior

The training criterion is the sum of adversarial, cycle-consistency, shared-latent-space, identity, and noise-utilization terms. The standard GAN loss for the OC domain is

$Y$2

The directional discriminator contributes

$Y$3

and the paper uses

$Y$4

Cycle-consistency is split into a one-to-one VC-domain cycle and a one-to-many extended OC-domain cycle: $Y$5

$Y$6

These are combined as

$Y$7

Geometry preservation is enforced by shared-latent-space losses in both directions: $Y$8

$Y$9

with

$Z$0

An identity loss keeps shading consistent in the VC$Z$1VC pass,

$Z$2

and a noise-utilization loss prevents $Z$3 from ignoring the appearance variable: $Z$4 The total objective is

$Z$5

The experimental setup uses OC/VC image pairs from 10 patients without pixel-wise ground-truth OC$Z$6VC alignment: 2000 images from 5 patients for training, 800 images from 2 patients for validation, and 1200 images from 3 patients for testing. Using textured VC as proxy ground truth, the reported per-pixel accuracy $Z$7 reached $Z$8, and the Dice coefficient was $Z$9. Qualitatively, the model highlights missing surfaces in green on real OC videos, matching “holes” found by Ma et al. in their reconstructed mesh. Neighboring frames yield consistent missing-surface masks without temporal smoothing, and the one-to-many appearance model produces multiple OC variants for the same geometry. The paper also states that code, data, and trained models will be released via the Computational Endoscopy Platform (Mathew et al., 2021).

4. Relative endoscopy on $N$0

In Chung-Ru Lee’s setting, $N$1 is a non-Archimedean local field of characteristic zero with ring of integers $N$2, uniformizer $N$3, and residue field of size $N$4. Let $N$5 be the unramified quadratic extension with Galois involution $N$6. The quasi-split unitary group in three variables is

$N$7

defined by

$N$8

where $N$9 is the anti-identity matrix of size $x\sim p_X(x)$0. On $x\sim p_X(x)$1 there is an involution

$x\sim p_X(x)$2

whose fixed points are $x\sim p_X(x)$3; the neutral component is written $x\sim p_X(x)$4. Instead of $x\sim p_X(x)$5, the paper works with

$x\sim p_X(x)$6

on which $x\sim p_X(x)$7 acts by conjugation. The paper states that $x\sim p_X(x)$8 is a spherical variety for $x\sim p_X(x)$9 of “type $y\sim p_Y(y)$0,” meaning that the stabilizer in $y\sim p_Y(y)$1 can be finite (Lee, 2021).

For regular semisimple $y\sim p_Y(y)$2, the $y\sim p_Y(y)$3-centralizer $y\sim p_Y(y)$4 is a torus and the $y\sim p_Y(y)$5-centralizer $y\sim p_Y(y)$6 is a finite abelian $y\sim p_Y(y)$7-group. Rogawski’s classification yields four types of tori, of which only types I–III give non-trivial $y\sim p_Y(y)$8. In those cases there is a finite group

$y\sim p_Y(y)$9

canonically isomorphic to

$z\sim p_N(z)$0

Its characters $z\sim p_N(z)$1 are the endoscopic characters.

With the basic function $z\sim p_N(z)$2, orbital integrals are normalized by

$z\sim p_N(z)$3

with $z\sim p_N(z)$4 and counting measure on $z\sim p_N(z)$5. The corresponding $z\sim p_N(z)$6-orbital integrals are

$z\sim p_N(z)$7

where $z\sim p_N(z)$8 ranges over rational representatives in the stable orbit of $z\sim p_N(z)$9. The relative fundamental lemma is formulated as the assertion that for each endoscopic character $Z$0 there is a matching function $Z$1 on a smaller symmetric space such that for every matching $Z$2,

$Z$3

The paper’s main achievement is the explicit evaluation of these $Z$4-orbital integrals for the basic function. Writing stable-orbit invariants

$Z$5

and

$Z$6

together with

$Z$7

the paper gives fully explicit combinatorial formulae for $Z$8 in each of five subcases determined by the relative sizes of the valuations $Z$9, and then sums them against $N$0. Lee states that this is the first time such a computation has appeared in the literature for spherical varieties with type $N$1-spherical roots. The paper does not yet identify the matching $N$2 and transfer factor $N$3, but it computes the right-hand side that is essential for observing transfer and thereby opens the way to stabilization of the relative trace formula, in line with the Sakellaridis–Venkatesh program (Lee, 2021).

5. Endoscopic transfer for unitary Lie algebras

Xiao studies a non-Archimedean local field $N$4 of characteristic zero, a quadratic extension $N$5, an $N$6-dimensional Hermitian space $N$7 over $N$8, the unitary group

$N$9

and its Lie algebra

$X$00

For a decomposition $X$01, the elliptic endoscopic groups are

$X$02

with dual groups

$X$03

and embedding $X$04 given by block-diagonal inclusion. The semisimple element

$X$05

with $X$06 entries $X$07 and $X$08 entries $X$09 defines the elliptic endoscopic datum. Matching of regular semisimple elements $X$10 and $X$11 is defined by equality of characteristic polynomials, equivalently by arising from the same maximal torus via embeddings conjugate under $X$12 (Xiao, 2018).

Transfer factors are normalized using a nontrivial additive character $X$13 and Haar measures. For matching regular-semisimple orbits $X$14, the Langlands–Shelstad transfer factor

$X$15

is characterized by the stable-conjugacy relation

$X$16

where $X$17 and $X$18 is the character attached to $X$19 via Tate–Nakayama duality. In the unitary case,

$X$20

For “nice” representatives $X$21 matching $X$22, Xiao shows

$X$23

where

$X$24

The main existence theorem states that for every $X$25, there exists $X$26 such that for every matching pair $X$27 of regular semisimple elements,

$X$28

and $X$29 if $X$30 does not match any $X$31. Xiao also proves Fourier-transform compatibility. If $X$32 and $X$33 are defined using the invariant forms $X$34 and the additive character $X$35, then there is an explicit nonzero constant

$X$36

such that whenever $X$37 and $X$38 match,

$X$39

By the work of Kazhdan–Varshavsky, existence of transfer together with Fourier compatibility implies the endoscopic fundamental lemma; in the unitary case this recovers the theorem of Laumon–Ngô for unramified $X$40. A distinctive feature of Xiao’s proof is that it is purely local and proceeds through the Jacquet–Rallis transfer, a nilpotent-orbit identity, and parabolic descent. The Jacquet–Rallis bridge compares orbital integrals on

$X$41

with those on

$X$42

using an explicit transfer factor $X$43. Nilpotent germ expansions and a finite Fourier inversion over $X$44 then recover the endoscopic $X$45-orbital integrals, while parabolic descent preserves orbital integrals and commutes with Fourier transform on the $X$46-factor (Xiao, 2018).

6. Comparative perspective

The two domains organize “equivalence” around different invariants and different modes of verification.

Setting	Invariant object	Mechanism
OC/VC colonoscopy	Shared latent space $X$47 for geometry	Encoders, decoders, adversarial/cycle/shared-latent/noise losses
Relative endoscopy on $X$48	Stable-orbit invariants, $X$49, $X$50-data	$X$51-orbital integrals of $X$52, transfer factors, matching elements
Unitary Lie algebras	Matching regular semisimple characteristic polynomials and $X$53	Stable orbital integrals, transfer, Fourier compatibility, Jacquet–Rallis descent

In the colonoscopy model, equivalence is constructive and generative. Geometry is encoded once and then re-rendered under multiple plausible OC appearances through the noise variable. Verification is empirical: per-pixel accuracy, Dice coefficient, visual agreement with reconstructed holes, and frame-to-frame stability.

In the two endoscopic-transfer settings, equivalence is spectral-orbital rather than generative. What is preserved is not a visual substrate but stable-orbit data, and verification takes the form of exact local identities among orbital integrals. The relevant outputs are explicit formulae for $X$54-orbital integrals, existence theorems for transferred test functions, and compatibility with Fourier transform.

This suggests that the word “equivalence” is overloaded across the literature. In one case it means that OC and VC images with matching colon geometry are identified in a shared latent representation and can be re-expressed in either modality. In the other, it means that orbital data on one group or symmetric space can be matched to endoscopic data on another through transfer factors and weighted orbital integrals. The shared theme is invariance under change of presentation; the underlying mathematics, algorithms, and applications are otherwise distinct.