**Image Analysis** Image Restoration

**Image Restoration** Image enhancement tries to improve subjective image quality. Image restoration tries to recover the original image. 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 EE4780

**Noise Models** Uniform noise Impulse (salt-and-pepper) noise EE4780

**Noise Models** Original Noisy images and their histograms 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 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. 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. EE4780

**Restoration When There is Only Noise** 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. 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).) EE4780

**Restoration When There is Only Noise** EE4780

**Restoration When There is Only Noise** Removing Periodic Noise with Band-Reject Filters Spikes are due to noise Periodic Noise Band-reject filter EE4780

**Restoration When There is Only Noise** Finding Periodic Noise from the Spectrum and Using Notch Filters Filter out these spikes Noise due to interference EE4780

**Image Restoration** Spatial domain: Frequency domain: EE4780

**Image Restoration** Inverse Filtering This could dominate signal. EE4780

**Image Restoration** EE4780

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

**Original image** EE4780

**Image Restoration** 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) 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.) 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: EE4780

**Least Squares Filtering** EE4780