Image Denoising using Locally Learned Dictionaries

Priyam ChatterjeePeyman Milanfar

Dept. of Electrical EngineeringUniversity of California, Santa Cruz

Computational Imaging VII – 20 Jan, 2009

20 Jan, 2009 Image Denoising using Locally Learned Dictionaries 2

Overview

Data Model

Kernel Regression for Denoising

Denoising with Locally Learned Dictionaries (K-LLD)

Results

Conclusions

20 Jan, 2009 Image Denoising using Locally Learned Dictionaries 3

Data Model

Pointwise data model

Patchwise model

Locally smooth functionto be estimated

Zero-mean I.I.D. noise

Observation

denoted as

20 Jan, 2009 Image Denoising using Locally Learned Dictionaries 4

Steering Kernel Regression (SKR)

Optimization problem

Solution: Nonlinear filters Polynomial basis

Data-dependent

weights

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SKR Weights

Weights based on pixel “self-similarity” in a local patch

Covariance matrix takes into account: orientation and strength of edges

Gradient Covariance

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SKR Weights

Note how the weights adapt to the underlying image structure

Noisy Noise-free

H. Takeda, S. Farsiu, and P. Milanfar, “Kernel Regression for Image Processing and Reconstruction”, IEEE Trans. on Image Processing, vol. 16, no. 2, pp. 349-366, February 2007.

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Now we extend it …..

is fixed order, everywhere -- not depending on underlying image structure

• Lower orders fit flat regions, higher order for texture and fine details

Global dictionary does not adapt to local image characteristics

• Dictionary atoms should capture underlying local image structure

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Denoising with Locally Learned Dictionaries (K-LLD)

Identify dictionary which best captures underlying geometric structure

Similar structures will have similar dictionary, similar weights

Cluster image based on geometric similarity (K-Means on the SKR weights)

Learn dictionary and order of regression for each cluster

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K-LLD: Algorithm Outline

Calculate

weights

Learn

dictionaries

Clustering

Iter

ate

Noisy Image

Kernel Regression

Denoised Image

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Class 1Class 1

Class KClass KClustering StageClustering Stage

K-LLD : Algorithm Outline

Dictionary Selection StageDictionary Selection Stage

Noisy ImgCalculate Steering

Weights

Calculate SteeringWeights

Coefficient Calculation Stage

Coefficient Calculation StageDenoised

Img

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SegmentImage

SegmentImage

Clustering Stage

K-MeansK-MeansClass 1Class 1

Class KClass K

K-LLD : Algorithm Outline

Dictionary Selection StageDictionary Selection Stage

Noisy ImgCalculate Steering

Weights

Calculate SteeringWeights

Coefficient Calculation Stage

Coefficient Calculation StageDenoised

Img

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Clustering Stage

Objective : Cluster image based on geometric similarity of underlying data

• Feature Selection

• What features capture data geometry ?

• Distance Metric

• What metric captures distance between features ?

• Clustering Algorithm

• What algorithm segments the image best ?

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K-Means for Clustering

Features : normalized steering wts

Distance Metric : L2

Initialization : Randomly initialize cluster centers

Run K-Means multiple times and select result that minimizes within-cluster distance

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Noi

se-f

ree

Noi

sy

Clustering the noise-free image

Clustering the noisy image

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SegmentImage

SegmentImage

Clustering Stage

K-MeansK-MeansClass 1Class 1

Class KClass K

K-LLD : Algorithm Outline

Dictionary Selection StageDictionary Selection Stage

Noisy ImgCalculate Steering

Weights

Calculate SteeringWeights

Coefficient Calculation Stage

Coefficient Calculation StageDenoised

ImgDictionary Selection Stage

PCAPCAFormDictionary

FormDictionary

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Dictionary Selection

Represent each patch in the kth cluster

Variable Proj.

Solved by PCA

Mean patch of k-th cluster

Enforceorthonormality

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Dictionary Selection

PCA in each cluster to form a dictionary

Describe data without fitting noise

Number of atoms based on cluster geometry • Clusters with flat regions need fewer atoms, finer

details need more

constant

Singular values

patch sizeNo. of atoms

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Example : House Image

Dictionary atoms for AWGN of std. dev. 15

Clu

ster

Ato

m 1

Ato

m 2

Ato

m 3

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Example : Noise-free clusteringC

lust

erA

tom

1A

tom

2A

tom

3A

tom

4

Few of the atoms in the dictionaries for different clusters

Brick facade

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Algorithm Outline

SegmentImage

SegmentImage

Clustering Stage

Class 1Class 1

Class KClass KK-MeansK-Means

Noisy ImgCalculate Steering

Weights

Calculate SteeringWeights

Coefficient Calculation Stage

Coefficient Calculation Stage

Denoised Img

Dictionary Selection Stage

PCAPCAFormDictionary

FormDictionaryKernel RegressionKernel Regression

Denoised Img

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Kernel Regression

Weighted least squares solution

Final estimate

Coefficient Calculation

center pixel of patch .

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Kernel RegressionKernel RegressionDenoised

Img

Algorithm Outline

SegmentImage

SegmentImage

Clustering Stage

Class 1Class 1

Class KClass KK-MeansK-Means

Noisy ImgCalculate Steering

Weights

Calculate SteeringWeights

Dictionary Selection Stage

PCAPCAFormDictionary

FormDictionary

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Iteration

Re-learn weights (features) from denoised image

Perform clustering of updated image using new features

Learn dictionary from updated image

Kernel regression on input noisy image• Preserves edges and finer structures

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K-LLD: Algorithm Outline

Calculate

weights

Learn

dictionaries

Clustering

Iter

ate

Noisy Image

Kernel Regression

Denoised Image

Original Noisy Image

Results – AWG noise (std dev 25)

K-LLD, MSE 96.95 SSIM 0.825

BM3D, MSE 88.82 SSIM 0.841Original Parrot Image

K-SVD, MSE 101.54 SSIM 0.826

SKR, MSE 99.96 SSIM 0.826

Noisy Image

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More ResultsM

SE

SS

IM

Results – Real noise & colorISKR K-LLDBM3D

Color Results

ISKR, Order 2

BM3DOriginal Image

K-LLD

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Thank you

P. Chatterjee and P. Milanfar, “Clustering-based Denoising with Locally LearnedDictionaries”, Accepted for publication in IEEE Trans. Image Processing

Available at:

http://www.ee.ucsc.edu/~milanfar

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Iterative Scheme

Why iterate ?

• Weights true to underlying structure in presence of lesser noise

• Better weights means better clustering

• Dictionary captures underlying data better when learned on less noisy image