Tirgul 4: Image Restoration in the Frequency Domain Example for Image Reconstruction (Tomography)‏

Degradation/Restoration Model Goal: Reconstruct a degraded image using a priori knowledge of the degradation phenomenon. Degradation Function H + Restoration filters

Degradation/Restoration Model Assumptions on H: Linearity (additivity & homogeneity)‏ Space invariance:  Linear, position-invariant degradation can be modelled as the convolution:

Degradation / Restoration Model Linear, position-invariant degradation can be modelled as the convolution: Equivalent frequency domain representation :

Estimating the Degradation function We need to have some information on the degradation function Observation Experimentation Mathematical modelling

Estimation by image observation Look for information in the image itself : search for small section of the image, s, containing simple structure (edge, point)‏ High signal to noise ratio Guess the image in that area:

Estimation by Experiment If we have the equipment used to acquire the degraded image we can obtain accurate estimation of the degradation Obtain an impulse response of the degradation using the same system setting

Estimation by modelling Mathematical model of degradation can be for example atmospheric turbulence Hufnagel & Stanley: k=0.00025 k=0.001 k=0.0025 Original

Estimation by modeling: Uniform motion blur

Estimation by modelling: Uniform motion blur Assuming the image was blurred by a uniform linear motion, between the image and the sensor, during image acquisition. The degradation : f(x)*recta(x)‏ In the Fourier domain: F(u)·sinc(u)‏

Noise Models Principal source of noise – acquisition / transmission Assumptions: White noise: Fourier spectrum of the noise is constant. Noise is independent in spatial coordinates. Uncorrelated with the image. Zero mean

Estimating White Noise Spectrum Assuming additive white noise, and given the linearity of Fourier transform, the transform of G will be sum of the transforms H∙F + N = G

Inverse Filtering (Deconvolution Filters)‏ Problem: H may have very small values. Conclusion: We need to find a way to deal with the noise N.

Minimum Mean Square Error (Wiener) Filtering Goal: have an inverse filter that deals with the noise better than inverse filtering. Try to minimize: – power spectrum

Minimum Mean Square Error (Wiener) Filtering K K can be chosen interactively to yield best visible results

(K is chosen interactively)‏ Examples Modelling atmosphere turbulence: Radially limited Inverse filtering Wiener filter (K is chosen interactively)‏ Inverse filtering

Examples Images corrupted by motion blur and additive noise. Image Inverse filtering Weiner Filtering Images corrupted by motion blur and additive noise. Noise Variance

Reconstruction Example: Computerized Tomography Goal : reconstruct 2D slices of 3D data given projection Computerized Tomography X-Ray : propagating beam Photons are lost from the beam as they are absorbed in the body Try to reconstruct linear attenuation coefficient representing the material

Parallel projection

Sum on f(x,y) along each ray AB: Radon Transform Sum on f(x,y) along each ray AB:

Fourier Slice Theorem Assume w.l.o.g. that Ө = 0; the projection of f(x,y) on the x axis is Thus, This is true for each angle, so

Fourier Slice Theorem

The Inverse Radon f(x,y) reconstruction algorithm: 1. Compute all of the projections pθ(r). 2. Apply Fourier transform on p, and Reconstruct F(u,v). 3.Apply inverse Fourier transform and get the reconstructed function f(x,y). Problem: Algorithm is very sensitive to noise.

Iterative Back Projection Start with initial guess of the image. For each ray – perform summation on the guess and on the real image. Spread the difference between guess and observation uniformly for all pixels along this ray. This process converges in many cases to the actual image .