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Some Blind Deconvolution Techniques in Image Processing

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## Some Blind Deconvolution Techniques in Image Processing

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### Some Blind Deconvolution Techniques in Image Processing

Tony Chan

Math Dept., UCLA

Joint work with Frederick Park and Andy M. Yip

Astronomical Data Analysis Software & Systems

Conference Series 2004

Pasadena, CA, October 24-27, 2004

Outline

Part I:

Total Variation Blind Deconvolution

Part II:

Simultaneous TV Image Inpainting and Blind Deconvolution

Part III:

Automatic Parameter Selection for TV Blind Deconvolution

Caution: Our work not developed specifically for Astronomical images

Blind Deconvolution Problem

=

+

Observed image

Unknown true image

Unknown point spread function

Unknown noise

Goal: Given uobs, recover both uorig and k

D

Total Variation Regularization- Deconvolution ill-posed: need regularization
- Total variation Regularization:

- The characteristic function of D with height h (jump):

- TV = Length(∂D)h
- TV doesn’t penalize jumps
- Co-area Formula:

TV Blind Deconvolution Model

(C. and Wong (IEEE TIP, 1998))

Objective:

Subject to:

- 1determined by signal-to-noise ratio
- 2 parameterizes a family of solutions, corresponds to different spread of the reconstructed PSF

- Alternating Minimization Algorithm:
- Globally convergent with H1 regularization.

Recovered Image

PSF

Blind

1= 2106, 2 = 1.5105

Blind v.s. non-Blind DeconvolutionObserved Image noise-free

non-Blind

True PSF

- An out-of-focus blur is recovered automatically
- Recovered blind deconvolution images almost as good as non-blind
- Edges well-recovered in image and PSF

Blind v.s. non-Blind Deconvolution: High Noise

Blind

Observed Image SNR=5 dB

non-Blind

True PSF

1= 2105, 2 = 1.5105

- An out-of-focus blur is recovered automatically
- Even in the presence of high noise level, recovered images from blind deconvolution are almost as good as those recovered with the exact PSF

1105

1104

Controlling Focal-LengthRecovered Images are 1-parameter family w.r.t. 2

Recovered Blurring Functions

(1 = 2106)

2:

0

The parameter 2 controls the focal-length

Generalizations to Multi-Channel Images

- Inter-Channel Blur Model
- Color image (Katsaggelos et al, SPIE 1994):

k1: within channel blur

k2: between channel blur

m-channel TV-norm (Color-TV)

(C. & Blomgren, IEEE TIP ‘98)

Examples of Multi-Channel Blind Deconvolution

(C. and Wong (SPIE, 1997))

Original image

Out-of-focus blurred blind non-blind

Gaussian blurred blind non-blind

- Blind is as good as non-blind
- Gaussian blur is harder to recover (zero-crossings in frequency domain)

Outline

Part I:

Total Variation Blind Deconvolution

Part II:

Simultaneous TV Image Inpainting and Blind Deconvolution

Part III:

Automatic Parameter Selection for TV Blind Deconvolution

Images Degraded by Blurring and Missing Regions

- Missing regions
- Scratches
- Occlusion
- Defects in films/sensors

- Blur
- Calibration errors of devices
- Atmospheric turbulence
- Motion of objects/camera

+

Blurring func.

Blurred Signal

Blurred + Occluded

=

Original Signal

Blurring func.

Blurred Signal

Blurred + Occluded

=

Problems with Inpaint then Deblur- Inpaint first reduce plausible solutions
- Should pick the solution using more information

=

Problems with Deblur then Inpaint

Original

Occluded

Support of PSF

- Different BC’s correspond to different image intensities in inpaint region.
- Most local BC’s do not respect global geometric structures

Dirichlet

Neumann

Inpainting

Coupling of inpainting & deblur

Inpainting take place

The Joint Model- Do --- the region where the image is observed
- Di --- the region to be inpainted

- A natural combination of TV deblur + TV inpaint
- No BC’s needed for inpaint regions
- 2 parameters (can incorporate automatic parameter selection techniques)

Restored

Zoom-in

Simulation Results (1)- The vertical strip is completed
- Not completed
- Use higher order inpainting methods
- E.g. Euler’s elastica, curvature driven diffusion

Simulation Results (2)

Original

Observed

Restored

Inpaint then deblur

(many ringings)

Deblur then inpaint

(many artifacts)

Inpainting B.C.

Periodic B.C.

Neumann B.C.

Boundary Conditions for Regular DeblurringOriginal image domain and artificial boundary outside the scene

Outline

Part I:

Total Variation Blind Deconvolution

Part II:

Simultaneous TV Image Inpainting and Blind Deconvolution

Part III:

Automatic Parameter Selection for TV Blind Deconvolution

(Ongoing Research)

Automatic Blind Deblurring (ongoing research)

observed image

Clean image

SNR = 15 dB

- Recovered images: 1-parameter family wrt 2
- Consider external info like sharpness to choose optimal 2

Problem: Find 2 automatically to recover best u & k

Motivation for Sharpness & Support

u

- Sharpest image has large gradients
- Preference for gradients with small support

Support of

Proposed Sharpness Evaluator

- F(u) small => sharp image with small support
- F(u)=0 for piecewise constant images
- F(u) penalizes smeared edges

u

Support of

Planets Example

Rel. errors in u (blue) and k (red) v.s. 2

1=0.02 (optimal)

Optimal Restored Image

Auto-focused Image

Proposed Objective v.s. 2

(minimizer of rel. error in u)

(minimizer of sharpness func.)

The minimum of the sharpness function agrees with that of the rel. errors of u and k

Satellite Example

Rel. errors in u (blue) and k (red) v.s. 2

1=0.3 (optimal)

Optimal Restored Image

Auto-focused Image

Proposed Objective v.s. 2

(minimizer of rel. error in u)

(minimizer of sharpness func.)

The minimum of the sharpness function agrees with that of the rel. errors of u and k

Potential Applications to Astronomical Imaging

- TV Blind Deconvolution
- TV/Sharp edges useful?
- Auto-focus: appropriate objective function?
- How to incorporate a priori domain knowledge?
- TV Blind Deconvolution + Inpainting
- Other noise models: e.g. salt-and-pepper noise

References

- C. and C. K. Wong, Total Variation Blind Deconvolution, IEEE Transactions on Image Processing, 7(3):370-375, 1998.
- C. and C. K. Wong, Multichannel Image Deconvolution by Total Variation Regularization, Proc. to the SPIE Symposium on Advanced Signal Processing: Algorithms, Architectures, and Implementations, vol. 3162, San Diego, CA, July 1997, Ed.: F. Luk.
- C. and C. K. Wong, Convergence of the Alternating Minimization Algorithm for Blind Deconvolution, UCLA Mathematics Department CAM Report 99-19.
- R. H. Chan, C. and C. K. Wong,Cosine Transform Based Preconditioners for Total Variation Deblurring, IEEE Trans. Image Proc., 8 (1999), pp. 1472-1478
- C., A. Yip and F. Park, Simultaneous Total Variation Image Inpainting and Blind Deconvolution, UCLA Mathematics Department CAM Report 04-45.

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