Method for reflection removal: user assisted Find I 1 and I 2 such - PowerPoint PPT Presentation

Method for reflection removal: user ‐ assisted • Find I 1 and I 2 such that: 1. I 1 and I 2 sum up to I 2. The gradient of I 1 at points in S 1 should match the gradient of I at those points. 3. The gradient of I 2 at points in S 2 should match the gradient of I at those points. Ajit Rajwade 1

Method for reflection removal: statistical model • Exploit a statistical property of a natural image. • The gradients are sparse! The image part with relationship ID rId2 was not found in the file. Ajit Rajwade 2

Method for reflection removal: statistical model • So the objective function becomes: The image part with relationship ID rId5 was not found in the file. Ajit Rajwade 3

Optimization algorithm • Given the statistical model for the gradient filter outputs, the function ρ is non ‐ convex. • The optimization procedure for this is not very easy. • The authors use a method called iteratively reweighted least squares (IRLS) . Ajit Rajwade 4

Segway: Least squares method • Consider the solution to the following problem: The image part with relationship ID rId4 was not found in the file. • This is a least squares problem, and it has a well ‐ known pseudo ‐ inverse based solution. • Now we will look at some flavours of least squares. Ajit Rajwade 5

Segway: Weighted least squares method • Now consider the solution to the following problem: The image part with relationship ID rId4 was not found in the file. • Here W is a n x n diagonal matrix containing weights which give different levels of importance to each entry of y . • The solution of this is again in terms of a pseudo ‐ inverse. Ajit Rajwade 6

Segway: Least p ‐ norm problem • Consider the solution to the following problem: The image part with relationship ID rId4 was not found in the file. • This has no known closed form solution! • Instead an iterative procedure has been proposed – called IRLS. Ajit Rajwade 7

Segway: IRLS • The IRLS at step t +1 involves a weighted least squares problem: The image part with relationship ID rId4 was not found in the file. Diagonal matrix of Weight for point i at weights at iteration t iteration t (for all points) • At t = 1, the weights are set to 1. • The weights are updated as follows: The image part with relationship ID rId4 was not found in the file. • This is done till convergence. Ajit Rajwade 8

Segway: IRLS • The weights are updated as follows: The image part with relationship ID rId4 was not found in the file. • Why these weights? Simply because the problem can be re ‐ written as follows: The image part with relationship ID rId4 was not found in the file. Ajit Rajwade 9

Optimization algorithm The image part with relationship ID rId4 was not found in the file. Ajit Rajwade 10

Back to the Optimization algorithm • The objective function is: The image part with relationship ID rId4 was not found in the file. • It can be expressed as: The image part with relationship ID rId4 was not found in the file. Ajit Rajwade 11

Method for reflection removal: actual statistical model used in the paper • The statistical model for the gradients of the image is chosen to be the following: The image part with relationship ID rId4 was not found in the file. z = gradient value • This is a mixture of two Laplacian distributions and it is seen to be sparser than a single Laplacian. Ajit Rajwade 12

Sample results The image part with relationship ID rId2 was not found in the file. Ajit Rajwade 13

Comparison: Laplacian and Sparse (mixture of two Laplacians) priors Ajit Rajwade 14

The image part with relationship ID rId2 was not found in the file. Comparison: Laplacian and Sparse (mixture of two Laplacians) priors Ajit Rajwade 15

The image part with relationship ID rId2 was not found in the file. Comparison: Laplacian and Gaussian priors. Notice the much better results For the Laplacian prior as compared to the Gaussian prior. Ajit Rajwade 16

Why study Natural Image Statistics: Bayesian Framework The image part with relationship ID rId4 was not found in the file. Unknown (to be Observation Known noise determined) operator signal The image part with relationship ID rId4 was not found in the file. The image part with relationship ID rId4 was not found in the file. Prior Model Likelihood on signal Posterior Bayes rule probability The image part with relationship ID rId4 was not found in the file. Ajit Rajwade 17

Bayesian Framework: To Estimate x • Minimum mean square error (MMSE) estimate: Prior is important! The image part with relationship ID rId4 was not found in the file. The image part with relationship ID rId4 was not found in the file. Integrate to 1 Ajit Rajwade 18

Bayesian Framework: To Estimate x • Maximum a posteriori (MAP) estimate: The image part with relationship ID rId4 was not found in the file. As y does not affect maximization w.r.t. x The MAP estimate asks the following question: Given the observation y, what x is the most likely, taking into account that we have prior information on x in the form of p(x)? If p(x) were a uniform distribution (or effectively we had no prior information about x), then MAP reduces to maximizing p(y|x) – which is called the maximum likelihood estimate. Ajit Rajwade 19

Simple Example: 1 The image part with relationship ID rId4 was not found in the file. Prior The image part with relationship ID rId4 was not found in the file. Likelihood Observed Value of y = 14. Determine x given y and the knowledge of the noise model (likelihood) and prior on x . Ajit Rajwade 20

Application in Denoising • Consider the following noise model: The image part with relationship ID rId4 was not found in the file. • Given y , and knowing σ , determine the underlying image x . • Exploit the prior (fact) that the image x has DCT coefficients which are Laplacian distributed. Ajit Rajwade 21

Application in Denoising • Let the DCT coefficients be given as follows: The image part with relationship ID rId4 was not found in the file. • So the estimation problem is The image part with relationship ID rId4 was not found in the file. Ajit Rajwade 22

Application in Deblurring • Consider the following noise model: The image part with relationship ID rId4 was not found in the file. • Given y , and knowing H and σ , determine the underlying image x . • Exploit the prior (fact) that the image x has DCT coefficients which are Laplacian distributed. Ajit Rajwade 23

Application in Deblurring • Let the DCT coefficients be given as follows: The image part with relationship ID rId4 was not found in the file. • So the estimation problem is The image part with relationship ID rId4 was not found in the file. Ajit Rajwade 24

Application in deblurring • A circulant matrix is a matrix where each row is a right circular shift of its preceding row in the following form: The image part with relationship ID rId4 was not found in the file. Ajit Rajwade 25

Gaussian instead of Laplacian prior • What would happen if you imposed a Gaussian prior on the DCT coefficients? The image part with relationship ID rId4 was not found in the file. Ajit Rajwade 26

Gaussian instead of Laplacian prior • Taking derivative w.r.t. θ , we get: This is the Wiener filter which we have seen last The image part with relationship ID rId4 was not found in the file. semester! The Wiener filter is the optimal linear filter regardless of the signal prior, which is what we proved in CS 663. For Gaussian likelihood and Gaussian prior, the Wiener filter is the optimal filter (among linear as well as non ‐ linear) in a MAP or MMSE sense. • However for natural images or image patches, the Laplacian prior on the DCT or wavelet coefficients, is better suited – and yields better results in denoising. Ajit Rajwade 27

The image part with relationship ID rId2 was not found in the file. The image part with relationship ID rId2 was not found in the file. The image part with relationship ID rId2 was not found in the file. The image part with relationship ID rId2 was not found in the file. i.e. solution with Laplacian prior Ajit Rajwade 28

Limitation of this model • For some images, a GGD with shape parameter less than 1 is more suitable to model the DCT coefficients than a Laplacian. • In such cases, the optimization problem becomes: The image part with relationship ID rId5 was not found in the file. • The problem however is that this is a non ‐ convex optimization problem – and hence the local minima are different from the global minimum. • With Laplacian or Gaussian priors, the problems were convex. Many convex problems have efficient solutions. Ajit Rajwade 29

Statistical Compressed Sensing • This is another view of compressed sensing based on Bayesian statistics. • Consider compressive measurements of the form: The image part with relationship ID rId4 was not found in the file. • Suppose . The image part with relationship ID rId4 was not found in the file. Ajit Rajwade 30

Statistical Compressed Sensing • Consider the MAP solution for x given y and Φ : The image part with relationship ID rId4 was not found in the file. Ajit Rajwade 31

Statistical Compressed Sensing • Consider the MAP solution for x given y and Φ : The image part with relationship ID rId4 was not found in the file. The latter expression follows by the Woodbury matrix identity. https://en.wikipedia.org/wiki/Woodbury_matrix_identity Ajit Rajwade 32