Image Analysis Image Restoration

Image Restoration Image enhancement tries to improve subjective image quality. Image restoration tries to recover the original image. Bahadir K. Gunturk EE4780

Noise Models Noise may arise during image due to sensors, digitization, transmission, etc. Most of the time, it is assumed that noise is independent of spatial coordinates, and that there is no correlation between noise component and pixel value. Noise may be considered as a random variable, its statistical behavior is characterized by a probability density function (PDF). Gaussian noise Bahadir K. Gunturk EE4780

Noise Models Uniform noise Impulse (salt-and-pepper) noise Bahadir K. Gunturk EE4780

Noise Models Original Noisy images and their histograms Bahadir K. Gunturk EE4780

Noise Models How to estimate noise parameters? If imaging device is available Take a picture of a flat surface. See the shape of the histogram; decide on the noise model. Estimate the parameters. (e.g., find mean and standard deviation.) When only images already generated are available Get a small patch of image with constant gray level Inspect histogram Estimate the parameters Bahadir K. Gunturk EE4780

Restoration When There is Only Noise Low-Pass Filters: Smoothes local variations in an image. Noise is reduced as a result of blurring. For example, Arithmetic Mean Filter is  Convolve with a uniform filter of size m-by-n. Bahadir K. Gunturk EE4780

Restoration When There is Only Noise Adaptive, local noise reduction filter Let be the noise variance at (x,y). Let be the local variance of pixels around (x,y). Let be the local mean of pixels around (x,y). We want a filter such that If noise variance is zero, it should return g(x,y). If local variance is high relative to noise variance, the filter should return a value close to g(x,y). (Therefore, edges are preserved!) If two variances are equal, the filter should return the average of the pixels within the neighborhood. Bahadir K. Gunturk EE4780

Restoration When There is Only Noise Bahadir K. Gunturk EE4780

Restoration When There is Only Noise Median Filter Replaces the value of a pixel by the median of intensities in the neighborhood of that pixel. Is very effective against the salt-and-pepper noise. Bahadir K. Gunturk EE4780

Restoration When There is Only Noise Adaptive Median Filter: The basic idea is to avoid extreme values Let z_min: minimum gray level value in a neigborhood of a pixel at (x,y). z_max: maximum gray level value… z_med: median… z(x,y): gray level at (x,y). Is z_med=z_min or z_med=z_max? (That is, is z_med an extreme value?) No: Is z(x,y) an extreme value? (Is z(x,y)=z_min or z(x,y)=z_max?) No: Output is z(x,y) Yes: Output is z_med. Yes: Increase window size (to find a non-extreme z_med) and go to the first step. (When a maximum allowed window size is reached, stop and output z(x,y).) Bahadir K. Gunturk EE4780

Restoration When There is Only Noise Bahadir K. Gunturk EE4780

Restoration When There is Only Noise Removing Periodic Noise with Band-Reject Filters Spikes are due to noise Periodic Noise Periodic noise arises typically from electrical or electromechanical interference during image acquisition. This is the only type of spatially dependent noise (considered in the book). Inspect the Fourier characteristics (look for spikes) to estimate the parameters of periodic noise. Band-reject filter Bahadir K. Gunturk EE4780

Restoration When There is Only Noise Finding Periodic Noise from the Spectrum and Using Notch Filters Filter out these spikes Periodic noise arises typically from electrical or electromechanical interference during image acquisition. This is the only type of spatially dependent noise (considered in the book). Inspect the Fourier characteristics (look for spikes) to estimate the parameters of periodic noise. Noise due to interference Bahadir K. Gunturk EE4780

Image Restoration Spatial domain: Frequency domain: Bahadir K. Gunturk EE4780

Image Restoration Inverse Filtering This could dominate signal. Bahadir K. Gunturk EE4780

Image Restoration Bahadir K. Gunturk EE4780

Image Restoration Cut off the inverse filter for large frequencies. (Signal-to-noise ratio is typically low for large frequencies.) Bahadir K. Gunturk EE4780

Image Restoration Minimum Mean Square Error (Wiener) Filtering: Find such that the expected value of error is minimized: Solution is Investigate this equation for different signal-noise ratios. Bahadir K. Gunturk EE4780

Original image Bahadir K. Gunturk EE4780

Image Restoration Bahadir K. Gunturk EE4780

Least Squares Filtering Find F(u,v) that minimizes the following cost function: The solution is (Unconstrained solution) (See the derivations in the classroom) Bahadir K. Gunturk EE4780

Least Squares Filtering Find F(u,v) that minimizes the following cost function: Choose a P(u,v) to have a smooth solution. (A high-pass filter would do the trick.) Bahadir K. Gunturk EE4780

Least Squares Filtering The frequency domain solution to this optimization problem is (Constrained solution) where P(u,v) is the Fourier Transform of p(x,y), which is typically chosen as a high-pass filter. Example: Bahadir K. Gunturk EE4780

Least Squares Filtering Bahadir K. Gunturk EE4780