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Unsupervised Medical Image Translation with Adversarial Diffusion Models

Imputation of missing images via source-to-target modality translation can improve diversity in medical imaging protocols. A pervasive approach for synthesizing target images involves one-shot mapping through generative adversarial networks (GAN). Yet, GAN models that implicitly characterize the image distribution can suffer from limited sample fidelity. Here, we propose a novel method based on adversarial diffusion modeling, SynDiff, for improved performance in medical image translation. To capture a direct correlate of the image distribution, SynDiff leverages a conditional diffusion process that progressively maps noise and source images onto the target image. For fast and accurate image sampling during inference, large diffusion steps are taken with adversarial projections in the reverse diffusion direction. To enable training on unpaired datasets, a cycle-consistent architecture is devised with coupled diffusive and non-diffusive modules that bilaterally translate between two modalities. Extensive assessments are reported on the utility of SynDiff against competing GAN and diffusion models in multi-contrast MRI and MRI-CT translation. Our demonstrations indicate that SynDiff offers quantitatively and qualitatively superior performance against competing baselines.


I. INTRODUCTION
Multi-modal imaging is key for comprehensive anatomical assessments in the human body [1]. Complementary tissue information captured by individual modalities serve to improve performance in downstream imaging tasks and diagnostic accuracy [2]. Unfortunately, broad adoption of multi-modal protocols is fraud with challenges due to economic and labor costs [3]- [6]. Medical image translation is a powerful solution that involves synthesis of a missing target modality under guidance from an acquired source modality [7]- [10]. This recovery is an ill-conditioned problem given nonlinear variations in tissue information across modalities [11]- [13]. At this juncture, learning-based methods are offering performance leaps by incorporating nonlinear data-driven priors to improve problem conditioning [14]- [17]. Learning-based synthesis involves network models that are trained to capture a prior on the conditional distribution of target given source images [18]- [20]. In recent years, generative adversarial network (GAN) models have been broadly adopted for translation tasks, given their exceptional realism in image synthesis [21]- [26]. A discriminator that captures information regarding the target distribution concurrently guides a generator to perform one-shot mapping from the source to the target image [27]- [31]. Based on this adversarial mechanism, state-of-the-art results have been reported with GAN-based translation in numerous tasks including synthesis across MR scanners [23], multi-contrast MR synthesis [21], [25], [27], [32], and CT synthesis [33], [34].
Although demonstratedly powerful, GAN models capture distributional information about the target modality indirectly through an interplay between the generator and discriminator subnetworks [35]. Thus, they provide an implicit characterization of the target distribution, without evaluating likelihood [36]. Such implicit characterization is amenable to learning biases including premature convergence of the discriminator and mode collapse, resulting in poor quality and diversity in synthesized images [37]. Recent computer vision studies have adopted diffusion processes as a promising alternative to GANs to improve sample diversity in generative modeling tasks [37], [38]. Yet, the potential of diffusion methods in medical image translation remains largely unexplored, partly owing to the substantial computational burden of image sampling in regular diffusion models [38].
Here, we propose a novel adversarial diffusion model for medical image synthesis, SynDiff, to perform efficient and high-fidelity modality translation (Fig. 1). Given the source image, SynDiff leverages conditional diffusion to generate the target image. Unlike regular diffusion models, SynDiff adopts a fast diffusion process with large step size for efficiency. Accurate sampling in reverse diffusion steps is achieved by a novel adversarial projector. To enable unsupervised learning, a cycle-consistent architecture is devised with bilateral diffusion processes between the source and target modalities (Fig.  2). Comprehensive demonstrations are performed for multicontrast MRI and MRI-CT translation. Our analyses clearly indicate the superiority of SynDiff against competing GAN and diffusion models. Code to implement SynDiff will be publicly available at https://github.com/icon-lab/SynDiff.

Contributions
• We introduce the first adversarial diffusion model for medical image synthesis to perform modality translation. • We propose an adversarial projector to capture reverse arXiv:2207.08208v1 [eess.IV] 17 Jul 2022 Fig. 1: a) Regular diffusion models gradually transform between actual image samples (x 0 ) and isotropic Gaussian noise (x T ) in several hundreds of time steps. Each forward step adds noise to the current image sample, whereas each reverse step denoises the image sample. A neural network is used for denoising in reverse diffusion steps. b) The proposed adversarial diffusion model instead performs fast transformation in few steps. Each forward step adds a greater amount to noise in correspondence with the larger step size. An adversarial projector is used for improved accuracy during the reverse diffusion process. A diffusive generator synthesizes a denoised image sample, while a discriminator distinguishes between actual and synthetic samples for the denoised image. transition probabilities over large step sizes for accelerated diffusion and thereby efficient image sampling. • We introduce a cycle-consistent architecture with bilateral diffusion processes to enable training on unpaired datasets of source and target modalities.

II. RELATED WORK
For medical image translation, conditional GAN models use a generator to perform one-shot source-to-target mapping, where the generator is trained via an adversarial loss computed with the help of a discriminator [23]. Adversarial loss terms are known to improve sensitivity to high-frequency details in tissue structure over canonical pixel-wise losses [21]. As such, GAN-based translation has been broadly adopted in many imaging applications. Augmenting adversarial loss with pixelwise losses, a first group of studies considered paired synthesis on training sets of source-target images matched across subjects [24], [26]- [30]. Other studies have adopted cycleconsistency or mutual information losses to enable learning from unpaired training sets for improved flexibility [21], [33], [39]- [44]. In general, enhanced spatial acuity and realism has been reported in target images synthesized with GAN models when compared to traditional or convolutional models [21]. That said, several well-known problems can potentially arise in GAN models including premature convergence of the discriminator before the generator is properly trained [31], and poor representational diversity due to mode collapse [35]. In turn, these problems can lower sample quality and diversity, thereby limiting generalization performance of GAN-based image translation.
As a recent alternative to GAN models, deep diffusion models have received interest for generative modeling tasks in computer vision [37], [38]. Starting from a pure noise sample, diffusion models generate image samples from a desired distribution through repetitive denoising across a large number of steps. Denoising is performed via a neural network architecture that is trained to maximize a correlate on data likelihood. Due to the gradual stochastic sampling process and explicit likelihood characterization, diffusion models can offer enhanced sample quality and diversity. Given this potential, diffusion-based methods have recently been adopted for medical image reconstruction [45]- [47] and denoising [48] tasks. Yet, these methods are characteristically based on unconditional diffusion processes devised to enhance the quality of single-modality images. Furthermore, prior methods are based on regular diffusion models that rely on a large number of diffusion steps for accurate image generation. This prolonged sampling process results in substantial computational burden during both training and inference with diffusion models.
In this study, we propose a novel adversarial diffusion model for improved efficiency and reliability during medical image translation. Unlike prior diffusion methods for reconstruction and denoising, SynDiff leverages a conditional diffusion model that receives a source-contrast image to anatomically guide the reverse diffusion process. Unlike regular diffusion models, SynDiff achieves efficient synthesis of target-contrast images via a fast diffusion process with large step size. An adversarial projector is introduced to maintain accuracy in reverse diffusion steps. Lastly, a novel cycle-consistent architecture is introduced to permit training on unpaired datasets. To our knowledge, SynDiff is the first adversarial diffusion model for medical image synthesis, and the first cycle-consistent diffusion model for unsupervised medical image translation in the literature.

A. Denoising Diffusion Models
Diffusion models map between pure noise samples and actual images through a bidirectional process over T time steps (Fig. 1a). The forward process involves repeated addition of Gaussian noise onto an input image x 0 ∼ q(x 0 ) to obtain a sample x T from an isotropic Gaussian distribution for sufficiently large T . The forward process is a Markov chain where the mapping from x t−1 to x t and the respective forward transition probability are: where β t denotes noise variance. The reverse diffusion process is also a Markov chain from x T onto x 0 , albeit each step aims to gradually denoise the samples. Under large T and thereby small β t , the reverse transition probability between x t−1 and x t can be approximated via a Gaussian distribution [49], [50]: Common diffusion models operationalize each reverse diffusion step as mapping through a neural network that provides estimates for µ and/or Σ. Training is then performed by minimizing a variational bound on log-likelihood: where E q denotes expectation over q, p θ is the network approximation to the reverse transition probability. The bound can be further decomposed as: where KL(q(x T |x 0 ) || p(x T )) is omitted since it does not depend on θ, and KL denotes Kullback-Leibler divergence.
A common parametrization focuses on µ while ignoring Σ: In Eq. 6, µ can be derived if the network estimates the added noise by minimizing the following loss [51]: where , x 0 and t are sampled from N (0, I), q(x 0 ) and U(0, T ), respectively. During inference, the reverse diffusion process is initiated with a random sample x T ∼ N (0, I). For each time step t ∈ T...1, µ is derived using Eq. 6 based on the network estimate θ , and x t−1 is sampled based on Eq. 3.

B. SynDiff
Here, we introduce a novel diffusion model for efficient, high-fidelity translation between source and target modalities of a given anatomy (Fig. 1b). SynDiff employs a fast conditional diffusion process with an adversarial projector for accurate reverse diffusion sampling over large step sizes. A cycle-consistent architecture is introduced to enable training on unpaired datasets of source-target images (Fig. 2). Working principles of SynDiff are detailed below.
1) Adversarial Diffusion Process: Regular diffusion models prescribe relatively large T such that the step size is sufficiently small to satisfy the normality assumption in Eq. 3, but this limits efficiency in image generation. Here, we instead propose fast diffusion with the following forward process: where k 1 is step size. The noise variance γ t is set as [52]: β min and β max denote the upper and lower bound on the noise variance in an exponential schedule.  Fig.1 requires anatomical guidance from a source image y B of modality B for the same anatomy, which is unavailable in the unpaired training set. Thus, SynDiff uses a non-diffusive generator to estimate a paired source imageỹ B for training purposes. a) The non-diffusive module comprises two generator-discriminator pairs to generate initial translation estimates for x A 0 →ỹ B (red) and x B 0 →ỹ A (teal). b) These initial translation estimatesỹ A,B are then used as guiding source-contrast images in the diffusive module. For cycle-consistency, the diffusive module also comprises two generator-discriminator pairs to generate denoised image estimates Guidance from a source image (y) is available during medical image translation, so a conditional reverse diffusion process is proposed. Note that, for k 1, there is no closed form expression for q(x t−k |x t , y) and the normality assumption used to compute Eq. 4 breaks down [38]. Here we introduce a novel adversarial projector to capture the complex transition probability q(x t−k |x t , y) for large k in our conditional diffusion model, as inspired by a recent report on natural image generation via unconditional diffusion models [53]. In SynDiff, a conditional generator G θ (x t , y, t) performs gradual denoising in each reverse step to estimatex t−k ∼ p θ (x t−k |x t , y). , y)). G θ is trained via a non-saturating adversarial loss [36]: where t ∼ U({0, k, ..., T }), and the discriminator arguments are abbreviated for brevity. D θ is also trained with a nonsaturating adversarial loss with gradient penalty [54]: where µ is the weight for the gradient penalty. Evaluation of Eqs. 11-12 require sampling from the unknown distribution q(x t−k |x t , y). Yet, Bayes' rule can be used to express it in terms of the forward transition probabilities: Using Eq. 8, it can then be shown that q(x t−k |x t , x 0 ) = N (x t−k ; µ(x t , x 0 ), γI) with the following parameters: where Eqs. 11-12 also require sampling from p θ (x t−k |x t , y). A trivial albeit deterministic sample would be the generator output, i.e.x t−k ∼ δ(x t−k − G θ (x t , y, t)). To maintain stochasticity, we instead operationalize the generator distribution as: θ denotes recursive projections through the generator. The main difference between Eqs. 13 and 16 is that the former is based on actual images at t = 0, whereas the latter is based on denoised images output by the generator. Following recursive sampling across multiple reverse diffusion steps, the eventual denoised image will be obtained via samplinĝ x 0 ∼ p θ (x 0 |x k , y) .
2) Cycle-Consistent Architecture: The ability to train on unpaired datasets of source and target modalities improves flexibility in learning synthesis models [21]. Note, however, that implementation of the diffusive generator in Eq. 16 requires paired source-target images within the same subjects. To enable unpaired training, we introduce cycle consistency in SynDiff to bilaterally translate between the two modalities (A, B). Because training images x A,B 0 are unmatched, we first employ two non-diffusive generators with parameters φ A,B to obtain initial translation estimatesỹ A,B [55]: These initial estimates are then used to condition two diffusive generators with parameters θ A,B in SynDiff: wherex A,B 0 are deterministic estimates for denoised images. For each modality, the correspondingx 0 can be used to sample from p θ (x t−k |x t , y) as described in Eq. 16. To express cycleconsistency loss,x 0 sampled from the diffusive generators can be compared against x 0 . Meanwhile, bilateral projections through non-diffusive generators are needed: The first two terms in Eq. 19 express cycles formed by nondiffusive generators; and they are critical to produce reasonable estimates of source images to condition the diffusive generators, especially during the initial phases of training. In contrast, the last two terms express cycles formed by diffusive generators; and they enforce SynDiff to better preserve the anatomical structure in the source contrast.
To prevent identity mapping in non-diffusive cycles, a nonsaturating adversarial loss similar to Eq. 11 is also adopted for the non-diffusive generators by including corresponding discriminators: φ receives x B 0 orỹ B as input. The overall loss for training generators is then given as: where λ is the regularization weight for cycle-consistency loss. Meanwhile, the non-diffusive discriminators are trained based on a non-saturating loss similar to Eq. 12, albeit omitting the gradient penalty. The overall discriminator loss is then:

A. Datasets
We demonstrated SynDiff for image synthesis on multi-contrast brain MRI data (IXI: https://braindevelopment.org/ixi-dataset/) and a multi-modal pelvic MRI-CT dataset [57]. Each analyzed dataset was three-way split to create training, validation and test sets with no subject overlap.

B. Competing Methods
We demonstrated Syndiff against several state-of-the-art diffusion and GAN models, all implemented for unpaired translation between source and target modalities. For each model, hyperparameter selection was performed to maximize performance on the validation set. The selected parameters included number of training epochs, learning rate for the optimizer, loss-term weightings for all models, along with number of time steps in diffusion models. 1) SynDiff: SynDiff was implemented with the following cross-validated hyperparameters: 50 epochs, 0.0001 learning rate, λ=0.5 relative weight for cycle-consistency loss, µ=0.5 relative weight for gradient penalty, T =1000, a step size of k=250, and a total of T k =4 diffusion steps. The lower and upper bounds on the noise variance schedule were adopted from [58] as β min =0.1, β max =20.
2) DDPM: A recent diffusion model proposed for improved sampling efficiency in image generation tasks was considered [58]. The architecture and loss functions were adopted from [58]. A conditional diffusion process was devised with guidance from the source-contrast image, and a cycle-consistent variant was implemented based on the same strategy as in SynDiff. Cross-validated hyperparameters were 50 epochs, 0.0001 learning rate, λ=1 relative weight for cycle-consistency loss, T =1000, a step size of k=1, and a total of 1000 diffusion steps. A noise schedule with β min =0.1, β max =20 was used.
3) cGAN: A cycle-consistent GAN model proposed for unsupervised medical image translation was considered [59]. The architecture and loss functions were adopted from [59]. cGAN comprised two generators with a ResNet backbone, and two discriminators with a cascade of convolutional blocks followed by instance normalization. Cross-validated hyperparameters were 100 epochs, 0.0002 learning rate constant for the first 50 epochs and linearly decaying to 0 afterwards, and λ cycle =100 relative weight for cycle-consistency loss. 4) UNIT: An unsupervised GAN model proposed for learning from unpaired source-target images was considered [60]. The architecture and loss functions were adopted from [60]. UNIT comprised two discriminators and two translators with a ResNet backbone in a cyclic setup. The translators contained parallel-connected domain image encoders and generators with a shared latent space. The discriminators contained a cascade of downsampling convolutional blocks. Cross-validated hyperparameters were 100 epochs, 0.0001 learning rate, λ r = 10, λ adv = 1, λ cycle = 10 relative weights for reconstruction, adversarial and cycle-consistency losses respectively.

C. Architectural Details
For cycle-consistent learning, SynDiff utilizes a diffusive and a non-diffusive module. The non-diffusive module comprises two generator-discriminator pairs to generate initial translation estimates on unpaired datasets. Meanwhile, the diffusive module also comprises two generator-discriminator pairs, one for each modality. During training, all four generator-discriminator pairs are simultaneously employed. During inference, only a single diffusive generator responsible for the desired source-to-target modality translation is used. The architectural details of diffusive and non-diffusive modules are provided below.
1) Non-Diffusive Module: Each generator in the nondiffusive module received a target-modality image as input, and it generated an estimate of the corresponding sourcemodality image of the same anatomy (which is unavailable in the training set). This initial estimate for source-modality was later used as anatomical guidance in the diffusive generators. A ResNet backbone was adopted with three encoding, six residual and three decoding blocks [61]. Meanwhile, the discriminators in the non-diffusive module served to prevent identity mapping during cycle-consistent training of the generators. Each discriminator received as input actual and synthetic source-modality images, unpaired across subjects. It then output a decision logit regarding whether the image sample is actual or synthetic. A convolutional backbone with six downsampling blocks was adopted. Each block contained two convolutional layers, and halved the feature map resolution.
2) Diffusive Module: Each generator in the diffusive module received as input a noisy image sample at time t and its time index along with a guiding source-modality image, and it output a denoised image sample at time t − k for the target modality. A UNet backbone was adopted with six downsampling and six upsampling blocks [62]. Each downsampling block used two residual subblocks followed by a convolutional layer to halve feature map resolution [61], while the channel dimensionality was doubled every other block. A temporal embedding was computed for each time index by projecting a 32dimensional sinusoidal position encoding through a two-layer multi-layer perceptron (MLP) [52]. The temporal embedding was added onto the feature maps in residual subblocks. Each upsampling block used two residual subblocks followed by a convolutional layer to double the feature map resolution, while the channel dimensionality was halved every other block. Meanwhile, each discriminator received as input either an actual or a synthetic sample for the denoised image at time t − k, time index and the noisy image sample at time t; and it output a decision logit regarding whether the image sample is actual or synthetic. A convolutional backbone with six downsampling blocks was adopted. Each block contained two convolutional layers, and halved the feature map resolution. An identical temporal embedding strategy to the diffusive generator was employed.

D. Modeling Procedures
All models were implemented in Python using the Py-Torch framework. Models were trained using the Adam optimizer with β 1 = 0.5, β 2 = 0.9. Models were executed on a workstation equipped with Nvidia RTX 3090 GPUs. Model performance was evaluated on the test set within each dataset. Performance was assessed via peak signal-to-noise ratio (PSNR), structural similarity index (SSIM) metrics. Prior to metric calculation, all images were normalized by their mean. Significance of performance differences between competing methods were assessed via non-parametric Wilcoxon signedrank tests (p<0.05). Fig. 3: SynDiff was demonstrated on the IXI dataset in a representative T 2 → T 1 synthesis task. Synthesized images from competing methods are displayed along with the source image and the ground-truth target image (reference). Compared to diffusion and GAN baselines, SynDiff achieves lower noise and artifact levels, along with a more accurate depiction of tissue structure. Fig. 4: SynDiff was demonstrated on the IXI dataset in a representative T 1 → T 2 synthesis task. Synthesized images from competing methods are displayed along with the source image and the ground-truth target image (reference). Compared to diffusion and GAN baselines, SynDiff achieves lower noise and artifact levels, along with a more accurate depiction of tissue structure.

A. Multi-Contrast MRI Translation
We first demonstrated SynDiff for unsupervised translation between separate MRI contrasts. SynDiff was compared against state-of-the-art diffusion (DDPM) and GAN models (cGAN, UNIT). Experiments were performed on brain images in the IXI dataset for T 1 → T 2 and T 2 → T 1 synthesis tasks. Quantitative metrics of synthesis performance are listed in Table I for competing methods. SynDiff yields the highest performance in both tasks (p<0.05). On average, SynDiff outperforms DDPM by 3.56dB PSNR and 3.80% SSIM, and GAN models by 1.05dB PSNR and 1.11% SSIM (p<0.05). Representative images are displayed for T 2 → T 1 in Fig. 3, and for T 1 → T 2 in Fig. 4. GAN models show residual noise and artifacts along with inaccurate depiction near tissue boundaries. DDPM shows a degree of spatial smoothing and residual artifacts. In comparison to baselines, SynDiff synthesizes target images with visibly lower noise and artifacts, and it achieves a high degree of anatomical accuracy in tissue depiction particularly manifest near diagnosticallyrelevant regions in CT images.

B. Cross-Modality Synthesis
We then demonstrated SynDiff for unsupervised translation between separate imaging modalities. SynDiff was compared against diffusion (DDPM) and GAN models (cGAN, UNIT). Experiments were performed on the pelvic dataset for the MRI → CT synthesis task. Quantitative metrics of synthesis performance are listed in Table II for competing methods. Overall, SynDiff yields the highest performance with 2.2dB higher PSNR and 4.7% higher SSIM than DDPM, and 1.28dB higher PSNR and 2.29% higher SSIM than GAN models (p<0.05). Representative images for MRI → CT are displayed in Fig. 5. In comparison to baseline methods, SynDiff synthesizes target images with lower noise and artifacts, and more Fig. 5: SynDiff was demonstrated on the multi-modal pelvic dataset in an MRI → CT synthesis task. Synthesized images from competing methods are displayed along with the source image and the ground-truth target image (reference). Compared to diffusion and GAN baselines, SynDiff achieves lower noise and artifact levels, along with a more accurate depiction of tissue structure. MRI

VI. DISCUSSION
GAN models have been widely adopted in medical image synthesis due to their exceptional realism, albeit they suffer from poor representational diversity that can limit their generalization performance. To alleviate this limitation, here we performed medical image translation based on adversarial diffusion modeling conditioned on a source modality. To our knowledge, this is the first adversarial diffusion model for medical image synthesis, and the first cycle-consistent diffusion model for unpaired medical image translation. We demonstrated SynDiff for missing modality synthesis in multicontrast MRI and MRI-CT tasks. We find that SynDiff outperforms state-of-the-art GAN models in unsupervised synthesis tasks in terms of qualitative and quantitative performance.
Regular diffusion models cast image generation as a gradual process starting at random noise and progressing with repeated denoising over thousands of time steps, resulting in poor sampling efficiency. In this study, we proposed to accelerate sampling over two orders-of-magnitude by leveraging a fast diffusion process with large step size. Increased step size between consecutive samples results in complex reverse transition probabilities, diverting away from the normality assumption typically used in diffusion models. To capture this complex distribution, we introduced an adversarial projector to perform accurate reverse diffusion sampling. A recent study in MRI reconstruction has proposed to initiate diffusion sampling not with random noise, but instead with an intermediate image [63]. For accelerated diffusion, the intermediate sample was taken as the image from a secondary reconstruction method. It may be possible to combine the benefits of our adversarial projector with initiation at an intermediate point based on a secondary synthesis method to gain further speed benefits during diffusion sampling.
There are several technical areas for further development that might help improve performance. Here, we considered synthesis tasks in which source and target modalities were unpaired across subjects. However, in some applications, paired datasets of source and target images from the same subjects might be available. In those cases, SynDiff can be adapted to perform paired training by substituting pixel-wise losses on spatially-registered images in place of cycle-consistency loss [21]. Performance improvements might also be viable by expanding the size of training datasets based on either a combination of paired and unpaired data [34], or collection of undersampled source-and target-modality acquisitions [64].
We primarily demonstrated SynDiff for one-to-one translation tasks between two modalities. It is possible that a single source modality may not contain sufficient information to reliably recover the target image. When multi-modality protocols are available, many-to-one translation can be performed to improve reliability in image translation [25]. To do this, SynDiff can be modified to include multiple source modalities as conditioning inputs to its generators [26], [28], [29], [65]. The generators in SynDiff that perform the reverse diffusion steps for denoising were based on convolutional backbones. Recent imaging studies have reported that transformer architectures with attention mechanisms offer improved sensitivity to long-range context in medical images during synthesis and beyond [65]- [67]. The strength and importance of contextual representations for progressive denoising remain to be demonstrated. Still, attention mechanisms might help enhance the generalization performance of SynDiff to atypical anatomy [68].

VII. CONCLUSION
Here we introduced a novel adversarial diffusion model for medical imaging to translate between source and target modalities. SynDiff leverages a fast diffusion process with large step size to conditionally synthesize target images, and an adversarial projector for accurate sampling in reverse diffusion steps. Unsupervised learning from unpaired data is achieved via a cycle-consistent architecture that embodies coupled diffusion processes for bilateral translation between the two modalities. SynDiff achieves superior translation quality compared to state-of-the-art GAN and diffusion models, so it holds great promise for high-fidelity medical image synthesis.