Measurement-conditioned Denoising Diffusion Probabilistic Model for Under-sampled Medical Image Reconstruction

1 Introduction

• Under-sampled medical reconstruction spans accelerated MRI, sparse-view or limited-angle CT, and digital breast tomosynthesis; most prior work seeks one sample from the posterior p(x | y).Here, x denotes the target image and y the under-sampled measurements.Source: §1 · p. 1
• DDPM and score-based models generate samples by reversing diffusion, and have been used for unconditional image generation and conditional tasks such as inpainting and super-resolution.The introduction also notes applications to audio, graphs, and image editing.Source: §1 · pp. 1–2
• MC-DDPM defines diffusion and sampling in measurement space, conditions diffusion on the under-sampling mask, and reports better evaluation metrics than baseline models and Chung et al.’s score-based method.The method can sample multiple reconstructions from the same measurements, enabling pixel-variance estimates of uncertainty.Source: §1 · p. 2, Figure 1 · p. 2

2.1 Denoising Diffusion Probabilistic Model

• DDPM uses a Markov chain with learned Gaussian transitions from a standard Gaussian latent variable to data, reversing a forward process that gradually adds Gaussian noise.Its variance schedule is fixed so that q(xT | x0) is approximately N(0, I).Source: §2.1 · pp. 2–3
• Sampling starts at xT from N(0, I) and repeatedly samples xt−1 until x0 is obtained; the forward process adds Gaussian noise according to a fixed variance schedule.The supplied background identifies β1, ..., βT as schedule constants.Source: §2.1 · p. 3

2.2 Under-sampled Medical Image Reconstruction

• The forward model maps an image x ∈ R^n to under-sampled measurements y ∈ R^m, with m < n, using a measuring matrix, a sampling pattern, and noise.A is usually invertible; PΩ represents the under-sampling pattern, and the CT example uses a Radon transform and limited-angle sinogram.Source: §2.2 · p. 3
• The reconstruction task is to estimate the posterior distribution of the image given the measurements and sampling matrix.The text notes that PΩ is usually omitted from the posterior notation.Source: §2.2 · p. 3

3 Method: Measurement-conditioned DDPM

• MC-DDPM estimates missing measurements conditioned on the sampling mask and observed measurements, then combines them with the known measurements to reconstruct the image.The method models the posterior over non-sampled measurements rather than reconstructing directly in the image domain.Source: §3 · pp. 3–4
• Diffusion adds Gaussian noise to non-sampled measurements while keeping sampled positions zero; the reverse process generates the missing measurements.The covariance confines noise to non-sampled positions.Source: §3 · p. 4
• Training optimizes a variational bound on negative log likelihood, with a reweighted simplified objective used in the training procedure.The paper presents the complete training procedure in Algorithm 1.Source: §3 · pp. 4–5
• Sampling iteratively generates the non-sampled measurements, then applies the inverse measuring transform to the sum of generated and observed measurements.Algorithm 2 returns the reconstructed image from the combined measurements.Source: §3 · p. 5
• Multiple posterior samples enable pixel-variance estimates to quantify reconstruction uncertainty.The variance is computed using a Monte Carlo approach.Source: §3 · p. 5

4.1 Experimental Setting

• Experiments use fastMRI single-coil PD and PDFS knee data, with 320 × 320 complex images, and compare PSNR/SSIM against U-Net and the score-based method.The MRI operator is a 2D Fourier transform; the model uses magnitude images for final reconstructions and pixel-wise variance.Source: §4.1 · p. 5, §4 · p. 5

4.2 Experimental Results

• For PD data at 4× and 8× acceleration, the proposed method’s mean of 20 reconstruction samples performs best versus zero-filled reconstruction and U-Net, showing more realistic structures and less zoomed-in error.Source: §4.2 · p. 5
• MC-DDPM outperforms the score-based method by 3.62/0.077 (4×) and 2.96/0.089 (8×) in PSNR/SSIM, using average U-Net-relative enhancements because testing volumes differ.The study also reports that MC-DDPM performs best against zero-filled reconstruction and U-Net in the displayed PD results at both acceleration factors.Source: §4.2 · pp. 5–6, Table 1 · p. 6, Figure 2 · p. 6
• On one PDFS volume, reducing sampling steps to 250 only slightly reduces PSNR, while increasing the sample count improves mean-image quality and appears to converge.The authors suggest 20 samples with 250 sampling steps as a possible efficiency-conscious choice.Source: §4.2 · p. 6, Figure 3 · p. 7, Figure 4 · p. 7

4.3 Discussion

• Under-sampling commonly reduces imaging cost or dosage; the authors suggest MC-DDPM may extend beyond MRI, while further investigation of its variance schedule is planned.Limited-angle or sparse-view CT are named as possible applications, and the schedule {βt} is identified as a hyperparameter potentially related to sampling quality and efficiency.Source: §4.3 · p. 7

• MC-DDPM is presented as a unified mathematical framework for medical image reconstruction from under-sampled measurements.Source: §5 · p. 7
• Its diffusion process operates in the measurement domain and is conditioned on the under-sampling mask.Source: §5 · p. 7
• The framework provides an uncertainty estimate, and its performance is demonstrated on accelerated MRI reconstruction.Source: §5 · p. 7

7.1 Derivation of Eq. 8, Eq. 9 and Training Objective in Sect. 3

• The derivation restricts the reverse conditional to non-sampled coordinates and obtains a reweighted MC-DDPM training objective from the variational bound.It derives q(yMc,t−1 | yMc,t, yMc,0, M, yM) and describes the final objective as a reweighted Lvlb.Source: §7.1 · pp. 10–12

7.2 More Details of Experiments

• Experiments use a U-Net with time embeddings and self-attention, batch size 48 for 35k steps, and a cosine schedule with β_t multiplied by 0.5.The first and last five slices are dropped to avoid noise-only training data; testing uses six randomly selected validation volumes for PD and PDFS.Source: §7.2 · p. 12

7.3 More Experimental Results

• Additional volume- and slice-level tables report PSNR, SSIM, NMSE, and MSE, but the supplied captions do not state their metric values.Table 2 covers volumes and Table 3 covers slices.Source: §7.3 · p. 12, Table 2 · p. 12, Table 3 · p. 12
• Figures show 4× and 8× reconstructions on PD and PDFS; the 4× PD figure includes mean/std from 20 samples, five samples, and PSNR/SSIM.The captions specify that yellow numbers indicate PSNR and SSIM.Source: §7.3 · p. 12, Figure 5 · p. 13, Figure 6 · p. 14