CS654: Digital Image Analysis Lecture 22: Image Restoration

Recap of Lecture 21 Image restoration in presence of only noise Image restoration in presence of only degradation Observation, experimentation and mathematical modeling Motion blur Restoration by inverse filtering

Outline of Lecture 22 Inverse filtering and its problems Pseudo Inverse filtering Constrained image restoration problem

Inverse filtering (error minimization) 2D Discrete Domain Representation Neglecting the noise component Approximate least square error: Unconstrained error minimization

Inverse filtering (error minimization) Equating to zero In frequency domain It doesn't perform well when used on noisy images.

Pseudo inverse filtering Equation of inverse filter in frequency domain Spectrum of the PSF Simulated impulseImpulse response Pseudo inverse filter

SVD approach to Pseudo-Inverse Image restoration model

Constrained image restoration SmoothnessRestoration

Minimization of error Equating to zero Emphasize restoration Emphasize smoothness

Constrained Restoration By applying Fourier transform matrices to both sides Constrained restoration results with Q = Laplacian and different γ values

Wiener Filters Assume: noise is zero mean and uncorrelated with the image

Weiner Filter Product of a complex quantity with its conjugate is equal to the magnitude of the complex quantity squared

Weiner filter

Weiner Filter Approximation of Weiner filter

Example Input image Full inverseRadially limitedWeiner Filter

Example Noise + Motion InverseWeiner Noise

Conclusion We considered several algebraic approaches to image restoration Constrained restoration imposes smoothness constraints and does well when noise is present Wiener filters model the noise/signal ratio to obtain a minimum mean square error restoration image