1. Lecture#4 Image reconstruction
Anat Levin
Introduction to Computer Vision Class Fall 2009 Department of Computer Science and App math, Weizmann Institute of Science.

2. Deconvolution ? =

3. Super resolution from one image? =

4. Super resolution from multiple image? =

5. Super resolution results from multiple shifted images
from one image
Super res
from multiple shifted images
deconvolution
Original sharp image

6. Deconvolution Input Gaussian prior with very low smoothness weight
Gaussian with reasonable smoothness weight

7. Deconvolution
Input
Sparse
Gaussian, freq domain
Gaussian

8. Inpainting
Gaussian prior
Sparse prior
Original image

9. Gaussian and Sparse priors- a toy example
Step edge
Smooth edge
cheaper
Gaussian
equal
Laplaceian
cheaper
Sparse

10. Natural image priors -|x|0.5 -|x|0.25 x x
Gaussian: -x2
Laplacian: -|x|
-|x|0.5
Log prob
-|x|0.25 x x
One strong property of natural images is the sparse derivatives distribution. If we plot the log histogram of derivatives in a natural image, we note that we can fit it with a parametric model of the form absolute derivative value to the power of alpha, when alpha is smaller than one. And an exponential distribution with alpha smaller than one is sparse.
Derivative histogram from a natural image
Parametric models
Derivative distributions in natural images are sparse:

11. Comparing deconvolution algorithms
(Non blind) deconvolution code available online:
Input
Gaussian prior
“spread” gradients
Sparse prior
“localizes” gradients
Richardson-Lucy

12. Comparing deconvolution algorithms “localizes” gradients
(Non blind) deconvolution code available online:
Input
“spread” gradients
“localizes” gradients
Richardson-Lucy
Gaussian prior
Sparse prior

13. Deconvolution ? =

14. Deconvolution is ill posed? =

15. Deconvolution is ill posed
Solution 1: = ?
Solution 2: = ?

16. Idea 1: Natural images prior What makes images special?
Unnatural
Image
gradient
Natural images have sparse gradients
put a penalty on gradients

17. Deconvolution with prior Equal convolution error
Derivatives prior 2 _ + ?
Low
Equal convolution error
And looking at the wrong solution, its convolution error is zero, but its gradient response is much higher 2 _ + ?
High