--- title: 'Endoscopic Equivalences: Imaging & Harmonic Analysis' url: https://www.emergentmind.com/topics/endoscopic-equivalences type: topic --- # Endoscopic Equivalences: Imaging & Harmonic Analysis Searching arXiv for the cited papers to ground the article in the literature. arxiv_search query: 2101.07280 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 [2101.07280]. 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 [2110.11890], [1802.07624]. 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, \[ E_X:X\to Z,\qquad E_Y:Y\to Z, \] two decoders, \[ G_X:Z\times N\to X,\qquad G_Y:Z\to Y, \] and two discriminators. \(D_X\) judges real versus fake OC images. \(D_{\mathrm{dir}}\) judges the correct direction of a translation pair. The design is asymmetric: \(G_X\) re-renders geometry in OC style using a noise vector, while \(G_Y\) 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 \(l=E_Y(y)\) or \(E_X(x)\) contains only geometry. Appearance is injected solely through the additional Gaussian noise input \(z\sim N(0,I)\). 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 \(Z\) if its colon geometry matches, and \(G_X\) or \(G_Y\) then re-renders that geometry in either OC or VC style [2101.07280]. ## 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 \[ L_{\mathrm{adv}_X}(G_X,D_X) = \mathbb E_{x\sim p_X}[\log D_X(x)] + \mathbb E_{y\sim p_Y,\,z\sim p_N}\left[\log\left(1-D_X(G_X(E_Y(y),z))\right)\right]. \] The directional discriminator contributes \[ L_{\mathrm{dir}}(G_X,G_Y,D_{\mathrm{dir}}) = \mathbb E_{x\sim p_X}\!\left[\log D_{\mathrm{dir}}(x,G_Y(E_X(x)))\right] + \mathbb E_{y\sim p_Y}\!\left[\log\left(1-D_{\mathrm{dir}}(G_X(E_Y(y),z),y)\right)\right], \] and the paper uses \[ L_{\mathrm{adv}}=L_{\mathrm{dir}}+L_{\mathrm{adv}_X}. \] Cycle-consistency is split into a one-to-one VC-domain cycle and a one-to-many extended OC-domain cycle: \[ L_{\mathrm{cyc}_Y} = \mathbb E_{y\sim p_Y}\left\|y-G_Y(E_X(G_X(E_Y(y),z)))\right\|_1, \] \[ L_{\mathrm{excyc}_X} = \mathbb E_{x\sim p_X}\left\|G_Y(E_X(x))- G_Y(E_X(G_X(E_Y(G_Y(E_X(x))),z)))\right\|_1. \] These are combined as \[ L_{\mathrm{cyc}}=\lambda_c[L_{\mathrm{cyc}_Y}+L_{\mathrm{excyc}_X}],\qquad \lambda_c=10. \] Geometry preservation is enforced by shared-latent-space losses in both directions: \[ L_{\mathrm{latent}_{Y\to X}} = \mathbb E_{y\sim p_Y}\left\|E_Y(y)-E_X(G_X(E_Y(y),z))\right\|_1, \] \[ L_{\mathrm{latent}_{X\to Y}} = \mathbb E_{x\sim p_X}\left\|E_X(x)-E_Y(G_Y(E_X(x)))\right\|_1, \] with \[ L_{\mathrm{latent}}=\lambda_{\mathrm{SLS}}[L_{\mathrm{latent}_{Y\to X}}+L_{\mathrm{latent}_{X\to Y}}],\qquad \lambda_{\mathrm{SLS}}=1. \] An identity loss keeps shading consistent in the VC\(\to\)VC pass, \[ L_{\mathrm{iden}}=\mathbb E_{y\sim p_Y}\left\|y-G_Y(E_Y(y))\right\|_1,\qquad \lambda_{\mathrm{iden}}=1, \] and a noise-utilization loss prevents \(G_X\) from ignoring the appearance variable: \[ L_{\mathrm{noise}} =\mathbb E_{l\sim p_Z,\,z_1,z_2\sim p_N} \left[\max\left(0,\|G_X(l,z_1)-G_X(l,z_2)\|_1-\alpha\right)\right], \qquad \alpha=0.1. \] The total objective is \[ L_{\mathrm{total}} = L_{\mathrm{adv}} + L_{\mathrm{cyc}} + L_{\mathrm{latent}} + \lambda_{\mathrm{iden}}L_{\mathrm{iden}} + L_{\mathrm{noise}}. \] The experimental setup uses OC/VC image pairs from 10 patients without pixel-wise ground-truth OC\(\to\)VC 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 \((TP+TN)/d\) reached \(81\%\), and the Dice coefficient was \(0.667\). 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 [2101.07280]. ## 4. Relative endoscopy on \(S=U_3/O_3\) In Chung-Ru Lee’s setting, \(F\) is a non-Archimedean local field of characteristic zero with ring of integers \(\mathcal O_F\), uniformizer \(\varpi\), and residue field of size \(q\). Let \(E/F\) be the unramified quadratic extension with Galois involution \(\theta\). The quasi-split unitary group in three variables is \[ G=U_3\subset \operatorname{Res}_{E/F}GL_3, \] defined by \[ G(R)=\{g\in \operatorname{Res}_{E/F}GL_3(R)\mid J_3\,\overline g^{-t}\,J_3^{-1}=g\}, \] where \(J_3\) is the anti-identity matrix of size \(3\). On \(G\) there is an involution \[ \sigma(g)=J_3\,g^{-t}\,J_3^{-1}, \] whose fixed points are \(O_3\); the neutral component is written \(G^1=SO_3\). Instead of \(G^1\backslash G\), the paper works with \[ S=\{\gamma\in G\mid \gamma=\sigma(\gamma)^{-1}\}, \] on which \(G^1\) acts by conjugation. The paper states that \(S\) is a spherical variety for \(G^1\) of “type \(N\),” meaning that the stabilizer in \(G^1\) can be finite [2110.11890]. For regular semisimple \(\gamma\in S(F)\), the \(G\)-centralizer \(G_\gamma\) is a torus and the \(G^1\)-centralizer \(G^1_\gamma=G_\gamma\cap G^1\) is a finite abelian \(2\)-group. Rogawski’s classification yields four types of tori, of which only types I–III give non-trivial \(G^1_\gamma\). In those cases there is a finite group \[ D_\gamma \cong \begin{cases} F^\times/N_{E/F}(E^\times), & \text{type I},\\[4pt] F^\times/(F^\times)^2, & \text{type II or III}, \end{cases} \] canonically isomorphic to \[ \mathfrak D(F,G^1_\gamma,G^1)=\ker\!\left[H^1(F,G^1_\gamma)\to H^1(F,G^1)\right]. \] Its characters \(\kappa\) are the endoscopic characters. With the basic function \(f=1_{S(\mathcal O_F)}\), orbital integrals are normalized by \[ O_\gamma(f)=\int_{G^1_\gamma(F)\backslash G^1(F)} f(g^{-1}\gamma g)\,dg, \] with \(\operatorname{vol}(SO_3(\mathcal O_F))=1\) and counting measure on \(G^1_\gamma(F)\). The corresponding \(\kappa\)-orbital integrals are \[ SO_\gamma^\kappa(f)=\sum_{\mu\in D_\gamma}\kappa(\mu)\,O_{\gamma_\mu}(f), \] where \(\gamma_\mu\) ranges over rational representatives in the stable orbit of \(\gamma\). The relative fundamental lemma is formulated as the assertion that for each endoscopic character \(\kappa\) there is a matching function \(f_H\) on a smaller symmetric space such that for every matching \(\gamma_H\), \[ \Delta(\gamma_H,\gamma)\,O_{\gamma_H}(f_H)=SO_\gamma^\kappa(f). \] The paper’s main achievement is the explicit evaluation of these \(\kappa\)-orbital integrals for the basic function. Writing stable-orbit invariants \[ \lambda_1=x+\nu^{1/2}y,\qquad \lambda_2=z,\qquad \lambda_3=x-\nu^{1/2}y, \] and \[ M_{ij}=v(\lambda_i-\lambda_j),\qquad N_{ij}=v(\lambda_i+\lambda_j), \] together with \[ \overline M=\max_{i