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

Deconvolution ? =

Super resolution from one image ? =

Super resolution from multiple image ? =

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

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

Deconvolution Input Sparse Gaussian, freq domain Gaussian

Inpainting Gaussian prior Sparse prior Original image

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

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:

Comparing deconvolution algorithms (Non blind) deconvolution code available online: http://groups.csail.mit.edu/graphics/CodedAperture/ Input Gaussian prior “spread” gradients Sparse prior “localizes” gradients Richardson-Lucy

Comparing deconvolution algorithms “localizes” gradients (Non blind) deconvolution code available online: http://groups.csail.mit.edu/graphics/CodedAperture/ Input “spread” gradients “localizes” gradients Richardson-Lucy Gaussian prior Sparse prior

Deconvolution ? =

Deconvolution is ill posed ? =

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

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

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