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External Prior Guided Internal Prior Learning for Real-World Noisy Image Denoising

Jun Xu, Lei Zhang, David Zhang

arXiv:1705.04505v2cs.CV

TL;DR

Real-world noisy images contain complex noise that simple Gaussian models struggle to describe, while external and noisy-image priors have complementary weaknesses. This paper learns clean-image external priors to guide adaptive internal priors, and reports better denoising performance than state-of-the-art methods across three real-world datasets.

  • Problem

    Real-world noise is complex and varies with scenes, cameras, and settings, while external priors may not adapt to an image and internal priors can be affected by noise.

    Source: §I · pp. 1–2

  • Method

    The method learns external priors from clean natural images and uses them to guide internal-prior learning, producing hybrid orthogonal dictionaries for denoising.

    Source: §I · p. 2, §V · p. 10

  • Results

    Experiments on three real-world noisy-image datasets demonstrated much better performance than state-of-the-art denoising methods.

    Source: §V · p. 10

  • Takeaways & Limitations

    Orthogonal dictionaries make patch encoding in the testing stage efficient, supporting an efficient denoising algorithm.

    Source: 2) Guided Orthogonal Dictionary Learning · p. 4

Abstract

from arXiv · show

Most of existing image denoising methods learn image priors from either external data or the noisy image itself to remove noise. However, priors learned from external data may not be adaptive to the image to be denoised, while priors learned from the given noisy image may not be accurate due to the interference of corrupted noise. Meanwhile, the noise in real-world noisy images is very complex, which is hard to be described by simple distributions such as Gaussian distribution, making real-world noisy image denoising a very challenging problem. We propose to exploit the information in both external data and the given noisy image, and develop an external prior guided internal prior learning method for real-world noisy image denoising. We first learn external priors from an independent set of clean natural images. With the aid of learned external priors, we then learn internal priors from the given noisy image to refine the prior model. The external and internal priors are formulated as a set of orthogonal dictionaries to efficiently reconstruct the desired image. Extensive experiments are performed on several real-world noisy image datasets. The proposed method demonstrates highly competitive denoising performance, outperforming state-of-the-art denoising methods including those designed for real-world noisy images.

I. INTRODUCTION

Real-world image noise is complex and often poorly handled by methods built for simpler noise models. The paper combines external clean-image priors with priors learned from the noisy image, shifting attention toward image-prior learning.

I. INTRODUCTION

  • Real-world noise varies across scenes, cameras, and settings, so AWGN-focused methods may leave noise or oversmooth image details.Noise-model-based approaches may also leave artifacts on real-world images.Source: §I · pp. 1–2
  • The method learns external priors from clean images, then uses them to guide internal-prior learning and form hybrid orthogonal dictionaries for denoising.The external priors are described as preserving fine-scale structure that is difficult to reproduce from noisy images.Source: §I · p. 2

A. Internal and External Prior Learning

  • External priors are efficient but not adaptive, whereas internal priors adapt to image content but can be corrupted by noise and slow to learn.The proposed method uses external priors to guide internal prior learning.Source: A. Internal and External Prior Learning · p. 2

B. Real-World Noisy Image Denoising

  • Rather than focus on explicit noise modeling, the paper develops image-prior learning for real-world denoising and reports highly competitive performance.The proposed approach models noise as locally Gaussian.Source: B. Real-World Noisy Image Denoising · p. 3

III. EXTERNAL PRIOR GUIDED INTERNAL PRIOR LEARNING FOR IMAGE DENOISING

The external prior is learned from groups of similar patches extracted from clean natural images. A GMM models these patch groups, and its component subspaces provide dictionaries and weights for the later guided learning stage.

A. Learn External Patch Group Priors

  • A patch group (PG) consists of a local patch and its M most similar patches searched within a surrounding W × W region.The patches are represented as vectors to form the group.Source: A. Learn External Patch Group Priors · p. 4
  • A GMM is learned to model patch-group priors from groups extracted from external natural images.The model has K Gaussian components.Source: A. Learn External Patch Group Priors · p. 4
  • SVD of each component covariance yields eigenvector matrices whose leading vectors form the external orthogonal dictionaries used to guide internal subdictionary learning.The singular values are also used as prior weights for weighted sparse coding.Source: A. Learn External Patch Group Priors · p. 4

B. Guided Internal Prior Learning

The method assigns noisy patch groups to externally learned subspaces, then learns an internal dictionary under the corresponding external dictionary's guidance. The hybrid orthogonal dictionaries combine general image structure with image-adaptive detail and support efficient denoising.

B. Guided Internal Prior Learning

  • The external prior guides both clustering of noisy patch groups and learning of their orthogonal dictionaries.Noisy local patches are grouped with similar patches before assignment to a subspace by maximum a-posteriori probability.Source: B. Guided Internal Prior Learning · p. 4, 1) Internal Subspace Clustering · p. 4
  • Orthogonal dictionaries avoid redundant overcomplete representations for patch groups in a subspace and make test-time patch encoding efficient.The paper notes that orthonormal dictionaries have zero mutual incoherence.Source: 2) Guided Orthogonal Dictionary Learning · p. 4
  • Each hybrid dictionary combines leading external eigenvectors with an internal subdictionary learned adaptively from noisy patch groups.The internal component is intended to capture input-specific latent structure not effectively represented by the general external component.Source: 2) Guided Orthogonal Dictionary Learning · p. 4
  • Weighted sparse coding uses an ℓ2 residual because patches in a group have similar content and their noise is assumed to have similar, roughly locally Gaussian statistics.Alternating updates optimize sparse codes and the internal subdictionary, initialized from the external dictionary.Source: 2) Guided Orthogonal Dictionary Learning · pp. 4–5
  • The dictionary-learning objective decreases at each alternating update, and the method was empirically found to usually converge in 10 iterations.The internal-dictionary update has a closed-form SVD solution, though solution uniqueness is not guaranteed in all cases.Source: 2) Guided Orthogonal Dictionary Learning · p. 5

C. The Denoising Algorithm

The algorithm iteratively forms patch groups, learns internal orthogonal dictionaries guided by external priors, and reconstructs the denoised image by aggregating recovered patches.

C. The Denoising Algorithm

  • For each patch group, the method uses sparse coding vectors and an orthogonal dictionary to reconstruct each latent clean patch.The group mean is added back to the reconstructed patch.Source: C. The Denoising Algorithm · pp. 5–6
  • Each iteration extracts patch groups from the current estimate, clusters them into subspaces, learns guided internal dictionaries, and recovers patches.The algorithm aggregates the recovered patches to form the next image estimate and repeats this process before returning the denoised image.Source: C. The Denoising Algorithm · p. 6

A. Implementation Details

The method uses fixed parameters across testing datasets captured with different sensors and camera settings, with selected values balancing denoising accuracy and computational cost.

A. Implementation Details

  • The method fixes its parameters across all testing datasets, despite differences in sensors and camera settings.The authors frame this as avoiding dataset-specific parameter tuning across those settings.Source: A. Implementation Details · p. 6
  • The external GMM prior uses 3.6 million patch groups from 24 Kodak PhotoCD images, with K=32 Gaussians and r=54 atoms per external sub-dictionary.The authors set the patch-group size to M=10 and the search window to W=31; they use p=6 to save computational cost.Source: A. Implementation Details · p. 6
  • λ=0.001 is fixed after the method showed good PSNR/SSIM performance over a range of λ values on dataset.Similar observations are reported for the other datasets, and the selected value works well across all three experimental datasets.Source: A. Implementation Details · p. 6, Figure 3 · p. 6

B. The Testing Datasets

The evaluation spans three real-world noisy-image datasets and compares the proposed method with external-only and internal-only priors and with competing denoisers. The proposed method reports better PSNR than competitors across the datasets, while guided internal priors combine the structural and texture strengths observed for the two individual priors.

B. The Testing Datasets

  • Dataset 1 uses repeated captures of 11 indoor scenes, while Dataset 2 provides 1,000 crops from paired noisy and lower-ISO, longer-exposure images across four cameras.Dataset 1 experiments use 60 crops from different shots than the 15 crops selected in the original dataset paper; DND's ground-truth images are unreleased, so evaluation uses website-reported averages.Source: B. The Testing Datasets · pp. 6–7
  • Dataset 3 adds 10 scenes captured by two cameras across five ISO settings, with realistic objects and 100 evaluation crops, addressing limitations in the scene contents of Datasets 1 and 2.Each scene was captured 500 times, allowing a mean image to serve as a kind of ground truth; the dataset is intended to support fairer evaluation.Source: B. The Testing Datasets · p. 7

C. Comparison among external, internal and guided internal priors

  • Guided Internal achieves better PSNR than External and Internal on 60 Dataset 1 images, combining External's large-structure recovery with Internal's fine-texture recovery.Internal is very slow because it requires online GMM learning, whereas Guided Internal is only a little slower than External.Source: C. Comparison among external, internal… · p. 7, Table I · p. 7

D. Comparison with State-of-the-Art Denoising Methods

  • The benchmark includes Gaussian-, Poisson-, and real-world-noise denoisers, but Cross-Channel is compared only on Dataset 1's 15 crops because its code is unavailable.For several methods, channel-specific Gaussian noise levels are estimated and supplied; discriminative models are retrained for noise levels from σ = 5 to σ = 50.Source: D. Comparison with State-of-the-Art… · pp. 7–8, Table II · p. 9
  • Across three datasets, the method reports better PSNR than competitors, with gains over the second-best method of 0.27 dB over CC, 0.43 dB over TNRD, and 0.16 dB over TNRD.The 0.27 dB comparison is on 15 Dataset 1 crops; 0.43 dB is on 60 additional Dataset 1 crops, and 0.16 dB is on Dataset 3.Source: D. Comparison with State-of-the-Art… · pp. 8, 10, Table II · p. 9
  • On Dataset 1, the proposed method preserves edges and textures better in visual comparisons, though it cannot clearly reproduce some fine hair details even when the mean-image reference is also unclear.The comparisons report that several methods retain noise or produce artifacts, while TNRD over-smooths the image.Source: D. Comparison with State-of-the-Art… · p. 8, Figure 7 · p. 8, Figure 8 · p. 8
  • For a 512×512 image, the proposed method takes about 24.1 seconds, versus 0.6 seconds for Neat Image and 5.2–152.2 seconds for the other compared methods.The runtime comparison used Matlab2014b on an Intel i7-5930K system; implementation language and parallelization differed across methods.Source: D. Comparison with State-of-the-Art… · p. 10

V. CONCLUSION

The paper combines external clean-image priors with adaptive internal priors learned from the noisy input, representing the hybrid priors as orthogonal dictionaries. Experiments on three real-world datasets show better quantitative and perceptual performance than state-of-the-art denoising methods.

V. CONCLUSION

  • Experiments on three real-world noisy image datasets show better quantitative and visual perceptual performance than state-of-the-art denoising methods.One dataset was newly constructed using different camera types and settings.Source: §V · p. 10
  • The method learns Gaussian Mixture Models from clean external images to guide adaptive internal-prior learning from the given noisy image.The resulting external-internal hybrid prior models are a set of orthogonal dictionaries for image denoising.Source: §V · p. 10

APPENDIX A CLOSED-FORM SOLUTION OF THE WEIGHTED SPARSE

The appendix derives a closed-form solution to weighted sparse coding by exploiting the orthogonality of the dictionary and separating the optimization across coefficients.

APPENDIX A CLOSED-FORM SOLUTION OF THE WEIGHTED SPARSE

  • Because D is orthogonal, the weighted sparse-coding problem reduces to an equivalent coefficient-space problem using z = D^T y.The weights λ_j are positive for j = 1, ..., 3p^2.Source: Appendix A · pp. 10–11
  • The coefficient-space objective separates into 3p^2 independent scalar minimization problems, one for each α_j.The derivation considers nonnegative and negative coefficient cases.Source: Appendix A · pp. 11–12
  • For the positive regime z_j ≥ λ_j/2 > 0, the solution is α̂_j = sgn(z_j)(|z_j| − λ_j/2), where sgn denotes the sign function.For z_j < −λ_j/2 < 0, the solution is α̂_j = sgn(z_j)(−z_j − λ_j/2).Source: Appendix A · p. 12

APPENDIX B PROOF OF THE THEOREM 1

The proof characterizes the internal subdictionary through orthogonality constraints and shows that an SVD yields an optimizing solution.

APPENDIX B PROOF OF THE THEOREM 1

  • The internal subdictionary D is constrained to have orthonormal columns and to be orthogonal to the external subdictionary E.The theorem's data matrices are A and Y, with E denoting the external subdictionary.Source: Appendix B · p. 12
  • An economy SVD of (I − EE^T)YA^T gives UΣV^T, and the proof obtains the candidate internal subdictionary as ˆD = UV^T.The projection removes the external-dictionary component before the decomposition.Source: Appendix B · p. 12
  • Under the stated orthogonality constraints, ˆD = UV^T attains the maximum trace objective Tr(Y^TDA).The proof uses the singular-value trace bound, with equality at D = UV^T.Source: Appendix B · p. 12

APPENDIX C PROOF OF THE THEOREM 2

The proof establishes the rank bound needed to analyze the SVD solution, then distinguishes full-rank uniqueness from rank-deficient non-uniqueness.

APPENDIX C PROOF OF THE THEOREM 2

  • The orthogonality of E implies rank(I − EE^T) ≥ 3p^2 − r, while the SVD rank is bounded by min{3p^2 − r, M}.The rank bound follows from E having r orthogonal columns.Source: Appendix C · p. 12
  • When the projected matrix has full rank, the SVD construction gives a unique solution ˆD = UV^T, including when singular values are repeated.The proof treats singular-vector orientations and concludes uniqueness for full rank.Source: Appendix C · pp. 12–13
  • When rank(Σ) < 3p^2 − r, additional solutions D = UΔV^T exist, with 2^(3p^2 − r − rank(Σ)) solutions for fixed U and V.The diagonal entries of Δ are ±1 in the null singular-value directions.Source: Appendix C · p. 13
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