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CNN-Based Projected Gradient Descent for Consistent Image Reconstruction

Harshit Gupta, Kyong Hwan Jin, Ha Q. Nguyen, Michael T. McCann, Michael Unser

arXiv:1709.01809v1cs.CV

TL;DR

Direct CNN regressors can reconstruct inverse-problem images efficiently but lack feedback enforcing consistency with measurements, a concern in biomedical imaging. The paper replaces PGD’s projector with a CNN and combines CNN prior injection with gradient-based consistency enforcement in a relaxed, convergent scheme. In sparse-view CT, RPGD outperforms TV and a recent CNN-based technique for noiseless and noisy measurements.

  • Problem

    CNN-based image regressors lack a feedback mechanism and assurance that reconstructions are consistent with measurements, which matters for diagnostic biomedical imaging.

  • Method

    RPGD alternates gradient-based measurement-consistency enforcement with a CNN trained to act as a projector onto the desired image-data set.

  • Results

    In sparse-view CT, RPGD outperforms TV and previous techniques for both noiseless and noisy measurements.

  • Takeaways & Limitations

    The framework combines measurement consistency and learned image priors, with potential application to super-resolution, deconvolution, and accelerated MRI.

Abstract

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We present a new method for image reconstruction which replaces the projector in a projected gradient descent (PGD) with a convolutional neural network (CNN). CNNs trained as high-dimensional (image-to-image) regressors have recently been used to efficiently solve inverse problems in imaging. However, these approaches lack a feedback mechanism to enforce that the reconstructed image is consistent with the measurements. This is crucial for inverse problems, and more so in biomedical imaging, where the reconstructions are used for diagnosis. In our scheme, the gradient descent enforces measurement consistency, while the CNN recursively projects the solution closer to the space of desired reconstruction images. We provide a formal framework to ensure that the classical PGD converges to a local minimizer of a non-convex constrained least-squares problem. When the projector is replaced with a CNN, we propose a relaxed PGD, which always converges. Finally, we propose a simple scheme to train a CNN to act like a projector. Our experiments on sparse view Computed Tomography (CT) reconstruction for both noiseless and noisy measurements show an improvement over the total-variation (TV) method and a recent CNN-based technique.

I. INTRODUCTION

Medical image reconstruction is an ill-posed inverse problem, especially under sparse, noisy measurements. Existing iterative and learning-based approaches incorporate priors, but direct CNN reconstructions lack guaranteed measurement consistency and feedback.

  • Motivation: Medical imaging can potentially reduce radiation dose or accelerate acquisition without compromising image quality through reconstruction algorithms that incorporate image priors.The paper formulates reconstruction using a forward model H and measurements y = Hx+n.
  • Motivation: Extreme imaging uses fewer, noisier measurements than unknown image values, creating a need for denoising and prior-guided reconstruction.The measurement count may be smaller than the number of unknowns, producing an ill-posed problem with many consistent images.
  • 1) Classical Algorithms:: Classical FBP-type methods are fast and effective with sufficient measurements and low noise but introduce artifacts in extreme imaging scenarios.Their artifacts are tied to the inversion step.
  • 2) Iterative Algorithms:: Variational methods balance measurement fidelity and image regularization, but convexity requirements restrict the prior knowledge that can be injected.The denoised measurement is y* = Hx*, and the selected reconstruction minimizes the regularizer among solutions admitting y*.
  • 2) Iterative Algorithms:: Plug-and-play ADMM alternates a measurement-consistency solver with prior reinjection, but the approach remains without theoretical justification in the cited discussion.Its nonlinear step has an effect akin to denoising.
  • 3) Learning-based Algorithms:: Direct CNN regressors learn image structure from examples but provide no worst-case performance guarantee or assurance that reconstructions remain measurement-consistent.Unlike iterative schemes, they lack feedback that imposes consistency, which raises concerns for diagnostic imaging.

A. Contribution

The paper combines CNN-based image priors with iterative projected gradient descent to enforce both learned-image structure and measurement consistency. It also develops a relaxed scheme designed to converge when the CNN is only an approximate projector.

  • A. Contribution: The CNN acts as a projector onto a set S that can be interpreted as the manifold of desired biomedical images.Its output is intended to have a stronger structural fit to the training data than its input.
  • A. Contribution: The relaxed PGD modifies the iteration because an approximate CNN projector may otherwise cause divergence.The relaxation parameters α_k are updated to ensure convergence, including when projector behavior is only approximate on unseen data.
  • A. Contribution: The proposed reconstruction alternates data-fidelity minimization with CNN projection to iteratively enforce measurement consistency and inject learned prior knowledge.The CNN is trained to map input images toward a set S representing desired images, while gradient descent addresses the data-fidelity term.
  • A. Contribution: Unlike direct CNN regression, the iterative scheme retains a feedback mechanism that reinforces consistency with the measurements.Direct regression approaches are described as lacking this feedback mechanism, while the proposed method alternates learned projection with data-consistency updates.
  • A. Contribution: The paper positions CNN-based PGD as an alternative to more complicated learned iterative or adversarial projector approaches.The authors contrast their simpler analysis and training scheme with unrolled methods and GAN-based projectors requiring extremely large datasets.

B. Constrained Least Squares

The reconstruction is formulated as constrained least squares over a set of desired images, combining measurement fidelity with strong prior knowledge. The theory characterizes local minimizers through fixed points and gives convergence conditions for classical and relaxed PGD.

  • B. Constrained Least Squares: The constrained least-squares formulation restricts reconstructions to a set S containing objects of interest, thereby encoding prior knowledge.This constraint serves as a regularizer and complements the measurement model.
  • B. Constrained Least Squares: When measurements are noisy, the constrained solution produces a point in S whose predicted measurements are as close as possible to the observations.The corresponding y* is treated as a denoised measurement.
  • B. Constrained Least Squares: For non-convex S, fixed points of the combined gradient-projector operator can characterize local minimizers under a suitable projector condition.If the condition holds globally, every fixed point is a solution of the constrained problem.
  • B. Constrained Least Squares: Orthogonal projection onto a finite union of closed convex sets satisfies the relaxed projector condition, whereas the stronger condition implies convexity of S.This broadens the applicable projector class beyond ordinary convex-set projections.
  • B. Constrained Least Squares: If the projector is Lipschitz with L < (λmax + λmin)/(λmax −λmin) and γ = 2/(λmax + λmin), PGD converges to a local minimizer regardless of initialization.The combined operator is then contractive and has a unique fixed point.
  • B. Constrained Least Squares: Because projectors cannot generally be contractive, the relaxed averaged iteration provides convergence when the projector is non-expansive and the combined operator has a fixed point.For γ < 2/λmax and any α ∈ (0, 1), the sequence converges to a local minimizer regardless of initialization.

III. RELAXATION WITH GUARANTEED CONVERGENCE

RPGD replaces the exact projector with a general nonlinear operator and relaxation parameters, guaranteeing convergence even when the CNN is not constrained to satisfy the projector’s Lipschitz condition.

  • Relaxed projected gradient descent: RPGD replaces the projector with a general nonlinear operator F and adds sequences {c_k} and {α_k} to control convergence and relaxation.The reconstruction update uses relaxed combinations of the current iterate and the nonlinear operator output.
  • Projector case: If F is an actual projector and relaxation parameters do not vanish, RPGD converges to a meaningful point.The projector-specific case strengthens the interpretation of the converged reconstruction.
  • Convergence guarantee: When {c_k} is asymptotically upper-bounded by C < 1, RPGD converges for every choice of F.This is the unconditional convergence statement of Theorem 3.
  • Convergence guarantee: If F is continuous and relaxation parameters remain above ε > 0, the limit x* is a fixed point of the iteration.Continuity is automatic when F is a CNN.
  • Motivation: Classical PGD converges to a local minimizer under a Lipschitz condition, but that condition is difficult to enforce on a trained CNN.Theorem 1 requires a Lipschitz constant below a bound that may be infeasible for projectors when H has a nontrivial null space.

IV. TRAINING CNN AS A PROJECTOR

The CNN is trained to behave like a projector by mapping perturbed samples toward training images in the desired image set. Multiple perturbation ensembles promote identity behavior, initialization compatibility, and training diversity.

  • Projector objective: For x in the target set S, projector training requires the CNN to map perturbed versions of x back toward x.The training set contains points drawn from S and an ensemble of N × Q perturbed points.
  • Perturbation design: The perturbation ensemble is designed to be diverse so the CNN does not overfit a single perturbation type.Perturbations are selected during training to broaden the inputs seen by the network.
  • Perturbation design: The perturbation based on A H x_q matches RPGD initialization with A y_q, while training only on it reproduces the CNN method of.Here A is a classical linear reconstruction algorithm such as FBP or BP.
  • Projector objective: Including unperturbed training points enforces the projector identity property; omitting them would instead yield an autoencoder.The defining property is P_S x = x for x ∈ S.
  • Perturbation design: The CNN’s own changing output is a dynamic perturbation that supplies T new perturbations per training point with only Q additional gradient computations per epoch.This perturbation is synchronized with the iterative reconstruction scheme.

A. Architecture

The method uses a residual U-net CNN and evaluates it on sparse-view, low-dose CT with clinically realistic abdominal scans, angular mismatch, and measurement noise.

  • Architecture: The CNN is a residual U-net with intrinsic layer skip connections and an extrinsic input-output skip connection.Its residual form is CNN = Id + Unet.
  • Training schedule: The three-stage training procedure progressively adds dynamic and identity perturbation ensembles before optimizing the full loss.Stages use J2, then J2 + J3, and finally J1 + J2 + J3.
  • CT setup: Sparse-view CT measurements are generated with the Radon transform, using 729 offsets per view and either 144 or 45 views.These settings correspond to ×5 and ×16 dosage reductions relative to 720 full views.
  • Dataset: The dataset contains 475 training images from nine patients and 25 test images from a separate patient, all from clinically realistic in vivo abdominal CT scans.The scans come from the AAPM Low Dose CT Grand Challenge dataset.
  • Noise and mismatch: Experiments introduce 0.05-degree angular AWGN perturbations and additional Gaussian sinogram noise to model forward-model mismatch and noisy measurements.The perturbations create a mismatch between the actual measurement process and H.
  • Evaluation: Performance is measured using regressed SNR, allowing scale and DC-offset correction before comparing the reconstruction with ground truth.The scalars a and b remove effects considered practically unimportant.

B. Comparison Methods

The study compares classical, variational, feed-forward CNN, and iterative CNN reconstructions under sparse-view CT conditions. RPGD is strongest in low-noise settings and remains competitive under higher noise.

  • Comparison methods: The comparison includes FBP, TV, FBPconv, and RPGD, with RPGD using the CNN trained as a projector inside Algorithm 1.FBP is direct inversion, TV uses constrained optimization, and FBPconv applies a CNN to FBP measurements.
  • Parameter selection: RPGD hyperparameters use C = 0.99, while RPGD40 uses C = 0.8; γ is tuned separately for each noise and dosage condition.One GPU iteration takes less than one second.
  • Evaluation: The reported SNRs are averaged over 25 test images, with separate low- and high-noise method variants.Low noise uses FBPconv and RPGD; high noise uses FBPconv40 and RPGD40.
  • Low-noise results: RPGD outperforms all other methods for both ×5 and ×16 dosage reductions in low-noise cases.The reconstruction SNRs are averaged over 25 test images.

FBPconv SNR 25.92

RPGD combines measurement-consistency updates with CNN-based projection and outperforms TV and FBPconv across sparse-view CT settings, including noisy measurements.

  • RPGD outperforms TV and FBPconv in image quality and measurement consistency for the ×16, no-noise case.The averaged sinogram SNRs are approximately 47 dB for TV, 57 dB for FBPconv, and 62 dB for RPGD.
  • 62 dB is the approximate reconstructed-sinogram SNR for RPGD in the ×16, no-noise case, compared with 47 dB for TV and 57 dB for FBPconv.
  • RPGD40 outperforms competing methods in low-view noisy cases and is more consistent than them in high-view cases.The compared dosage reductions are ×16 and ×5.
  • RPGD40 exceeds FBPconv40 by around 3.5 dB at 45-dB noise and 6 dB at 35-dB noise in the ×16 case.FBPconv40 degrades significantly when testing noise differs from its 40-dB training condition, whereas RPGD40 is more robust to this difference.
  • The proposed method alternates measurement-consistency enforcement with CNN projection and is reported to converge while using a training set of 475 images.The framework is proposed for sparse-view CT and is suggested for other inverse problems including super-resolution, deconvolution, and accelerated MRI.

APPENDIX

The appendix establishes convergence by showing that the relaxed iterates form a Cauchy sequence and converge to a fixed point, which can correspond to a local or global constrained solution.

  • The proof of convergence shows that the iterate differences contract asymptotically, making the sequence Cauchy in R^N.Completeness of R^N then guarantees convergence to some point x∗.
  • If the relaxation factors remain bounded below, their monotone sequence converges to a positive limit.
  • The iteration converges to a fixed point of Gγ when the nonlinear operator F is continuous.Continuity of F implies continuity of Gγ, allowing the limit to pass through the update equation.
  • A fixed point of Gγ is a local minimizer of the constrained least-squares problem when the projector satisfies condition (6).
  • When the stronger projector condition (7) holds, the fixed point is a solution of the constrained problem rather than only a local minimizer.

B. Proof of Proposition 2

The proof shows that the relevant projection condition forces the constraint set to be convex, and establishes the condition for finite intersections of closed convex sets.

  • Assuming a non-convex set produces a convex combination outside the set, contradicting the projection condition and forcing convexity.
  • For a finite intersection of closed convex sets, the orthogonal projection onto the intersection is also the projection onto each component set.
  • If some component sets do not contain the projected point, a sufficiently small neighborhood excludes their intersection from the local analysis.
  • Intersecting the relevant component sets with that neighborhood preserves convexity and establishes condition (6) locally.

D. Proof of Theorem 1

The theorem proves convergence of the unrelaxed iteration by showing that Gγ is contractive, yielding a unique fixed point that is also a local minimizer.

  • The contraction threshold is expressed using the largest and smallest eigenvalues of H^T H.
  • When L < (λmax + λmin)/(λmax − λmin), Gγ is contractive and the iterates converge from every initialization.
  • The contraction guarantees a unique fixed point x∗ of Gγ.
  • Because the projector satisfies condition (6), the unique fixed point is also a local minimizer of the constrained least-squares problem.

E. Proof of Theorem 2

Under γ < 2/λmax, the proof establishes that Gγ is non-expansive. The Krasnosel’skiĭ-Mann theorem then yields convergence to a fixed point that is also a local minimizer.

  • The proof assumes γ < 2/λmax, where λmax is the largest eigenvalue of H^T H.
  • The eigenvalue bound shows that the gradient-related operator is non-expansive under this step-size condition.
  • Combining this bound with the non-expansiveness of P_S establishes the corresponding distance inequality for all x, z ∈ R^N.
  • Because Gγ is non-expansive and has a nonempty fixed-point set, the Krasnosel’skiĭ-Mann theorem applies to iteration (8).
  • Iteration (8) converges to a fixed point of Gγ, which Proposition 1 identifies as a local minimizer of (3).
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