Iterated Denoising for Image Recovery

Onur G. Guleryuzonur@danbala.poly.edu

To see the animations and movies please use full-screen mode.Clicking on pictures to the left of PSNR curves should start the movies.

There are also reminder notes for some slides.

Presentation given at DCC 02.

Overview•Problem definition.

•Main Algorithm.

•Rationale.

•Choice of transforms.

•Many simulation examples, movies, etc.

•Brought code. Can run for other images, for your images, etc.If interested, please find me during breaks or evenings.

•Errata for manuscript.

Notices:

Problem Statement

Image

LostBlock

Use surrounding spatial information to recover lost block via overcomplete denoising with

hard-thresholding.*

Generalizations: Irregularly shaped blocks, partial information, ...

Pretend “Image + Noise”

Applications: Error concealment, damaged images, ...

What is Overcomplete Denoising with Hard-thresholding?

x

y

DCT (MxM) tilings

Image

Hard threshold coefficients with T

Partially denoised result

1Hard threshold

coefficients with TPartially

denoised result 2.

.

.

Average partially denoised resultsfor final denoised image.

Utilized transform will be very important!

Examples(Figure 1 in the paper)

+9.37 dB

+8.02 dB

+11.10 dB

+3.65 dB

Main Algorithm IDenoising with hard-thresholding using overcomplete transforms

Recover layer P by mainly using information from layers 0,…,P-1

(Figure 2 in the paper)

Main Algorithm II• Assign initial values to layer pixels.

for i=1: number_of_layers

recover layer i by overcomplete denoising with threshold T

end

T=T- dT

• T=T0

• while ( T > T )F

•end

thk DCT block

*Main Algorithm III

x

y

DCT (MxM) tiling 1

Outer border of layer P

Image

Lost block o (k)y

o (k)x

Hard threshold block k coefficients if

o (k) < M/2y

o (k) < M/2x

OR

(Figure 3 in the paper)

(Figure 4 in the paper)Example DCT Tilings and

Selective Hard Thresholding

Rationale: Denoising and Recovery

Main intuition: Keep coefficients of high SNR, zero out coefficients of low SNR.

ecc ˆoriginal transform coefficient

error

Assume that the transform yields a sparse image representation:

ec ec ~ˆ

Hard thresholding removes more noise than signal.c

Rationale: Other AnalogiesBand limited reconstructions via POCS:

Set of bandlimited (low pass) signals

Set of possible signals given the available

information.

.

.

.

Assumes low frequency Fourier coefficients are important and zeros out high frequencies coefficients.

This work: Adaptively change sets at each iteration. Let data determine the important coefficients and

which coefficients to zero out.

Best subspaces to zero-out in a POCS setting. Optimal linear estimators. Sparse transforms.

Properties of Desired Transforms

•Periodic, approximately periodic regions:

Transform should “see” the period

Example: Minimum period 8 at least 8x8 DCT, ~ 3 level wavelet packets.

•Edge regions (sparsity may not be enough):

Transform should “see” the slope of the edge.

k

kNngns )()(

Periodic Example(Figure 1 in the paper)

DCT 9x9

+11.10 dB

Periodic Example

(period=8)

(Figure 5 in the paper)

DCT 8x8

Perf. Rec.

Periodic Example(Figure 6 in the paper)

DCT 16x16

+3.65 dB

Periodic Example

DCT 24x24

+5.91 dB

“Periodic” Example

DCT 16x16

+7.2 dB

“Periodic” Example

DCT 24x24

+10.97 dB

Edge Example

DCT 8x8

+25.51 dB

Edge Example(Figure 6 in the paper)

Complex wavelets

+9.37 dB

Edge Example(Figure 6 in the paper)

Complex wavelets

+16.72 dB

Edge Example(Figure 6 in the paper)

DCT 24x24

+9.26 dB

Edge Example(Figure 1 in the paper)

Complex wavelets

+8.02 dB

Unsuccessful Recovery Example(Figure 7 in the paper)

DCT 16x16

-1.00 dB

Partially Successful Recovery Example

(Figure 7 in the paper)

DCT 16x16

+4.11 dB

Edges and “Small Transforms”

DCT 4x4

-1.06 dB

Edges and “Small Transforms”

+5.56 dB

DCT 4x4

Edge Example(Figure 6 in the paper)

DCT 24x24

+9.26 dB