Optical Coherence Tomography Harmonization with Anatomy-Guided Latent Metric Schrödinger Bridges
Authors: Shuwen Wei
Samuel W. Remedios
Blake E. Dewey
Zhangxing Bian
Shimeng Wang
Junyu Chen
Bruno Michel Jedynak
shiv saidha
Peter A. Calabresi
Aaron Carass
Jerry L Prince
Venue: NeurIPS 2025 poster ()
Keywords: Schrödinger Bridge
Harmonization
Optical Coherence Tomography
Anatomy
Metric
Primary area: deep_learning
TL;DR: We propose a latent metric Schrödinger bridge (LMSB) framework to improve the anatomical consistency of OCT image harmonization.

Abstract: Medical image harmonization aims to reduce the differences in appearance caused by scanner hardware variations to allow for consistent and reliable comparisons across devices. Harmonization based on paired images from different devices has limited applicability in real-world clinical settings. On the other hand, unpaired harmonization typically does not guarantee anatomy consistency, which is problematic because anatomical information preservation is paramount. The Schrödinger bridge framework has achieved state-of-the-art style transfer performance with natural images by matching distributions of unpaired images, but this approach can also introduce anatomy changes when applied to medical images. We show that such changes occur because the Schrödinger bridge uses the square of the Euclidean distance between images as the transport cost in an entropy-regularized optimal transport problem. Such a transport cost is not appropriate for measuring anatomical distances, as medical images with the same anatomy need not have a small Euclidean distance between them. In this paper, we propose a latent metric Schrödinger bridge (LMSB) framework to improve the anatomical consistency for the harmonization of medical images. We develop an invertible network that maps medical images into a latent Euclidean metric space where the distances among images with the same anatomy are minimized using the pullback latent metric. Within this latent space, we train a Schrödinger bridge to match distributions. We show that the proposed LMSB is superior to the direct application of a Schrödinger bridge to harmonize optical coherence tomography (OCT) images.

Optical Coherence Tomography Harmonization with A Anatomy-Guided Latent Metric Schrödinger Bridges
Shuwen Wei
Samuel W. Remedios'
Blake E. Dewey?
Zhangxing Bian'
Shimeng Wang'
Junyu Chen²
Bruno M. Jedynak
Shiv Saidha²
Peter A. Calabresi
Aaron Carass
Jerry L. Prince
Johns Hopkins University
JJohns Hopkins School of Medicine
Portland State University
Abstract
Medical image harmonization aims to reduce the differences in appearance caused by scanner hardware variations to allow for consistent and reliable comparisons across devices. Harmonization based on paired images from different devices has limited applicability in real-world clinical settings. On the other hand, unpaired harmonization typically does not guarantee anatomy consistency, which is problem- atic because anatomical information preservation paramount. The Schrödinger bridge framework has achieved state-of offte-of-the-arr style transfer performance with natural images by matching distributions of unpaired images, but this approach can also introduce anatomy changes when applied to medical images. We show that such changes occur because the Schrödinger bridge uses the square of the Euclidean distance between images the transport cost in entropy-regulariz optimal transport problem. Such a transport cost not appropriate for measuring anatomical distances, as medical images with the same anatomy need not have small Euclidean distance between them. In this paper, we propose a latent metric Schrödinger bridge (LMSB) framework to improve the anatomical consistency for the harmonization of medical images. We develop an invertible network that maps medical images into a latent Euclidean metrie space where the distances among images with the same anatomy are minimized using the pullback latent metric. Within this latent space, train Schrödinger bridge to match distributions. We show that the proposed LMSB is superior to the direct application of a Schrödinger bridge harmonize optical coherence tomography (OCT) images.
Introduction
Optical coherence tomography (OCT) (Huang al., 1991) is a medical image modality that uses low- coherence interferometry to visualize micro structures in biological tissue. Its current primary clinical uses are for retina and cornea health monitoring and disease diagnosis (Petzold al., 2010; Saidha et al., 2015; Rothman et al., 2019), but different hardware configurations and scanning settings cause image variations in the forms of contrast, speckle, and noise (Wei al., 2023). For example, Zeiss Cirrus OCT quickly scans the retina in exchange for low contrast and signal-to-noise ratio (SNR). whereas Heidelberg Spectralis OCT slowly scans the retina (via multiple image acquisitions) in exchange for superior contrast and SNR (via averaging of the multiple acquisitions). This type of domain shift caused by different scanners or acquisition parameters creates challenge in the development and deployment of robust and generalizable medical image analysis methods (Guan and Liu, 2021), affecting multi-center center studies and longitudinal analyses where imaging data are pooled
"Corresponding author: swei 14@ jhu.edu
code: https: com /om//huuen--e Wei/lmsb
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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from different sources. To address domain shift, medical image harmonization, which seeks to map images from one domain another (i.e., from Cirrus to Spectralis) while preserving the underlying anatomy, has emerged as an important preprocessing strategy (Mirzaalian et al., 2016; Zhu et al., 2017; Liu et al., 2017: Huang et al., 2018; Wrobel al., 2020; Park et al., 2020; Zuo et al., 2021a,b; Tian et al., 2022; Cackowski et al., 2023; Zuo et al., 2023, 2024).
Harmonization as a Coupling Problem Harmonization between two datasets can be formulated as a coupling problem. Let (Ru F., P.) with i E {0, } be two probability spaces, where the S2 i are two sets, the F; _ 20' s are two or algebrass on their respective sets, and the P/'s, map the Fi's to [0, 1], are probability measures. A coupling between these two probability spaces is the new probability space (o.1 Po.1). where 2o,1 = So x Q1 and Po.1 : Fo,1 + [0. 1] such that projo Po.1 Po and Po. = Pt. Here, projo Ro + Un and proj, Ra + S2 are defined as the projection maps, and # indicates the pushforward of the probability measure. naive coupling is the independent coupling Po.1 Po @ P1. which means that given an image from one probability space, we arbitrarily sample an image from the other probability space as the harmonization result. This is obviously not good coupling for preserving anatomy, which is undesirable because preserving anatomical information is critical for consistent analyses. To find a good coupling, additional constraints or assumptions must be applied. Generative adversarial networks (GANs) (Goodfellow et al., 2020) find a deterministic coupling by adversarial training without further constraints, and thus do not preserve anatomy. Both the optimal transport (OT) (Monge, 1781; Kantorovich, 1942; Villani, 2008) and Schrödinger bridge (SB) (Schrödinger, 1931, 1932; Léonard, 2014) approaches find an optimal coupling assuming a transport cost and reference path measure, respectively. However, some commonly assumed transport cost and reference path measure for OT's and SBs are not correct for medical image harmonization because images with the same anatomy but different contrast do not necessarily have small Euclidean distance.
Main Contributions In this paper, we propose new method for OCT image harmonization that explicitly reduces anatomical shifts. This method, named anatomy-guided latent metric Schrödinger bridge (LMSB), follows from two key contributions. First, we provide full analysis of the anatomy shift issue of applying SBs directly to harmonization and show several theoretical solutions to address this issue. Second, implement maps OCT images to a latent Euclidean metric space via anatomy guidance. We call the pullback metric the latent metric which minimizes the distances among OCT images with the same anatomy. We then train a SB to map between distributions in this learned latent Fuclidean metric space. The whole training process does not require any paired training data. We demonstrate LMSB on the harmonization of OCT images and show that it achieves better performance in anatomy preservation than conventional SBs and other baseline methods.
theory by training an invertible neural network (INN) that
2
Related Work
Medical Image Harmonization Traditional medical image harmonization methods can be divided into supervised and unsupervised methods. Although supervised harmonization methods yield good results (Tian et al., 2022; Zuo et al., 2023), the paired images they require for training are not widely available. Unsupervised harmonization methods such as CycleGAN (Zhu et al., 2017), UNIT (Liu et al., 2017), MUNIT (Huang et al., 2018), and CUT (Park et al., 2020) do not require paired data, but they generally suffer from anatomical changes during the harmonization process.
Optimal Transport OT (Monge, 1781; Kantorovich, 1942; Villani, 2008) is method to find an optimal coupling that minimizes an overall transport cost. For discrete measures, OT can be solved exactly, but is computationally expensive. A landmark paper by (Cuturi, 2013) shows that an entropy-regularized version of OT can be solved more efficiently using the Sinkhorn algo- rithm (Sinkhorn, 1967). Deep learning approaches such as Wasserstein GANs (Arjovsky al., 2017) have been established for solving OT in its Kantorovich duality form (Kantorovich, 1940). However, they are difficult to train and some constraints are difficult to implement. For example, when the transport cost is the Euclidean distance, a bounded Lipshitz continuity condition needs to be satisfied in the Kantorovich duality form (Kantorovich, 1940). But this condition can only be approximated with either spectral normalization (Miyato et al., 2018), weight clipping (Arjovsky et al., 2017), or a penalty on the gradient (Gulrajani et al., 2017). This also makes it hard to generalize to different transport costs using deep learning framework.
2

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Diffusion Schrödinger Bridge SB can be solved by iterative projection fitting (IPF) (Kullback, 1968; Rüschendorf and Thomsen, 1993; Rüschendorf, 1995). which projects to the path measure that satisfies the initial distribution and the final distribution iteratively. However, IPF is com- putationally expensive because of the high-dimensional nature of the path measure. Diffusion SB (DSB) (De Bortoli et al., 2021) uses novel implementation of IPF which takes advantage of diffusion models (Sohl-Dickstein et al., 2015; Ho et al., 2020; Song et al., 2021b) by breaking down the path measure and optimizing over smaller pieces of a Markov chain through time reversal of the stochastic process (Anderson, 1982). A dual form of IPF called iterative Markovian fitting (IMF) was applied to DSB (Shi et al., 2023; Peluchetti, 2023) and has demonstrated improved performance. Compared with IPF, IMF projects the path measure into Markov measure and reciprocal class iter- atively, while preserving the initial and final distribution. Due to its state-of-thee the-art performance, DSB (De Bortoli et al., 2021; Shi et al., 2023) has found many applications in natural images, such as image restoration by image-to-image SB (I²SB) (Liu et al., 2023), and style transfer by dual diffusion implicit bridge (DDIB) (Su et al., 2023). However, DSB still introduces anatomical shifts when directly applied to medical images.
3 Latent Metric Schrödinger Bridge
We introduce the dynamic SB used in DSB, and show that it is equivalent to the solution of a static SB and the solution of an cntropy-regularin this engropy-regularrvved OT is not correct for medical image harmonization to preserve anatomy. We then propose the LMSB framework for harmonization, which improves anatomical consistency.
OT. We make the observation that the assumption behind
Dynamic Schrödinger Bridge We assume that 22g = 2 = Rd in the coupling problem setup in Sec. 1. Let Ω C([O, I], Rd) be all continuous -d-luleed paths on the unit time interval [0, 1]. We construct a probability space (2, P) by adding the o-algebra- F C 2 and the path measure F [0, 1J. P(2)) is the set of all path measures, II(Po, P1) C P(2) the subset of path measures with their marginal densities at t = 0 and = 1 being Po and P1. respectively, and M(2) C P(22) the subset of path measures that are Markovian. The dynamic SB problem (Schrödinger, 1931, 1932; Léonard, 2014) finds the optimal path measure psB E (I P1) with respect to a reference path measure Q MMS2) by minimizing their Kullback-Letie Leibler (KL) divergence.
SSB
arg min (P || Q), PEI((F.PP))
(1)
where Q E ((2) is a Markov path measure of random process X, described by the forward stochastic differential equation (SDE), dX,- = fx(X))dt + g,dW,, where Xo ~ Po without loss of
generality, f: [0, 1] Rd + Rd and g [0, 1] + R are the drift and diffusion coefficients, and W, is
the standard Wiener process.
Static Schrödinger Bridge A connection between the dynamic and static SB problems was established by (Föllmer, 1988). The static SB problem finds the optimal joint law PS, at times f = 0 and t = by,
PSB arg min DKL. || Qo.1); P...
(2)
where T(Po, P3) is the set of joint laws at times f. = and 1 with their marginals being Po and Pt, respectively, and Q0, is the joint law of Q at times = 0 and = 1. It can be shown that the solution p&B B to Eq. can be decomposed into a mixture of bridges as psR_ = -P Q where PSS is the solution to Eq. 2; conversely if PSR is the solution to Eq. 2, the mixture of bridges psR_ = P 1o.1 is the solution to Eq. 1. Here, the mixture of bridges pSB_ = Q,, is short notation for PSB (.) fg=xE= @ (-)r x1) PS (dro, dr1), where Q is the diffusion bridge of Q conditioned on the initial condition æa at 0 and the final condition 21 t 1. Therefore, the solutions to Eq. 1 and Eq. 2 are equivalent.
Matropy-Regulaiizee Optimal Transport Reference path measure Q is associated with random process X, without drift, i.e., d.X, = g,dW, where ft = O. then Eq. 2 can be derived as,
SSR = arg min Po (P Rd
||1 2 Po. (dzo, dx1) - 20² 2 (Po..))
(3)
3

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(a) Issue
(b) Solution
(c) Solution #2
(d) Solution #3
TpU U
Ty V
T
0
Contrast
Contrast
df
Contrast
Contrast
df
2
Target contrast
cantraa
Target contrest
Target contrast
Anatomy
Anatomy
f(p)
f(p) Reduced
Anatomy 0 of
f(r) Reduced
Anatomy 0 DE
Anatomy shift
D5,
No anatomy shift df
anatormy shift
anatomy shift
Figure 1: Local view of the anatomy and contrast geometry. The smooth map f U - V induces a local linear map of between the tangent space T and the tangent space T(()) V. The tangent vectors # and df( ( point to the direction where the anatomy changes. The tangent vectors a Del
DE 8 point to the direction where the contrast changes. The blue dashed lines show the same target contrast, and they are parallel to the anatomy change direction. The green arrows connect the black points to the blue points with the shortest distance. The red dash lines are parallel to the contrast change direction, and the red arrows show the anatomy shift amount. (a) Anatomy shift is introduced if we use the Euclidean metric in the image intensity coordinate system. (b) Solution #1: find f that flattens the metric, which eliminates the anatomy shift. However, this is not always possible for an open neighborhood of p for an arbitrary metric. (c) Solution #2: find a f that shrinks the distance along the contrast direction, which reduces the anatomy shift. (d) Solution #3: find a that stretches the distance along the anatomy direction, which also effectively reduces anatomy shift.
and df aaga
where o² =²= dt, and H is the entropy of the joint law Po.1- We observe that the first term in Eq. is the same objective as the OT formulation. Therefore, SB with a non-drift Brownian motion as the reference path measure, is equivalent an regropy-regularized OT problem, with the transport cost being the square of Euclidean distances. However, using this transport cost is not suitable for medical image harmonization where preserving anatomy is essential.
Issues of Euclidean Metric and Potential Solutions We illustrate the issue of using a Euclidean metric in the image intensity coordinate system between two images in IR2 as an example. Suppose Vp E IR there is an open neighborhood U C R2, such that p U and there is coordinate system E: U + R² where { and &2 are the anatomy coordinate and contrast coordinate, respectively. At the same time, there is another coordinate system on U which uses the image intensity value from each pixel (in this case two pixels) as a coordinate, which we denote as 5' : + R22. It is unlikely
that & and g' are the same. Now if we equip U with the metric tensor g' = y @ ded i.e., an Euclidean metric on the coordinate system e", then the tangent vectors á उद, and å are orthogonal, उe
but and ag, 0g might not be. This is illustrated in Fig. I(a), where ãc, ãe_ are not orthogonal under this metric. Because &1 and & are the anatomy coordinate and contrast coordinate, Dé, points
and
8 to the direction where anatomy changes, and BEg points to the direction where contrast changes. If we try to find the closest point in the target contrast space (blue dashed line), can do orthogonal projection as shown by the green arrow, and the blue point is the solution with the shortest distance. However, it introduces an anatomy shift as indicated by the red dashed line and the red arrow.
To eliminate the anatomy shift, we need & and 8 to be orthogonal. It is sufficient to achieve de.
ø5 this by choosing g ōtjde @ de' as the metric tensor, and we can find a smooth map f whose pushforward metric of g is Euclidean metric on the new coordinate system, as shown in Fig. 1(b). This ideal because the anatomy shift is totally eliminated. However, such map may not exist to flatten the metric on an open neighborhood of p. unless the Riemann curvature tensor of this metric space vanishes (Lee, 2019), which depends on how the metric tensor g is defined in the open neighborhood of p. Alternatively. there are ways to reduce the anatomy shift. One way is to find map f that maps images into a latent Euclidean metric space, whose pullback metric shrinks the distances along the contrast direction. As shown in Fig. ((C), the target contrast moves closer to the black point, which reduces the anatomy shift. Another way is to find map that maps images into a latent Fuclidean metric space, whose pullback metric stretches the distances along the anatomy direction. As shown

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DSBM
LMSB
Anatomy
Anatomy
Anatomy #1
Anatomy #2
Contrast #2
Contrast #2
Latent Contrast
DSSM
DSBM
Contrast #1
Contrast #1
(
Figure 2: Global view of the anatomy and contrast geometry. (a) Diffusion Schrödinger Bridge Matching (DSBM): Solid lines indicate the same anatomy and dashed lines indicate the same contrast. Directly applying DSBM between Contrast #1 and #2 results in a shift in anatomy, as shown by the green arrows. (b) Latent Metric Schrödinger Bridge (LMSB): A Latent Contrast space is constructed. An invertible f is trained to map both contrasts to the Latent Contrast, as shown by the orange arrows. Applying DSBM to anatomies within the Latent Contrast reduces anatomical shift. The right panel of OCT images show examples of the same anatomy in Contrast #1 (bottom) and #2 (top), as well as the anatomy within the Latent Contrast.
in Fig. (dd) in this case, although the target contrast space does not get closer, the anatomy shift is still effectively reduced, by comparing the red arrow with the tangent vector df (
In practice, medical images live in a higher dimension space. It is almost impossible to decouple the coordinates into an anatomy component and a contrast component if we use the intensity values in each pixel as the coordinate system for medical images. Therefore, we directly apply SBs for medical image harmonization, or equivalently entropy-regularized OT's with the square of Euclidean distances as the transport cost, it is very likely that they shift the anatomy. Therefore, is not a good idea to use the square of Euclidean distances as the transport cost in the OT formulation. A potential solution is to use an alternative transport cost, such as local normalized cross correlation (LNCC), which is often used in medical image registration to evaluate the alignment of anatomical structures (Chen et al., 2025). The LNCC compresses the distances along the contrast direction, as shown by df Bay in Fig. Icc), and thus effectively reduces anatomical shift even if it exists. However, an OT problem with an arbitrary transport cost is difficult solve using deep learning frameworks, and thus is not practically useful. Another alternative method is to find a coordinate transformation of a metric space whose pushforward metric is the Euclidean metric, and then do the OT in the new coordinate system. However, this is generally not possible because an arbitrary metric space is generally not equivalent to the Euclidean metric globally or even locally in an open neighborhood. unless the Riemann curvature tensor of this metric space vanishes (Lee, 2019). However, it still inspires our proposed method LMSB, which tries find a coordinate transformation to compress the distance along the contrast direction, which we explain in the following section.
0
Latent Metric Schrödinger Bridge The core idea of the LMSB is to find an invertible map f that maps the images into a latent Euclidean metric space, where the distances among the images with the same anatomy are minimized using the pullback latent metric. Then, we use the DSB matching (DSBM) to match the distribution in the new coordinate system to minimize the anatomical shift. The difference between DSBM and LMSB is shown in Fig. 2. The solid lines indicate the same anatomy. The dashed lines indicate the same contrast.
Invertible Neural Network We build deep learning based invertible network f using the idea of affine coupling that is implemented in normalizing flow (Dinh al., 2015; Rezende and Mohamed, 2015; Dinh et al., 2017; Kingma and Dhariwal, 2018; Stimper et al., 2023). The structure of the invertible network is shown in Fig. 3. The network first splits the input images w into two parts w [We, Wa] through a checkerboard decomposition. Then the affine coupling layers map them to [ze, žn| = [w, + f2(w, + fi (w.)), W. + fi (w.)).
the end, [zc, za] are merged to get the final output [ze, =e). The affine coupling layers Si and f2 use two U-Nets (Ronneberger et al., 2015) with the same structure but different weights. Both the split and merge operations are invertible, For
the affine coupling layers, it can be inverted by [We, wa] = [ze f2(2z), za fi(zo f=(z)))]-
5

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We
Ze
split
affine coupling
merge
f
f2
We
Zg
Figure 3: Structure of the invertible neural network f. The input split into parts [w,, wal through a checkerboard decomposition, and goes through two affine coupling layers fi and f2 to generate [Ze: zzl. and merges together produce the output
(e)
Figure 4: (a) Cirrus OCT B-scan. (b) Spectralis OCT B-scan. (c) Segmentation label map of (a). (d) Segmentation label map of (b). (a, e) and (b, d) are used as constructed paired images (w, 2) for training Eq. After training, the invertible map takes (a) and (b) (e) and (f), respectively.
Training Objective {(u,, H- with the same anatomy but different contrasts, that we can minimize the following training objective,
To train the invertible map f. we first assume that we have paired images
arg min E... [lI((w) [[v(()-((v)]]) f(w)))
where (u, u)'s randomly sampled from the paired images {(4,, u,)) )- However, in practice, we only have unpaired images {u: ) and {or) and we do not want less general method that requires paired images. To solve the issue, we note the training objective in Eq. 4 is bounded by,
E [I((w))((v))
- [(w)]]
E... [II(u)) =1I] E.. [1((v)) -2l
(5)
where we add latent variable of the same dimension to each image pair (u, v). we use the triangle inequality [u] f(v)ll <. [u] 2|| (v) z|I. and marginalize the unrelated random variables. Therefore, we find a common z for images with the same anatomy but different contrasts, we can train the invertible map f by an alternative training objective,
arg min Eu, [l((w) =1|
(6)
where (we 2s's 'ss are randomly sampled from a constructed paired dataset ((w; =))) in which w's are from {4) (vr H2 and 2 constructed for each w. There are many choices for a when building the paired dataset {(w, 2,) tm2 A sufficient condition for Eq. the same anatomy as w within a latent contrast, so that images with the same anatomy but different contrasts all map close to the same z. We call the space of the Latent Contrast space, shown in Fig. 2(b). A straightforward design is to use segmentation label map of w as where we assign an unique intensity to different pixel labels. Examples of paired w's and z's are shown in Fig. Intuitively, the invertible map f brings both Cirrus and Spectralis B-scans closer to common latent contrast while preserving the anatomy using segmentation label maps so that the anatomy shift, i.e., segmentation change, is reduced during harmonization.
hold that has

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4 Experiments
Dataset The OCT dataset consists of 388 Cirrus volumes and 338 Spectralis volumes. The 388 Cirrus volumes come from 194 subjects (388 eyes in total). The 338 Spectralis volumes come from 165 subjects (269 eyes in total). Note that there are repeated scans on the same eye for Spectralis. Training and testing splits are done by subject: 352 Cirrus volumes from 176 subjects (352 eyes) and 307 Spectralis volumes from 156 subjects (252 eyes) for training and 36 Cirrus volumes from 18 subjects (36 eyes) and 31 Spectralis volumes from 9 subjects (17 eyes) for testing. Our experiments are in 2D and operate on B-scans independently. Because each Cirrus volume contains 128 B-scans and each Spectralis OCT volume contains 49 B-scans over the same field of view 6 x 6 mm? the Cirrus volumes have denser B-scan sampling. Therefore, to reduce anatomical redundancy across individual Cirrus volume B-scans, we extract every third B-scan. Specifically, in the training dataset, we extract 15, 000 B-scans in the training dataset and 1, 500 B-scans in the testing dataset for both Cirrus and Spectralis, resulting in 30, 000 training B-scans and 3, 000 testing B-scans with no subject data leakage between train and test splits.
The original size of Cirrus B-scans is 1024 x 512 (axial x lateral) and the original size of Spectralis B-scans is 496 1024 (axial x lateral). The axial resolution of Cirrus B-scans is 2 pm um/pizel, and the axial resolution of Spectralis B-scans is 4 amppiell. Both B-scans cover 6 mm lateral scanning range. We crop the Cirrus B-scans axially to 512 512 and resize them 128 x 512, and we crop the Spectralis B-scans axially to 256 x 1024 and resize them to 128 x 512, such that they have the same image size, digital resolution, and field of view.
Network Training We trained the invertible neural network / with 15, 000 Cirrus and 15, 000 Spectralis OCT B-scans in the training set. We constructed a paired dataset ((w,, 2;);;- 30,000 by generating segmentation label map 2 for each B-scan w using deep learning based retinal OCT segmentation algorithm (He et al., 2019, 2021, 2023). Examples of these paired images are shown in Fig. 4, where Cirrus OCT B-scan and its segmentation label map is shown in Fig. 4(a) and Fig. 4(c), respectively, and Spectralis OCT B-sean and its segmentation label map is shown in Fig. 4(b) and Fig. 4(d). respectively. We then trained f using this constructed paired dataset by optimizing the training objective in Eq. 6. There are two training strategies we used to make the invertible network more robust. First, we augmented the paired dataset by adding Gaussian noise to w, with the paired being unchanged. Second, we constrained the space of the invertible map to volume preserving maps, i... Jacobian determinant of f is everywhere. We note that this constraint was imposed by our network structure in Fig. 3. Training results are shown in Fig. 4(e) and Fig. 4(f) for the Cirrus and Spectralis OCT B-scans, respectively. Both the Cirrus and Spectralis OCT B-scans are not perfectly mapped to the segmentation label maps, and some detailed anatomical structures such as vessel and shadows are preserved. This a desirable property because perfectly mapping into the segmentation label map will result in poor invertibiliy. Without the use of the noise augmentation and volume preserving map, we will not get this property because the network will overfit on the training data.
We trained both DSB in the original image domain, which we denote as DSBM, and in the latent contrast space which we denote as LMSB. We adapted the implementation in DSBM (Shi al.. 2023) for both methods and use the exact same parameters for fair comparison. Specifically, we ran 30 steps of IMF in total to solve DSBM and LMSB. For the first IMF step, we used independent coupling as the intial coupling, and we ran 10, 000 iterations both forward and backward. For the remaining IMF steps, we ran 2, 500 iterations both forward and backward, and we cached simulation trajectories every 1, 250 iterations. We set the number of diffusion steps to 100. We let the diffusion coefficient be constant at each diffusion step and choose の² = 0.1. After training, we compared their performance with two sampling strategies: 1) stochastic differential equation (SDE); and 2) ordinary differential equation (ODE). The difference between SDE and ODE is that SDE preserves the path measure of stochastic process, but ODE only preserves the marginal density of the path measure. Furthermore, ODE is deterministic and invertible, which is desirable in many applications, but SDE is stochastic and not invertible.
We also trained a dual diffusion implicit bridge (DDIB) (Su et al., 2023) as another method for comparison. To map between two distributions using DDIB, we trained two independent de- noising diffusion probabilistic models (DDPMs) (Ho et al., 2020) on both Cirrus and Spectralis training sets with 1, 000 diffusion steps. After training, we use the denoising diffusion implicit model (DDIM) (Song et al., 2021a) sampling strategy to convert an image from one domain to a latent Gaussian noise, and then to an image in another domain. DDIM can also be categorized as an
7

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Table I: Mean MAE (Std. Dev.) comparison (N = 1, 500) across nine retinal boundaries for both Spectralis to Cirrus and Cirrus to Spectralis harmonization. Bold numbers indicate the best result in that row for that subtask. Asterisks indicate statistical significance (i.e., paired t-test comparing 1s and 2nd best results from the other two methods gave p-value 0.05). Key: ILM: internal limiting membrane; RNFL: retinal nerve fiber layer; GCL: ganglion cell layer; IPL: inner plexiform layer; INL: inner nuclear layer; OPL: outer plexiform layer; ONL: outer nuclear layer; ELM: external limiting membrane; IS: inner segment; OS: outer segment; RPE: retinal pigment epithelium complex; BM: Bruch's 's membrane; AVG: Average.
Spectralis to Cirrus
Cirrus to Spectralis
DDIB
DSBM
LMSB
DDIB
DSBM
LMSB
ODE
SDE
ODE
SDE
ODE
ODE
SDE
ODE
SDE
ODE
0.82
0.47
0.50
0.27
0.26*
1.29
0.64
0.61
0.29
0.23*
ILM
(0.79)
(0.23)
(0.36)
(0.28)
(0.32)
(2.35)
(1.01)
(0.85)
(0.48)
(0.52)
RNFL
1.19
0.73"
0.75
0.89
0.88
1,64
0.71
0.69
0.78
0.73
GCL
(0.95)
(0.29)
(0.34)
(0.46)
(0.45)
(2.41)
(1.98)
(2.33)
(0.51)
(0.54)
IPL
0.96
0.75
0.71
0.62
0.54*
1.22
0.66
0.61
0.55
0.47*
-INL
(0.74)
(0.32)
(0.33)
(0.33)
(0.34)
(1.91)
(1.91)
(2.15)
(0.59)
(0.56)
INL
0.90
0.52
0.47
0.45
0.39*
1.06
0.52
0.46
0.43
0.34
-PP.
(0.72)
(0.22)
(0.23)
(0.24)
(0.27)
(1.56)
(1.59)
(1.85)
(0.74)
(0.68)
OPL
0.96
0.66
0.64
0.52
0.48*
1.09
0.60
0.54
0.50
0.41*
-ON..
(0.72)
(0.26)
(0.26)
(0.26)
(0.29)
(1,29)
(1.29)
(1.44)
(0.89)
(0.84)
0.71
0.29
0,29
0.30
0.31
0.83
0.35
0.33
0.33
0.30
ELM
(0.74)
(0.16)
(0.19)
(0.17)
(0.22)
(0.99)
(0.81)
(0.84)
(1.12)
(1.11)
IS
0.70
0.26
0.25
0.27
0.26
0.81
0.32
0.30
0.30
0.27
-OS
(0.75)
(0.14)
(0.17)
(0.15)
(0.22)
(0.94)
(0.67)
(0.68)
(1.20)
(1.20)
OS
0.81
0.34
0.35
0.31*
0.31
0.85
0.34
0.33
0.30
0.28
-RPE
(0.74)
(0.12)
(0.15)
(0.16)
(0.27)
(0.94)
(0.65)
(0.62)
(1.25)
(1.20)
0.93
0.45
0.44
0.43
0.41*
0.93
0.40
0.38
0.42
0.39
BM
(0.75)
(0.15)
(0.18)
(0.23)
(0.33)
(0.93)
(0.61)
(0.58)
(1.17)
(1.12)
0.89
0.50
0.49
0.45
0.43*
1.08
0.51
0.47
0.43
0.38*
AVG
(0.65)
(0.13)
(0.17)
(0.18)
(0.21)
(1.37)
(1.16)
(1.24)
(0.86)
(0.84)
ODE sampling strategy. In practice, we do not go through all 1,000 diffusion steps, but stop early at 500 steps for balance between sampling quality and harmonization quality (Wei et al., 2026).
Anatomy Consistency Comparison We compared the performance of our proposed LMSB with the DSBM and the DDIB by evaluating the anatomical consistency before and after harmonization. To do so, we ran all the methods on the testing dataset that contains 1, 500 Cirrus B-scans and 1, 500 Spectralis OCT B-scans to generate Cirrus B-scans from Spectralis B-scans and vice versa as two harmonization tasks. We then applied the deep learning based retinal OCT segmentation method (He et al., 2019, 2021, 2023) to identify nine retinal boundaries. We computed the mean absolute error (MAE) for these boundary locations before and after harmonization. The results are summarized in Table 1. We see that LMSB ODE achieves the best performance in preserving anatomy during harmonization.
MAE evaluation may be subject to segmentation bias or errors. Therefore, we computed the LNCC between the OCT images before and after harmonization to evaluate structural similarity, which is shown in Table 2. Higher LNCC means higher structural similarity. We see that for both harmonization subtasks, LMSB outperforms DSBM with either SDE or ODE as sampling strategies, and outperforms DDIB with ODE sampling strategy. Moreover, when comparing between SDE and ODE sampling strategies, we find that ODE sampling preserves the anatomy much better than SDE. We believe this is due to the speckle in OCT images, which is preserved better by ODE.
8

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Table 2: Mean LNCC (Std. Dev.) comparison (N = L, 500) on both Spectralis to Cirrus and Cirrus to Spectralis harmonization. Bold numbers indicate the best result in that row for that subtask. Asterisks indicate statistical significance (i.e., paired t-test comparing 1 and 2nd best results gave p-value < 0.05).
Spectralis to Cirrus
Cirrus to Spectralis
DDIB
DSBM
LMSB
DDIB
DSBM
LMSB
ODE
SDE
ODE
SDE
ODE
ODE
SDE
ODE
SDE
ODE
0.65
0.16
0.63
0.17
0.66"
0.63
0.16
0.67
0.16
0.69
LNCC
(0.08)
(0.01)
(0.05)
(0.02)
(0.05)
(0.05)
(0.02)
(0.04)
(0.02)
(0.03)
We also show a qualitative comparison in Fig. 5. The bottom rows show enlarged images of the region indicated by the yellow rectangle in the top two rows. The red solid lines and the yellow dashed lines show the retinal boundary segmentation on the original image and the harmonized image, respectively, The yellow arrows show locations where segmentation on the harmonized anatomy disagrees with the original segmentation. We see that for LMSB, the yellow dashed lines are closer to the red solid lines, which mean LMSB preserves the anatomy better than DSBM and DDIB during harmonization.
5 Discussion
There are potential limitations in the LMSB approach. First, the theory shows that LMSB can reduce anatomical changes during harmonization, but may not totally eliminate them (Solution #1) as discussed in Fig. 1. This is because not all metric spaces are isomorphic to Euclidean metric spaces. potential solution is to find a coordinate transform that flattens the metric as much as possible, which we have not explored.
Second, the method constructs a reference anatomy with latent contrast by using the segmentation label map, but is not clear whether this latent contrast space is optimal. Specifically, as discussed in Fig. I. we can either compress the distance along the contrast direction (Solution #2) stretch the distance along the anatomy direction (Solution #3) to reduce the anatomical shift. We chose to compress the distance along the contrast direction, but we have not explored how to stretch the distance along the anatomy direction. For example, the segmentation label map used this paper assigns unique numbers to different retinal layers, and the number is inereasing from the top layer to the bottom layer. This is not necessarily optimal. To further stretch the distance along the anatomy direction, we can change the segmentation label maps such that the label number differences between adjacent retinal layers become larger, which effectively stretches the distance along the anatomy direction. It remains to be investigated whether this helps further improve the performance of LMSB.
Third, a different segmentation bias for Cirrus and Spectralis retinal OCT images may affect the LMSB performance when constructing the latent contrast space because it violates the assumption in Eq. 5, i.e., the latent variable 2 should be the same for an image pair with the same anatomy but different contrast. Note that a consistent segmentation bias for both Cirrus and Spectralis, the other hand, does not affect the LMSB performance.
Fourth, we have not evaluated the LMSB performance using real paired B-scans because individual B-scans are not well aligned even for paired OCT volumes due to the difficulty of optical alignment during acquisition. Moreover, registering two OCT volumes is not straightforward (Chen et al., 2014; Rcaungamornrat et al., 2018). This is because deformable registration is required due to different geometrical distortions between Cirrus and Spectralis, and highly anisotropic digital resolutions and field strength variations further complicate the registration process. There are work that demonstrated improved OCT volume interpolation along the slow axis using deformable registration and generative models (Wei et al., 2025), which may benefit OCT volume registration but it remains to be investigated. Further developments registering two OCT volumes and evaluations real paired B B-scans should be conducted the future.
Fifth, we have only demonstrated the LMSB performance using retinal OCT images, which have relatively simple anatomical structure. Theoretically, LMSB can generalize to other imaging modali- ties because the training of LMSB only requires unpaired datasets and corresponding segmentation

---

Spectralis
Original Spectralis
DSBM
LMSB SDE
LNCC:
DDIB ODE MAE: 0.41
LNCO: 0.70
DSBM ODE MAE: 0.49
LNCC: 0.63
LMSB ODE MAL: 0.31 LNCC: 0.60
(b)
Spectralis
Original Cirrus
DSBM MAE: 0.44
LNCC,
LMSB SDE
MAR: LNCC:
DOIR ODE
MAR: 0.50
ENCE: 0.68
PSHM OIR
MAEG 0.40
ENCE: 0.67
LMSS ODEE
MAIS 0:26 LIDEC 0.71
Figure 5: Qualitative comparison between DDIB, DSBM, and LMSB for (a) Spectralis to Cirrus harmonization and (b) Cirrus to Spectralis harmonization. The bottom rows show enlarged images of the region indicated by the yellow rectangle in the top two rows. The red solid lines show the retinal boundary segmentation on the original anatomy. The yellow dashed lines show the retinal boundary segmentation on the harmonized anatomy by different methods. The yellow arrows show locations where segmentation on the harmonized anatomy disagrees with the original segmentation.
label maps. However, it may require different preprocessing steps for other imaging modalities. The segmentation label map may require more careful design when the anatomy becomes more complex. Further experiments involving more complex anatomies and other imaging modalities should be studied in the future.
6 Conclusion
In this paper, we proposed anatomy-guided latent metric Schrödinger bridge (LMSB) framework to improve the anatomical consistency for medical image harmonization. We trained an invertible network that maps OCT images into latent Euclidean metric space where the distances among medical images with the same anatomy are minimized using the pullback latent metric. A diffusion Schrödinger bridge then trained to match the distribution in this learned latent Euclidean metric space. We demonstrated our method on OCT image. We showed that the proposed LMMSB method achieves better harmonization performance while preserving the anatomy than directly applying the Schrödinger bridge and other unsupervised harmonization methods.
10

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Acknowledgments and Disclosure of Funding
This work is supported in part by the NIH through NEI grant R01-EY032284 (PI: J.L. Prince), NINDS grant R01-NS082347 (PI: P.A. Calabresi), as well NIA grants R01-AG021155 (PI: S.C. Johnson) and R01-AG027161 (PI: S.C. Johnson). This material is partially supported by the National Science Foundation grant number 2136228 (PI: J. Gopalakrishnan), Graduate Research Fellowship grant number DGE-174689) (S. Remedios), and Johns Hopkins University Discovery Grant (PI: A. Carass).
J.L. Prince and A. Carass have received royalties from JuneBrain Inc. P.A. Calabresi is PI on grants to JHMI from Genentech and the Myelin Repair Foundation, and has received personal consulting Honoria from Idorsia, Spolia Therapeutics, Novartis, and Lilly. The other authors have no competing interests to declare.
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Appendices
Proof of Volume Preservation
15
B Ablation Study
15
B.1
Comparison of INN Outputs
15
B.2 Comparison of INN Invertibility
16
B.3 Comparison of Anatomy Preservation
17
Additional Qualitative Results
18
D Additional Quantitative Results
21
D.1 Domain Consistency Comparison
21
D.2 Reducing Inter-Device Measurement Bias
E Computational Resources
22
14

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A Proof of Volume Preservation
In this section, we provide a proof that the invertible neural network (INN) f that we constructed is volume-preserving We note that the determinant of the induced local linear map df can be derived as,
det(df) = det (id
d/2 det id
id dfi
0 =1 id
(7)
where id is the identity map, f1 and f2 are maps in the affine coupling layers in the INN, dfi and dfz are the induced local linear maps from fi and f2, and we abuse the use of the notations id, dfi and 2f2 to be the corresponding matrix representations. Because the determinant of df is 1 everywhere, the invertible map is
B Ablation Study
For the INN, we constrained the invertible map to be volume-preserving and we augmented the input images by adding Gaussian noise during network training. To demonstrate the importance of the volume preservation and the data augmentation, we conducted the following ablation studies. The first ablation implements the INN without volume preservation, but with data augmentation, which we denote as w/o Volume Preservation. This is done by adding a scaling operation to each affine coupling layer. The second ablation implements the INN without data augmentation, but with volume preservation, which we denote as w/o Data Augmentation. We denote LMSB-INN (Ours) as the implementation of the INN with both volume preservation and data augmentation.
B.1 Comparison of INN Outputs
The INN training results of these ablation experiments are summarized in Table 3. We calculated the root mean square error (RMSE) between the outputs of the INNs and the segmentation label maps. We also calculated the local normalized cross correlation (LNCC) between the outputs of the INNs and the original input images. Lower RMSEs indicate that the INN outputs are closer to the segmentation label maps. Higher LNCCs indicate that the INN outputs have more structure similarity to the original input images. From Table 3, we see that without volume preservation, the INN produces results that are almost identical to the segmentation label maps, but have a low structural similarity the original input images. This will lead to a poor invertibility of the INN, which we will demonstrate later. The volume preservation constraint helps improve the structural similarity between the INN outputs and the original input images. Moreover, with both volume preservation and data augmentation, the INN maps images to a neighborhood of the segmentation label map, and the INN outputs have the highest structural similarity with the original input images. Furthermore, by comparing the INN results between Spectralis and Cirrus, we see that INN produces noiser outputs with a higher structure similarity when using Cirrus B-scans as input.
We show some qualitative results in Fig. 6. Column I shows the input images to the INNs, including two Spectralis B-scans in Rows 1-2 and two Cirrus B-scans in Rows 3-4. Columns 2-4 show the output images of the INNs trained from w/o Volume Preservation, w/o Data Augmentation, and LMSB-INN (Ours), respectively. Column 2 shows almost identical results the segmentation label maps that we constructed for the input images. This means that without volume preservation, the trained INN maps images with the same anatomy but different contrasts to almost the same output, i.e, the segmentation label map. The results in Column 3 shows more different output images between using Cirrus B-scans and Spectralis B-scans as input. The INN outputs are much noiser when using the Cirrus B-scans as inputs than using the Spectralis B-scans as input. This improves the invertibility of the INN. However, there are still a lot of detailed anatomical structures missing from the results in Column 3, for example, vessels and shadows. This is because the segmentation label maps that we constructed only contain the segmentation of different retinal layers, and do not include segmentation of vessels and shadows. Therefore, the constructed segmentation label maps are not perfect anatomical representations of the input images. This issue alleviated by adding Gaussian noise to the input images as data augmentation, because it reduces the detailed anatomical structures from the original input images and makes the corresponding segmentation label maps more representative. We can see the results of LMSB-INN (Ours) with both volume preservation and data augmentation from Column 4, where the detailed anatomical structures of the vessels and shadows are well preserved for both the Cirrus and Spectralis input images.
15

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Table 3: Mean RMSE (Std. Dev.) and mean LNCC (Std. Dev.) comparison (N = 1, 500) on the results of the INNs trained from w/o Volume Preservation, w/o Data Augmentation, and LMSBB INN (Ours), respectively. Examples of the INN output images are shown in Fig. 6. The RMSEs were calculated between the outputs of the INNs and the target segmentation label maps. Lower RMSEs indicate that the INN outputs are closer to the segmentation label maps. The LNCCs were calculated between the outputs of the INNs and the original input images. Higher LNCCs indicate that the INN outputs have more structural similarity as the original input images.
w/o Volume Preservation
w/o Data Augmentation
LMSB-INN (Ours)
RMSE
0.013 (0.002)
0.029 (0.005)
0.056 (0.006)
Spectralis
LNCC
0.205 (0.012)
0.412 (0.066)
0,540 (0.080)
RMSE
0.014 (0.008)
0.063 (0.010)
0.078 (0.007)
Cirrus
LNCC
0.102 (0.011)
0.596 (0.011)
0.794 (0.020)
Volume Preservaioo
Data Augmentation
LMSB-INN (Ours)
Figure 6: INN outputs. Column I shows the input images to the INNs, including two Spectralis B- B-scans Rows 1-2 and two Cirrus B-scans in Rows 3-4. Columns 2-4 show the output images of the INNs trained from w/o Volume Preservation, w/o Data Augmentation, and LMSB-INN (Ours), respectively.
B.2 Comparison of INN Invertibility
To test the invertibility of the INNs, we added small Gaussian noise with a standard deviation of 0.05 to the INN output images, and then applied the inverse of the INN to obtain the corresponding reconstructed images. The reconstruction results are summarized in Table 4. We calculate the RMSE and the LNCC between the reconstructed images and the original input images. Lower RMSEs and higher LNCCs indicate a better invertibility of the INNs. From Table 4, we see that without volume preservation, the RMSES are big and the LNCCs are small, which indicates a poor invertibility. The volume preservation constraint helps improve the invertibility. Moreover, with both volume preservation and data augmentation, the reconstructed images achieve the lowest RMSE and the highest LNCC. Furthermore, the RMSEs for the LMSB-INN (Ours) are 0.05, which is identical to the standard deviation of the added Gaussian noise. This suggests that a well regularized invertibility is achieved when using both volume preservation and data augmentation.
We also show some qualitative results in Fig. 7. Column I shows the input images to the INN, the same as Column in Fig. 6. Columns 2-4 show the reconstructed images of the INNs trained from w/o Volume Preservation, w/o Data Augmentation, and LMSB-INN (Ours), respectively. From the results in Column 2, we see that the reconstruction results are very different from the original input images Column Here, Column uses the same colorbar as Column 1 for consistent comparison. This indicates that without volume preservation, the invertibility INN is diminished, where a small error in the latent space leads to a big error in the image space. The results in Columns 3-4 show better invertibility. This is because with volume preservation, the INN establishes bijection between a neighborhood in the image space and neighborhood in the latent space with the same volume size, which makes the invertibility of the INN more robust. Comparing the results in Columns 3-4, we
16

---

Table 4: Mcan RMSE (Std. Dev.) and mean LNCC (Std. Dev.) comparison (N = 1, 5,500 on the invertibility of the INNs trained from w/o Volume Preservation, w/o Data Augmentation, and LMSB-INN (Ours), respectively. We added small Gaussian noise with a standard deviation of 0.05 to the INN output images, and then applied the inverse of the INNs to obtain the corresponding reconstructed images. Examples of the reconstructed images are shown in Fig. 7. Both the RMSEs and LNCCs were calculated between the reconstructed images and the original input images. Lower RMSEs and higher LNCCs indicate better invertibilitty of the INNs. Bold numbers indicate the best result in that row. Asterisks indicate statistical significance (i.e., paired t-test comparing 1 and 2 best results gave p-value < 0.05).
w/o Volume Preservation
w/o Data Augmentation
LMSB-INN (Ours)
RMSE
44.420 (1.138)
0.106 (0.011)
0.050 (0.000)
Spectralis
LNCC
0.029 (0.002)
0.289 (0.061)
0.418 (0.057)
RMSE
44.680 (1.241)
0.070 (0.007)
0.050 (0.000)
Cirrus
LNCC
0.026 (0.001)
0.496 (0.031)
0.618 (0.047)
Iupul
Volumu Proserte vo
Data Auguwelatiun
IMSB-INN (Own)
Figure 7: INN invertibility. Column shows the input images the INNs, the same as Column I in Fig. 6. We added small Gaussian noise with a standard deviation of 0.05 to the INN output images that are shown in Columns 2-4 in Fig. 6, and then applied the inverse of the INNs to obtain the corresponding reconstructed images. Columns 2-4 show the reconstructed images of the INNs trained from w/o Volume Preservation, w/o Data Augmentation, and LMSB-INN (Ours), respectively.
that the invertibility is further improved with data augmentation. This because by adding Gaussian noise to the input images as data augmentation, it provides the information where the neighborhood should be centered.
B.3 Comparison of Anatomy Preservation
We evaluate the anatomy preservation of LMSB harmonization using the INNs trained from w/o Volume Preservation, w/o Data Augmentation, and LMSB-INN (Ours), respectively. For each INN, we trained LMSB using the same training parameters as we described in Network Training. The harmonization results are summarized in Table 5. We calculated the LNCCs between harmonized and original images. Higher LNCCs indicate a better preservation of the anatomical structure. From Table 5, preservation helps improve the anatomy preservation. Morcover, with both volume preservation and data augmentation, the harmonized images achieve the highest LNCC, for both Spectralis to Cirrus and Cirrus to Spectralis harmonization, and for both SDE and ODE sampling strategies.
that without volume preservation, the harmonization results are poor. The volume
We also show some qualitative results in Fig. 8. Column 1 shows the original input images before harmonization, the same as Column in Fig. 6. Rows 1-2 show two examples of Spectralis to Cirrus harmonization, and Rows 3-4 show two examples of Cirrus to Spectralis harmonization. From the results in Column 2, we see that without volume preservation, the poor invertibility leads to a bad harmonization performance. Here, Column 2 uses the same colorbar as Column for consistent
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Table 5: Mean LNCC (Std. Dev.) comparison (N = 5000)) on the LMSB harmonization results using the INNs trained from w/o Volume Preservation, w/o Data Augmentation, and LMSB-INN (Ours), respectively. Examples of the harmonized images are shown in Fig. 8. The LNCCs were calculated between the harmonized images and the original input images. Higher LNCCs indicate a better preservation of the anatomical structure. Bold numbers indicate the best result in that row. Asterisks indicate statistical significance (i.e., paired I-test comparing 1 and 2nd best results gave p-value < 0.05).
w/o Volume Preservation
w/o Data Augmentation
LMSB-INN (Ours)
SDE
ODE
SDE
ODE
SDE
ODE
Spectralis
0.045
0.234
0.147
0.518
0.165
0.662
to Cirrus
(0.004)
(0.063)
(0.013)
(0.042)
(0.016)
(0.052)
LNCC
Cirrus
0.037
0.239
0.139
0.510
0.164
0.692
Spectralis
(0.011)
(0.083)
(0.014)
(0.044)
(0.019)
(0.030)
Input
wie Volume Preservation
win Data Augmentation
IMSB-INN (Ours)
Figure 8: Anatomy preservation of LMSB harmonization with the ODE sampling strategy. Column 1 shows the original input images before barmonization, the same as Column 1 in Fig. 6. Rows 1-2 show two examples of Spectralis to Cirrus harmonization, and Rows 3-4 show two examples of Cirrus to Spectralis harmonization. We trained a LMSB for each INN that was trained from w/o Volume Preservation, w/o Data Augmentation, and LMSB-INN (Ours). We used the same training parameters for LMSB, which we described in Network Training. Columns 2-4 show the harmonized images of the LMSBs trained from w/o Volume Preservation, w/o Data Augmentation, and LMSB-INN (Ours), respectively.
comparison. From the results in Column 3, that without data augmentation, although most retinal layer structures are preserved before and after harmonization, the detailed anatomical structures that are not in the segmentation label maps are changed, such vessels and shadows. The results of LMSB with both volume preservation and data augmentation are shown in Column 4, where we see that not only the retinal layers but also the vessels, shadows, and choroids are well preserved before and after harmonization.
C Additional Qualitative Results
We demonstrate the harmonization quality of our proposed method LMSB with ODE sampling on several examples, with Spectralis to Cirrus harmonization in Fig. 9 and Cirrus to Spectralis harmonization in Fig. 10. From these qualitative results, we see that the retinal anatomy before and after harmonization is well preserved, including anatomical structures such as retinal layers, vessels, shadows, and choroids. We note that some of these structures, such as vessels, shadows, and choroids, do not appear in the segmentation label maps that were used to train the INN. Moreover, from the Cirrus to Spectralis harmonization results Fig. 10, we see that many anatomical structures such as vessels, shadows, and choroids that are difficult to visualize in the original Cirrus B-scan become clearer the harmonized B-scan with the Spectralis contrast. This suggests that the proposed harmonization method, LMSB, may benefit downstream tasks such as vessel segmentation.
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Spectralis Cirrus
Original
Harmenized
Figure 9: Random examples of Spectralis to Cirrus harmonization by LMSB using the ODE sampling strategy. The left column shows the original Spectralis OCT B-scans. The right column shows the synthetic Cirrus OCT B-scans that are harmonized from the left column.
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Cirrus to Spectralis
Original
Harmonized
Figure 10: Random exmaples Cirrus to Spectralis harmonization by LMSB using the ODE sampling strategy. The left column shows the original Cirrus OCT B-scans. The right column shows the synthetic Spectralis OCT B-scans that are harmonized from the left column.
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Table 6: FID comparison for both Spectralis to Cirrus and Cirrus to Spectralis harmonization. Before harmonization, the FID between the 1, 500 Cirrus Bcacans and the 1, 500 Spectralis B-scans in the testing dataset is 140.53. After harmonization, the FIDs were calculated between the 1, 500 synthetic B-scans and the 1, 500 original B-scans with the target contrast. Bold numbers indicate the best result for that subtask.
Spectralis to Cirrus
Cirrus to Spectralis
DDIB
DSBM
LMSB
DDIB
DSBM
LMSB
ODE
SDE
ODE
SDE
ODE
ODE
SDE
ODE
SDE
ODE
FID
48.48
18.48
24.26
17.24
25.34
50.97
34.99
31.39
41.72
32.97
D
Additional Quantitative Results
D.1 Domain Consistency Comparison
To measure the domain consistency, we calculated Fréchet inception distance (FID) before and after harmonization with respect to the testing dataset. Before harmonization, the FID between the 1, 500 Cirrus B-scans and the 1, 500 Spectralis B-scans in the testing dataset is 140.53. We conducted both Spectralis to Cirrus harmonization and Cirrus to Spectralis harmonization. After harmonization, the FIDs were calculated between the 1, 500 synthetic B- Bcaans and the 1. 500 original B-scans with the target contrast. The FID results are shown in Table We found that FID improves after harmonization with respect to the target contrast for all the harmonization methods. Moreover, by comparing the FIDs of different methods after harmonization, we can see that both DSBM and LMSB have better FID than DDIB. LMSB has slightly larger FID than DSBM, which is expected because the LMSB does not directly apply harmonization between two original data domains.
D.2 Reducing Inter-Device Measurement Bias
To evaluate the impact of the LMSB on this inter-device bias. we performed a downstream segmenta- tion and retinal layer thickness measurement on 115 pairs of Cirrus and Spectralis volumes collected from 59 subjects and 115 eyes, which is separate from the training and testing dataset used in this paper. These 115 pairs of scans were completed contemporaneously. The retinal layer thickness measurements from these two devices disagree in general, which is an important clinical issue in OCT imaging when comparing the measurements from different devices on monitoring subject that scanned on different machines during the time course the study.
We conducted the following harmonization experiments using LMSB with ODE sampling strategy on the paired OCT dataset: 1) Spectralis to Cirrus harmonization; and 2) Cirrus to Spectralis harmonization. In the first experiment, we harmonize the 115 Spectralis volumes to the Cirrus scanner. We computed the retinal layer thickness measurement differences between synthetic Cirrus (synC) volumes and corresponding original Cirrus (orgC) volumes, and we compared them with the differences between original Spectralis (orgS) volumes and original Cirrus (orgC) volumes. The retinal layer thickness measurements were done by first applying a deep learning segmentation algorithm and then computing the average layer thickness fovea. The second experiment is similar to the first with the exception that we harmonize the 115 Cirrus volumes to the Spectralis scanner. We report difference in thickness measurements between the synthetic Spectralis (synS) volumes and corresponding original Spectralis (orgS) volumes, as well as the differences between original Cirrus (orgC) volumes and original Spectralis (orgS) volumes. The results are shown in Table 7, with the mean and the standard deviation of the signed difference of the retinal layer thickness measurement.
5 mmm square centered on the
For both results, we see that harmonization helps reduce the inter-device bias for all retinal layers including the most important retinal layer GCIPL. Moreover, paired t-tests show statistical significance (p-value < 0.05) for most cases. We need to highlight that the proposed method was originally developed such that the thickness measurements from original volumes and harmonized volumes are consistent. However, from the results in Table 1, we see that we still do not obtain the exact anatomy consistency with the proposed method. From these additional experiments on the paired dataset, we see that this remaining anatomy inconsistency actually helps reduce the inter- bias.
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Table 7: Mean difference (Std. Dev.) comparison (N = 115) in units of pmm for thickness measure- ments across eight retinal layers before and after harmonization for both Spectralis Cirrus and Cirrus to Spectralis harmonization. Before harmonization, the thickness measurement differences were derived from the two original OCT volumes. After harmonization, the thickness measurement differences were derived from the synthetic and corresponding original OCT volumes. Bold numbers indicate reduced inter-device thickness measurement bias that row for that subtask. Asterisks indi- cate statistical significance (i.e., paired t-test comparing before and after harmonization gave p-value < 0.05). Key: RNFL: retinal nerve fiber layer; GCIPL: ganglion cell and inner plexiform layer; INL: inner nuclear layer; OPL: outer plexiform layer: ONL: outer nuclear layer; IS: inner segment; OS: outer segment; RPE: retinal pigment epithelium complex; orgC: original Cirrus; orgS: original Spectralis; synC: synthetic Cirrus; synS: synthetic Spectralis.
Spectralis to Cirrus
Cirrus to Spectralis
orgS orgC
synC
orgC
orgC orgS
synS orgS
RNFL
9.66 (2.70)
5.44" (5.35)
-9.66 (2.70)
-3,23* (3.37)
GCIPL
-4.75 (1.31)
-1.89* (2.26)
4.75 (1.31)
2.66" (1.92)
INL
(1.19)
-1.79* (1.92)
2.64 (1.19)
1.09 (1.57)
OPL
0.63 (0.66)
-0.16" (1.38)
-0.63 (0.66)
0.26* (0.75)
ONL
(1.668
-2.33* (2.64)
4.20 (1.68)
0.48 (2.35)
IS
-0.31 (0.41)
-0.66* (1.04)
0.31 (0.41)
0.24 (0.48)
OS
1.16 (1.04)
-0.88 (1.27)
-1.16 (1.04)
0.38 (0.81)
RPE
1.64 (1.23)
0.27 (0.98)
-1.64 (1.23)
0.74* (1.26)
E
Computational Resources
The proposed model LMSB requires only a single GPU with 48 GB of memory for training and 15 GB for evaluation when the batch size is 16. However, in practice we used several different GPUs in parallel train and test different models with different hyperparameters. To train all the models including the proposed model, comparison models and ablation models, we used seven GPUs in total, including four NVIDIA A40 (48 GB), two NVIDIA RTX A6000 (48 GB), and one QUADRO RTX 8000 (48 GB).
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Reviews:
[{"review_id": "EN4ZKClohS", "reviewer": "Reviewer_ZM1d", "summary": "This paper tackles the long-standing “anatomy-shift” problem that occurs when applying Schrödinger Bridge (SB) or entropy-regularized optimal transport directly to medical images. In this manuscript, the authors proposed anatomy guided latent metric Schrödinger Bridge (LMSB) to achieve optical coherence tomography (OCT) image harmonization. The authors demonstrated that directly using SB might cause anatomy shift, which could affect the downstream tasks. The main technical contributions is mapping OCT images to a latent Euclidean metric space via anatomy guidance, which was demonstrated to have better performance than DDIB and DSBM.", "questions": "1. Why did authors choose segmentation label maps as the sole anatomy-preserving coordinate? Have them explored alternative learned metrics.\n2. Can author address my concern on the dataset preparation?\n3. Can author acquire scans of the same retinal region in a single participant with both OCT devices and display the harmonized output and the true target image side by side?", "limitations": "Yes", "rating": 4, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n1. The paper formalizes how pixel-space Euclidean costs confound contrast variation with anatomical differences.\n2. Training a network to “flatten” anatomy into a Euclidean metric, followed by SB in that space, is conceptually elegant and empirically effective.\n3. The proposed method does not require paired data, which would be very helpful in the field of medical imaging.\n4. Compared to other pipelines, the proposed method displayed expecptional improved. And there's almost no anatomy shift for images generated by LMSB ODE.\n\nWeaknesses:\n\n1. I have a concern about the dataset. For the dataset preparation, the authors acquired 16, 500 Cirrus B-scans sampled from 387 Cirrus OCT volumes, and 16, 500 Spectralis B-scans sampled from 337 Spectralis OCT volumes. And then, they split the B-scans into 15,000 and 1,500 for each OCT device. The field of view of OCT is quite small (usually 3x3, 6x6 or 12x12 mm2) compared to other imaging modalities like ultrasound and CT. Consequently, the B-scans within a single volume are often highly similar, especially those that are adjacent or otherwise close to one another. Hence, I feel like splitting the dataset by B-scans rather than the volumes is a kind of data leakage and cannot reflect the performance in the 'real world' setting.\n2. The current validation is insufficient. I suggest acquiring scans of the same retinal region in a single participant with both OCT devices and then registering the two volumes (both two devices mentioned in the manuscript can automately take OCT images located the central fovea of the retina, so the subsquent registration is not quite hard). This setup would enable a direct, voxel-wise comparison between the synthesized and ground-truth images, clearly revealing any discrepancies.\n3. This works relies on deep learning segementation of the retina layers, so any bias or error in the segmentation could propagate and adversely affect the subsequent LMSB stage.\n4. For the inner retina, the current OCT/ophthalmology studies focus a lot on RNFL-GCL as it's related to glucoma (biomaker: thickness of the layer) or other nuerodegenration diseases (biomarker: morphology of the retina ganglion axon bundles). Unfortunately, the proposed method does not deliver satisfactory performance on this critical layer.", "was_revised": true, "final_justification": "The authors have addressed most of my concerns except registering two volumes from different devices, which might be difficult to complete within a few weeks. Hence, I changed the score from 3 to 4.", "rebuttal": "We thank the reviewer for the thorough list of the strengths and constructive comments about the weaknesses of our paper. We address the concerns as follows.\n\n**1. Training and testing data leakage.**\n\nWe split the training and testing datasets by volumes so there is no data leakage. Reviewer rEQb also brought up a similar concern. For clarity, we will modify the following details in the final paper. \n\nThe dataset consists of 388 Cirrus volumes and 338 Spectralis volumes. The 388 Cirrus volumes come from 194 subjects (388 eyes in total). The 338 Spectralis volumes come from 165 subjects (269 eyes in total). Note that there are repeated scans on the same eye for Spectralis. Training and testing splits are done by subject: 352 Cirrus volumes from 176 subjects (352 eyes) and 307 Spectralis volumes from 156 subjects (252 eyes) for training and 36 Cirrus volumes from 18 subjects (36 eyes) and 31 Spectralis volumes from 9 subjects (17 eyes) for testing. Our experiments are in 2D and operate on B-scans independently. Because each Cirrus volume contains 128 B-scans and each Spectralis OCT volume contains 49 B-scans over the same field of view 6x6 mm2, the Cirrus volumes have denser B-scan sampling. Therefore, to reduce anatomical redundancy across individual Cirrus volume B-scans, we extract every third B-scan. Specifically, in the training dataset, we extract 15,000 B-scans in the training dataset and 1,500 B-scans in the testing dataset for both Cirrus and Spectralis, resulting in 30,000 training B-scans and 3,000 testing B-scans with no subject data leakage between train and test splits.\n\n**2. Validation on paired B-scans.**\n\nWe agree with the reviewer that paired B-scans would lead to improved validation. Reviewer tKW4 also brought up a similar concern. However, creating paired B-scans using registration as described by the reviewer is not straightforward. This is due to the following three aspects: 1) Deformable registration is required due to the difference of optical setups and post alignment methods between the two devices. The retinal curvature between two different scanners is different within the same scanning region, and thus there is no affine relationship between volumes acquired from the two different scanners. 2) Registering highly anisotropic volumes is inaccurate and would hinder validation. For example, in our dataset, Cirrus volumes contain 128 B-scans, each with 512 A-scans, but Spectralis volumes contain 49 B-scans, each with 1024 A-scans. Even with affine registration, this kind of anisotropy would lead to egregious aliasing artifacts that would corrupt analysis. 3) Field bias differs between the two scanners. Spectralis volumes often have abrupt field variations, i.e., the mean intensity may change abruptly between adjacent B-scans, in which case an aligned B-scan after registration may not look like a Spectralis B-scan. This adds to the difficulty of creating real paired B-scans. \n\nHowever, we do plan to continue work to improve validation of this technique for OCT volumes. Additional evaluations are out-of-scope for this work, but we will mention this limitation in our discussion and discuss how future work can investigate and evaluate the method further.\n\n**3. Segmentation bias or error may affect harmonization.**\n\nA consistent segmentation bias for both Cirrus and Spectralis B-scans does not affect the subsequent harmonization, and an accurate segmentation is not required for training the invertible neural network. \n\nThe first statement can be seen from Eq. 5, where the only requirement for the latent variable z is that it should be the same for images with the same anatomy but different contrast. Therefore, a consistent segmentation bias for both Cirrus and Spectralis B-scans does not affect the subsequent harmonization. The second statement can be seen from Fig. 4, where we can see that both the Cirrus and Spectralis B-scans are not perfectly mapped to the segmentation label maps and some details such as vessel and shadows are preserved. This suggests that an accurate segmentation is not required. Both have been discussed in the original paper. \n\nHowever, a different segmentation bias between Cirrus and Spectralis B-scans will affect the subsequent LMSB harmonization because it violates the assumption in Eq. 5, for example, if a Cirrus segmentation is biased upwards and a Spectralis segmentation is biased downwards. We will add this as a limitation of our method to the discussion section. \n\n**4. Performance on RNFL-GCL and application to ophthalmology.**\n\nWe agree with the reviewer that RNFL-GCL is important in many applications in ophthalmology but emphasize that our paper is mainly a methodology development and evaluation that tries to improve anatomical consistency in harmonization instead of applying it to a specific application. As an aside, in our group we are also interested in some potential important applications other than RNFL-GCL, including hyper-reflective foci (HRF) detection and GCL-IPL separation. HRFs refer to small bright spots seen in OCT in different retinal layers that do not cast shadows, and it is recognized as a biomarker of retinal degeneration. GCL and IPL are often grouped together in retinal OCT segmentation because they have subtle differences in tissue properties, and the independent segmentation of GCL and IPL would introduce a more important biomarker from GCL alone in ophthalmology. For Cirrus B-scans, both HRF and the GCL-IPL boundary are ambiguous because of speckle and noise. In some of the results of Cirrus to Spectralis harmonization in Fig. 10 in the supplemental material, we observe that both HRF and the GCL-IPL separation becomes more visible in the synthetic Spetralis B-scans, which may benefit HRF detection and GCL-IPL segmentation from the original Cirrus B-scans. We will add these observations when presenting the results in Fig. 10. Moreover, the other comparison methods also do not perform well on RNFL-GCL boundary compared to other retinal boundaries, which is interesting and worthy of further investigation. We will add this discussion and observation when presenting the results in Table 1. \n\n**5. Reasons for choosing segmentation label map and alternative learned metrics.**\n\nThe reason for choosing segmentation label map is that it is a sufficient and straightforward design that satisfies the condition of Eq. 5, i.e., the latent variable z should be the same for images with the same anatomy but different contrast. Moreover, this choice is easy to generalize to other modalities and anatomies because the proposed method is potentially applicable when segmentation label maps exist. \n\nAlternative learned metrics include variants of the segmentation label maps. The current segmentation label map assigns unique numbers to different retinal layers, and the number is increasing from the top layer to the bottom layer. This is not necessarily optimal. As we discussed in the original paper, we can either compress the distance along the contrast direction or stretch the distance along the anatomy direction to reduce the anatomical shift. We did the former approach. But to further stretch the distance along the anatomy direction, we can change the segmentation label maps such that the label number differences between adjacent retinal layers become larger, which effectively stretches the distance along the anatomy direction. While we have not explored these alternative learned metrics, we instead provided a discussion about the potential alternative learned metrics in the discussion."}, {"review_id": "ZSd3SwLAEV", "reviewer": "Reviewer_rEQb", "summary": "Existing image harmonization approaches distort the underlying anatomy (often severely in my experience.) This approach uses the existing annotations of anatomy to suppress this issue in the framework of a Diffusion Schrödinger Bridge. The result is a meaningful improvement in faithfulness.", "questions": "The paper states that the dataset consisted of 16500 slices from each machine, coming from ~330 scans, and split into train and test set at a ration of 15000 slices to 1500 slices. Please confirm whether the train-test split was done at the scan level or the slice level- the phrasing makes it sound concerningly like the split was done at the slice level, which would introduce heavy test set contamination.", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "There is a great deal of math going on, but digging to the root of it it appears that the latent variable z is just the segmentations associated with the data, and so the correspondence that is being preserved is just that the segmentation shouldn't change. This is admirable and of course the goal, but it could have been stated more clearly. The specific way of using segmentations as a flow matching target is clever and well integrated to the schroedinger bridge framework.\n\nOn the other side of clarity, Figure 1 is excellently done, and was central to my ability to grok the paper's mathematical content.", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the commendation. We are glad to hear that Fig. 1 helps improve the clarity of the mathematical content. We address the concerns and clarify some misunderstandings as follows. \n\n**1. The correspondence that is being preserved is just that the segmentation shouldn’t change.**\n\nAlthough this is almost true, we want to highlight that both Cirrus and Spectralis B-scans are not perfectly mapped to the segmentation label maps, which can be seen from Fig. 4. The harmonization then applies to the latent representation of the original images through the invertible neural network, which effectively reduces the change of segmentation but not necessarily eliminate it. Therefore, we cannot say that the segmentation does not change. \n\n**2. The core idea could have been stated more clearly.**\n\nFollowing the previous response, we plan to give an intuitive explanation of the core idea as follows: Intuitively, the invertible neural network brings both Cirrus and Spectralis B-scans closer to a common latent contrast while preserving the anatomy using segmentation label maps so that the anatomy shift, i.e., segmentation change, is reduced during harmonization. We will add this intuitive explanation to the final paper.\n\n**3. Test set contamination.**\n\nWe split the training and testing datasets by volumes so there is no data leakage. Reviewer ZM1d also brought up a similar concern. For clarity, we will modify the following details in the final paper. \n\nThe dataset consists of 388 Cirrus volumes and 338 Spectralis volumes. The 388 Cirrus volumes come from 194 subjects (388 eyes in total). The 338 Spectralis volumes come from 165 subjects (269 eyes in total). Note that there are repeated scans on the same eye for Spectralis. Training and testing splits are done by subject: 352 Cirrus volumes from 176 subjects (352 eyes) and 307 Spectralis volumes from 156 subjects (252 eyes) for training and 36 Cirrus volumes from 18 subjects (36 eyes) and 31 Spectralis volumes from 9 subjects (17 eyes) for testing. Our experiments are in 2D and operate on B-scans independently. Because each Cirrus volume contains 128 B-scans and each Spectralis OCT volume contains 49 B-scans over the same field of view 6x6 mm2, the Cirrus volumes have denser B-scan sampling. Therefore, to reduce anatomical redundancy across individual Cirrus volume B-scans, we extract every third B-scan. Specifically, in the training dataset, we extract 15,000 B-scans in the training dataset and 1,500 B-scans in the testing dataset for both Cirrus and Spectralis, resulting in 30,000 training B-scans and 3,000 testing B-scans with no subject data leakage between train and test splits."}, {"review_id": "jzp1oejegO", "reviewer": "Reviewer_tKW4", "summary": "This paper develops a method, namely anatomy-guided latent metric Schrödinger bridge (LMSB), for OCT image harmonization to reduce anatomical shifts. The authors provide theoretical, quantitative, and qualitative analysis of the proposed algorithm on a private dataset.", "questions": "1. Will the pre-processing of the images affect the performance of the algorithm? What is the original size of the OCT images?\n2. The authors may consider adding more evaluation metrics, such as PSNR, SSIM or some Edge-preserving metrics, to better demonstrate the performance of the algorithm. Even though the pairs are not pixel-wise matched, the authors may consider using registration algorithms before calculating these metrics.\n3. The authors may consider collecting feedback from the experts to validate the performance of the algorithm.", "limitations": "The authors may consider discussing the limitations of the societal impacts, not just the technical impacts.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 4, "strengths_and_weaknesses": "Strengths:\n1. The proposed LMSB framework enhances anatomical consistency in medical image harmonization by mapping images to a learned latent Euclidean space using an invertible neural network.\n2. The authors provide a rigorous theoretical analysis of why conventional Euclidean transport costs in SB and optimal transport are unsuitable for medical images.\n\nWeakness: \n1. The method is only demonstrated on retinal OCT images, which have relatively simple and regular anatomical structures.\n2. There is no evidence that LMSB can generalize to other imaging modalities, or even for other OCT vendors. For example, the two OCT systems in the paper (Zeiss Cirrus OCT and Heidelberg Spectralis OCT ) are SDOCT systems. Will the algorithm be effective between SDOCT and SSOCT?\n3. The authors claim that the proposed algorithm is for the image harmonization task. However, the evaluation metrics include MAE and LNCC. Can these two metrics measure the domain consistency between two modalities?", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the thorough list of the strengths and constructive comments about the weaknesses of our paper. We address the concerns as follows.\n\n**1. The proposed method is only demonstrated on retinal OCT images.**\n\nWe agree with the reviewer that this is a limitation of the paper and we pointed out this limitation in the discussion. Our application was to retinal OCT because that is what our group is interested in. However, the proposed method is a general framework and can generalize to other modalities. Please see the following response for the generalizability of the proposed method. \n\n**2. Can the proposed method generalize to other imaging modality or other OCT vendors?**\n\nTheoretically yes, LMSB can generalize to other imaging modalities or other OCT vendors. Currently the training of LMSB only requires unpaired datasets and segmentation label maps associate with the data. The unpaired datasets are easier to obtain than paired datasets in clinical settings, and the segmentation label maps are theoretically available if we train segmentation models on both datasets. \n\n**3. Can MAE and LNCC measure domain consistency between two modalities?**\n\nWe agree with the reviewer that measuring the domain consistency is an important metric. MAE and LNCC however measures the anatomical consistency instead of domain consistency. We use these two metrics because improving the anatomical consistency is the goal of the paper. To measure the domain consistency, we calculated Fréchet inception distance (FID) before and after harmonization with respect to the testing dataset. We found that FID improves after harmonization with respect to the target contrast. Moreover, we compare the FIDs of different methods after harmonization as shown in the following table, where we can see that both DSBM and LMSB have better FID than DDIB. LMSB has slightly larger FID than DSBM, which is expected because the LMSB does not directly apply harmonization between two original data domains. We will add this FID evaluation to the final paper. \n\n| | |Spectralis|to|Cirrus| | |Cirrus|to|Spectralis| |\n|:-:|:-:|:-:|:-:|:-:|:-:|:-:|:-:|:-:|:-:|:-:|\n| |DDIB|DSBM|DSBM|LMSB|LMSB|DDIB|DSBM|DSBM|LMSB|LMSB|\n| |ODE|SDE|ODE|SDE|ODE|ODE|SDE|ODE|SDE|ODE|\n|Before|140.53|140.53|140.53|140.53|140.53|140.53|140.53|140.53|140.53|140.53|\n|After|48.48|18.48|24.26|**17.24**|25.34|50.97|34.99|**31.39**|41.72|32.97|\n\n\n**4. Will the pre-processing affect the algorithm? What is the original size of OCT images?**\n\nWe agree with the reviewer’s concern over the impact of pre-processing. For this reason, the preprocessing techniques we chose were minimal: only cropping and resizing to have both Cirrus and Spectralis B-scans of the same image size, digital resolution, and field of view. We do not perform any other processing steps that could affect the image intensities, such as histogram equalization or bias field correction. The intent was exactly to address the reviewer’s concern that excessive preprocessing would affect the domain appearance of each scan.  \n\nThe original size of Cirrus B-scans is 1024x512 (axial x lateral) and the original size of Spectralis B-scans is 496x1024 (axial x lateral). The axial resolution of Cirrus B-scans is 2 um/pixel, and the axial resolution of Spectralis B-scans is 4 um/pixel. Both B-scans cover a 6 mm lateral scanning range. We crop the Cirrus B-scans axially to 512x512 and resize them to 128x512, and we crop the Spectralis B-scans axially to 256x1024 and resize them to 128x512, such that they have the same image size, digital resolution, and field of view. We will add these details to the final paper. \n\n**5. Adding alternative metrics such as PNSR SSIM for paired B-scans after registration.**\n\nWe agree with the reviewer that paired B-scans would lead to improved validation. Reviewer ZM1d also brought up a similar concern. However, creating paired B-scans using registration as described by the reviewer is not straightforward. This is due to the following three aspects: 1) Deformable registration is required due to the difference of optical setups and post alignment methods between the two devices. The retinal curvature between two different scanners is different within the same scanning region, and thus there is no affine relationship between volumes acquired from the two different scanners. 2) Registering highly anisotropic volumes is inaccurate and would hinder validation. For example, in our dataset, Cirrus volumes contain 128 B-scans, each with 512 A-scans, but Spectralis volumes contain 49 B-scans, each with 1024 A-scans. Even with affine registration, this kind of anisotropy would lead to egregious aliasing artifacts that would corrupt analysis. 3) Field bias differs between the two scanners. Spectralis volumes often have abrupt field variations, i.e., the mean intensity may change abruptly between adjacent B-scans, in which case an aligned B-scan after registration may not look like a Spectralis B-scan. This adds to the difficulty of creating real paired B-scans. \n\nHowever, we do plan to continue work to improve validation of this technique for OCT volumes. Additional evaluations are out-of-scope for this work, but we will mention this limitation in our discussion and discuss how future work can investigate and evaluate the method further.\n\n**6. Feedback from experts.**\n\nWe agree with the reviewer that feedback from experts is important. We have extensive medical expertise in our team, specifically we have two medical doctors with 55 years of experience between them. They have provided extensive feedback throughout the development and testing processing of the proposed work."}, {"review_id": "Ygh4lINeNg", "reviewer": "Reviewer_ymJv", "summary": "The paper introduces a new method, the latent metric Schrödinger bridge (LMSB), which improves anatomical consistency for the harmonisation of medical images. They train an invertible network that maps the OCT images into a latent Euclidean space where distances between images with the same anatomical structure are minimized, The authors show that the method was able to harmonise OCT images.", "questions": "1. Could the authors elaborate on how the method would apply to other medical images? The paper discusses the case of OCT images. Would there be any problems in applying it to MRI or Ultrasounds? Would the variability of the acquisition of ultrasound make it harder to apply the proposed method ? In the limitations section, the authors mention that the complexity of the anatomy might add complexity to the method. Can the authors elaborate on that point?\n2. Could the authors provide a more thorough statistical analysis of the quantitative results presented?\n3. Can the authors expand on what computational resources were used in this work?\n4. Line 54: please elaborate which assumptions from OT and SB are not correct for medical image harmonization.\n5. The authors report the MAE as the evaluation metric, could the authors clarify why this is the only metric reported instead of FID and more traditional generative metrics?", "limitations": "yes", "rating": 4, "confidence": 1, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strengths:\n- Comparison to other methods\n- The paper tackles the challenge of domain shift caused by different scanners or acquisition parameters\n\nWeaknesses:\n- There is a lack of statistical significance that supports the main claim of the paper.", "was_revised": true, "final_justification": "Thanks for the response. I think it's important to include the FID results in the paper, and it's also good to hear that you'll be adding a discussion on the generalizability of the pre-processing steps in the final version. That said, I'll keep my score as is.", "rebuttal": "We thank the reviewer for the constructive comments that helps improve the quality of our paper. We address the concerns as follows.\n\n**1. Statistical analysis.**\n\nFollowing the reviewer’s suggestion, we conducted a statistical analysis by one-side paired T-test to see if the best mean value (the bold number) is statistically better than the second best mean value, and the test results show a statistical significance for most cases in both Table 1 and Table 2. \n\nNote that in the original paper, we have reported the variance of each metric in both Table 1 and Table 2, and our test dataset is relatively large (N=1,500 B-scans), which already shows some level of statistical significance. We will add the additional one-side paired T-test results to the final paper. \n\n**2. Generalization to other medical images, such as MRI and ultrasounds.**\n\nTheoretically LMSB can generalize to other imaging modalities. Because the training of LMSB only requires unpaired datasets and segmentation label maps associated with the data. Unpaired datasets are easier to obtain than paired datasets in clinical settings for both MRI and ultrasound, and the segmentation label maps are theoretically available if we train segmentation models on both datasets. \n\n**3. Would the variability of the acquisition of ultrasound make it harder to apply the proposed method?**\n\nThe variability of the acquisition of ultrasound could potentially complicate the pre-processing steps but should not affect the application of the proposed method. Currently, we have two pre-processing steps for OCT, including a crop and a resize, to have both OCT B-scans from two devices of the same image size, digital resolution, and field of view. With ultrasound, the acquisition could be much more complex, and the dataset may contain images with quite different image sizes, image resolutions, fields of view, etc, which complicates the pre-processing steps. However, once these pre-processing steps are solved, the application of the proposed method should be the same. We will add a discussion on the generalizability of the pre-processing steps to the final paper. \n\n**4. How does complexity of anatomy affect the proposed method?**\n\nThe complexity of the anatomy would affect the proposed method. As we discussed in Fig. 1 in the original paper, we can either compress the distance along the contrast direction or stretch the distance along the anatomy direction to reduce the anatomical shift. We did compress the distance along the contrast direction by training an invertible neural network, but we didn’t optimize the distance along the anatomy direction. The current segmentation label map assigns unique numbers to different retinal layers, and the number is increasing from the top layer to the bottom layer. This is not necessarily optimal. For example, if we have more layers, then the contrast between adjacent layers in the segmentation label map becomes lower, and the same anatomy shift would lead to a smaller Euclidean distance, which effectively compresses the distance along the anatomy direction and thus will affect the performance of the proposed method. We will add elaborate on the corresponding discussion in the final paper.\n\n**5. Computational resources.**\n\nThe proposed model required only a single GPU with 48GB of memory for training and 15GB for evaluation when the batch size is 16. However, in practice we used several different GPUs in parallel to train and test different models with different hyperparameters. To train all the models including the proposed model, comparison models and ablation models, we used seven GPUs in total, including four NVIDIA A40 (48GB), two NVIDIA RTX A6000 (48GB), and one QUADRO RTX 8000 (48GB). We will add these details about the computational resources we used to the final paper. \n\n**6. Assumption from OT and SB for medical image harmonization.**\n\nThe assumption behind OTs and SBs is that images with the same anatomy and different contrast should have small Euclidean distances. This is because the OTs and SBs are trying to find an optimal coupling that minimizes the overall transport cost, where the transport cost is the square of Euclidean distance between images. This assumption is not correct because images with the same anatomy but different contrast do not necessarily have small Euclidean distance, and thus OTs and SBs may not find a correct correspondence between them. We have discussed this in detail in “issues of Euclidean metrics and potential solutions” in the original paper, but we will add a brief explanation around line 54 in the final paper. \n\n**7. Traditional generative metrics such as FID.**\n\nWe agree with the reviewer that traditional generative metrics such as FID are important. We use MAE and LNCC to measure anatomical consistency because improving the anatomical consistency is the goal of the paper. To measure the domain consistency, we calculated Fréchet inception distance (FID) before and after harmonization with respect to the testing dataset. We found that FID improves after harmonization with respect to the target contrast. Moreover, we compare the FIDs of different methods after harmonization as shown in the following table, where we can see that both DSBM and LMSB have better FID than DDIB. LMSB has slightly larger FID than DSBM, which is expected because the LMSB does not directly apply harmonization between two original data domains. We will add this FID evaluation to the final paper. \n\n| | |Spectralis|to|Cirrus| | |Cirrus|to|Spectralis| |\n|:-:|:-:|:-:|:-:|:-:|:-:|:-:|:-:|:-:|:-:|:-:|\n| |DDIB|DSBM|DSBM|LMSB|LMSB|DDIB|DSBM|DSBM|LMSB|LMSB|\n| |ODE|SDE|ODE|SDE|ODE|ODE|SDE|ODE|SDE|ODE|\n|Before|140.53|140.53|140.53|140.53|140.53|140.53|140.53|140.53|140.53|140.53|\n|After|48.48|18.48|24.26|**17.24**|25.34|50.97|34.99|**31.39**|41.72|32.97|"}]

Author rebuttal:

Meta-review: (a) Scientific Claims and Findings:
This paper addresses the anatomy-shift problem in medical image harmonization, which arises when using Schrödinger Bridge (SB) or entropy-regularized optimal transport directly on images. The authors propose a novel anatomy-guided latent metric Schrödinger Bridge (LMSB), which introduces an invertible neural network to map optical coherence tomography (OCT) images into a latent Euclidean space guided by anatomical segmentation maps. Key findings: LMSB reduces anatomical distortions while harmonizing OCT images across devices. It achieves superior anatomical consistency compared to DDIB and DSBM baselines. Quantitative and qualitative results demonstrate improved fidelity in preserving anatomical structures. The method is theoretically generalizable to other modalities such as MRI and ultrasound.

(b) Strengths: 
1. Novelty and significance: Introduces the concept of an anatomy-guided latent metric in SB for harmonization, directly tackling a long-standing weakness of prior approaches.
2. Clear motivation: Identifies and corrects the flawed assumption in standard OT/SB (Euclidean closeness of same-anatomy images), making the contribution well-founded.
3. Methodological rigor: Strong combination of theory, algorithmic innovation, and empirical validation.
4. Comprehensive evaluation: Includes theoretical discussion, quantitative experiments (MAE, LNCC, FID), and qualitative analysis on a sizeable OCT dataset.
5. Clinical relevance: Preserves anatomy critical for downstream clinical tasks, with medical experts involved in the validation process.
6. Generality: Framework is extensible beyond OCT, applicable to other medical imaging modalities.

(c) Weaknesses and Missing Elements:
1. Limited modality scope: Demonstrations are restricted to retinal OCT; broader empirical validation (e.g., MRI, ultrasound) would strengthen claims.
2. Metric limitations: Anatomical metrics (MAE, LNCC) dominate evaluation; while FID was added in rebuttal, additional perceptual or task-driven metrics would be useful.
3. Preprocessing reliance: Some dependence on resizing/cropping steps, which may be more complex in other modalities such as ultrasound.
4. Anatomy label map encoding: Current simple numbering scheme for layers may not optimally capture inter-layer relationships.
5. Paired validation: Registration-based paired scan evaluation (e.g., PSNR/SSIM) is missing due to technical challenges.

(d) Reasons for Acceptance
The paper makes a clear conceptual advance: introducing anatomy-guided latent metrics into Schrödinger Bridge models to mitigate anatomical shifts. It provides convincing empirical results on a substantial OCT dataset, with statistical validation (t-tests) and complementary metrics (FID) confirming robustness. The work has high potential impact in clinical imaging, where anatomical fidelity is crucial. The methodology is general and extensible, making it relevant beyond the specific OCT case. The rebuttal addressed all reviewer concerns thoroughly, strengthening confidence in the results and claims.

Overall, this is a well-rounded contribution that advances both theory (metric design in OT/SB) and practice (harmonization for medical imaging).

(e) Rebuttal Period and Discussion

Reviewers raised several points:
1. Statistical validity of results: Authors added one-sided paired t-tests, confirming statistical significance.
2. Generality to other modalities: Authors argued LMSB is theoretically generalizable; challenges in preprocessing were acknowledged and discussed.
3. Ultrasound variability: Clarified as a preprocessing issue, not a limitation of LMSB itself.
4. Anatomical complexity encoding: Authors acknowledged current label encoding is suboptimal and promised elaboration.
5. Computational resources: Authors provided detailed GPU usage and clarified feasibility on a single GPU.
6. Assumptions of OT/SB: Authors reiterated why Euclidean distances are problematic and how LMSB corrects this.
7. Evaluation metrics: Added FID to measure domain consistency; results show LMSB competitive with baselines while preserving anatomy better.
8. Dataset splitting and contamination concerns: Authors clarified subject-level train/test splits with no leakage.
9. Preprocessing details: Authors provided full crop/resize pipeline and original image resolutions.
10.Expert validation: Noted two medical doctors contributed throughout development.
All concerns were addressed convincingly, and reviewers unanimously agreed on acceptance after rebuttal.

The paper introduces a principled and impactful method with clear clinical and methodological contributions.
Decision: Accept (poster)