Image Restoration PDF

Image Restoration Image restoration attempts to reconstruct or recover an image that has been degraded by a degradation phenomenon. As in image enhancement, the ultimate goal of restoration techniques is to improve an image in some predefined sense. Image restoration vs. image enhancement...

Image Restoration Image restoration attempts to reconstruct or recover an image that has been degraded by a degradation phenomenon. As in image enhancement, the ultimate goal of restoration techniques is to improve an image in some predefined sense. Image restoration vs. image enhancement Image restoration Image enhancement 1. is an objective process is a subjective process 2. formulates a criterion of involves heuristic procedures goodness that will designed to manipulate an image yield an optimal estimate of the in order to satisfy the human desired result visual system 3. Techniques include noise Techniques include contrast removal and deblurring (removal stretching of image blur) Like enhancement techniques, restoration techniques can be performed in the spatial domain and frequency domain. For example, noise removal is applicable using spatial domain filters whereas deblurring is performed using frequency domain filters. A Model of Image Degradation & Restoration As shown in the next figure, image degradation is a process that acts on an input image f(x,y) through a degradation function H and an additive noise η(x,y). It results in a degraded image g(x,y) such that: where h(x,y) is the spatial representation of the degradation function and the symbol “ * ” indicates convolution. Note that we only have the degraded image g(x,y). The objective of restoration is to obtain an estimate of the original image. We want the estimate to be as close as possible to the original input image and, in general, the more we know about H and η, the closer will be to f(x, y). A model of the image degradation/restoration process In the frequency domain, this model is equivalent to: The approach that we will study is based on various types of image restoration filters. We assume that H is the identity operator, and we deal only with degradations due to noise. Noise and its characteristics Noise in digital images arises during: Acquisition: environmental conditions (light level & sensor temperature), and type of cameras and/or transmission – interference in the transmission channel To remove noise we need to understand the spatial characteristics of noise and its frequency characteristics (Fourier spectrum). Generally, spatial noise is assumed to be independent of position in an image and uncorrelated to the image itself (i.e. there is no correlation between pixel values and the values of noise components). Frequency properties refer to the frequency content of noise in the Fourier sense. Noise Models Spatial noise is described by the statistical behavior of the gray-level values in the noise component of the degraded image. Noise can be modeled as a random variable with a specific probability distribution function (PDF). Important examples of noise models include: 1. Gaussian Noise also known as white Gaussian noise 2. Rayleigh Noise 3. Gamma Noise 4. Exponential Noise 5. Uniform Noise 6. Impulse (Salt & Pepper) Noise  Impulse (Salt & Pepper) Noise The PDF of impulse noise is given by Impulse noise PDF If b > a , then gray level b appears as a light dot (salt), otherwise the gray level a appears as a dark dot (pepper). Salt and pepper noise is a common type of noise that can corrupt digital images in image processing. It is characterized by the presence of white and black pixels scattered randomly throughout the image, which resemble grains of salt and pepper. This type of noise can degrade the quality of images and make it difficult to extract meaningful information from them. Salt and pepper noise can occur for various reasons, such as errors in the image acquisition process, transmission errors, or electronic interference. It is important to remove or reduce this noise to improve the quality and reliability of the image data. There are several techniques used to address salt and pepper noise in image processing: Median Filter Adaptive Median Filter Mean Filter Gaussian Filter Image Morphology Non-Local Means Filter Determining noise models The simple image below is well-suited test pattern for illustrating the effect of adding various noise models. Gaussian Rayleigh Gamma Exponential Uniform Salt & Pepper Images and histograms from adding Gaussian, Rayleigh, Gamma, Exponential, Uniform, and Salt & Pepper noise. To determine the noise model in a noisy image, one may select a relatively small rectangular sub-image of relatively smooth region. The histogram of the sub-image approximates the probability distribution of the corrupting model of noise. This is illustrated in the figure below. (a) (b) (c) (d) (e) (f) (a) Gaussian noisy image. (b) sub-image extracted from a. (c) histogram of b (d) Rayleigh noisy image. (e) sub-image extracted from d. (f) histogram of e Image restoration in the presence of Noise Only When the only degradation present in an image is noise, the degradation is modeled as: and Spatial filtering is the method of choice in situations when only additive noise is present. Spatial filters that designed to remove noise include: 1. Order Statistics Filters: e.g. Min, Max, & Median 2. Adaptive Filters: e.g. adaptive median filter Order-Statistics Filters We have used one of these filters (i.e. median) in the image enhancement. We now use additional filters (min and max) in image restoration. Min filter This filter is useful for finding the darkest points in an image. Also, it reduces salt noise as a result of the min operation. Max filter This filter is useful for finding the brightest points in an image. Also, because pepper noise has very low values, it is reduced by this filter as a result of the max operation. Adaptive Filters The previous spatial filters are applied regardless of local image variation. Adaptive filters change their behavior using local statistical parameters in the mask region. Consequently, adaptive filters outperform the non-adaptive ones. Adaptive median filter The median filter performs well as long as the spatial density of the impulse noise is not large (i.e. Pa and Pb less than 0.2). Adaptive median filtering can handle impulse noise with probabilities even larger than these. Moreover the adaptive median filter seeks to preserve detail while smoothing non-impulse noise, while the median filter does not do. The adaptive median filter aims to replace f(x,y) with the median of a neighborhood up to a specified size as long as the median is different from the max and min values but f(x,y)=min or f(x,y)=max. Otherwise, f(x,y) is not changed. Consider the following notation: Sxy = mask region (neighborhood sub-image) zmin = minimum gray level value in Sxy zmax = maximum gray level value in Sxy zmed = median of gray levels in Sxy zxy = gray level at coordinates (x, y) Smax = maximum allowed size of Sxy The adaptive median filtering algorithm works in two levels A and B as follows: Level A: A1 = zmed - zmin A2 = zmed - zmax If A1 > 0 AND A2 < 0, Go to level B Else increase the window size If window size 0 AND B2 < 0, output zxy Else output zmed. The next figure shows an example of filtering an image corrupted by salt- and-pepper noise with density 0.25 using 7×7 median filter and the adaptive median filter with Smax = 7.  (a) (b) (c) (a) Image corrupted by salt&pepper noise with density 0.25. (b) Result obtained using a 7×7 median filter. (c) Result obtained using adaptive median filter with Smax = 7. From this example, we find that the adaptive median filter has three main purposes: 1. to remove salt-and-pepper (impulse) noise. 2. to provide smoothing of other noise that may not be impulsive. 3. to reduce distortion, such as excessive thinning or thickening of object boundaries.