Digital Image Processing: Image Restoration - Duong Anh Duc

Digital Image Processing: Image Restoration - Duong Anh Duc includes Image Restoration; Restoration vs. Enhancement; Degradation Model; Gaussian noise; Erlang(Gama) noise; Exponential noise; Impulse (salt-and-pepper) noise; Plot of density function of different noise models.

Digital Image Processing Image Restoration 21/11/15 Duong Anh Duc - Digital Image Processing Image Restoration  Most images obtained by optical, electronic, or electro-optic means is likely to be degraded  The degradation can be due to camera misfocus, relative motion between camera and object, noise in electronic sensors, atmospheric turbulence, etc  The goal of image restoration is to obtain a relatively “clean” image from the degraded observation  It involves techniques like filtering, noise reduction etc 21/11/15 Duong Anh Duc - Digital Image Processing Restoration vs Enhancement  Restoration: A process that attempts to reconstruct or recover an image that has been degraded by using some prior knowledge of the degradation phenomenon  Involves modeling the degradation process and applying the inverse process to recover the original image  A criterion for “goodness” is required that will recover the image in an optimal fashion with respect to that criterion  Ex Removal of blur by applying a deblurring function  21/11/15 Duong Anh Duc - Digital Image Processing Restoration vs Enhancement  Enhancement:  Manipulating an image in order to take advantage of the psychophysics of the human visual system  Techniques are usually “heuristic.”  Ex Contrast stretching, histogram equalization 21/11/15 Duong Anh Duc - Digital Image Processing (Linear) Degradation Model g(m,n) = f(m,n)*h(m,n) + (m,n) G(u,v) = H(u,v)F(u,v) + N(u,v) f(m,n) : Degradation free image g(m,n) : Observed image h(m,n) : PSS of blur degradation (m,n) : Additive Noise 21/11/15 Duong Anh Duc - Digital Image Processing (Linear) Degradation Model Problem: Given an observed image g(m,n) , to recover the original image f(m,n) , using knowledge about the blur function h(m,n) and the characteristics of the noise (m,n) ?  We need to find an image ^f (m,n) , such that the error f (m,n) - ^f (m,n) is “small.” 21/11/15 Duong Anh Duc - Digital Image Processing Noise Models  With the exception of periodic interference, we will assume that noise values are uncorrelated from pixel to pixel and with the (uncorrupted) image pixel values  These assumptions are usually met in practice and simplify the analysis  With these assumptions in hand, we need to only describe the statistical properties of noise; i.e., its probability density function (PDF) 21/11/15 Duong Anh Duc - Digital Image Processing Gaussian noise  Mathematically speaking, it is the most tractable noise model  Therefore, it is often used in practice, even in situations where they are not well justified from physical principles  The pdf of a Gaussian random variable z is given by: where z represents (noise) gray value, m is the mean, and s is its standard deviation The squared standard deviation is usually referred to as variance  For a Gaussian pdf, approximately 70% of the values are within one standard deviation of the mean and 95% of the values are within two standard deviations of the mean 21/11/15 Duong Anh Duc - Digital Image Processing Rayleigh noise  The pdf of a Rayleigh noise is given by:  The mean and variance are given by:  This noise is “one-sided” and the density function is skewed 21/11/15 Duong Anh Duc - Digital Image Processing Erlang(Gama) noise  The pdf of Erlang noise is given by: where, a > 0, b is an integer and “!” represents factorial  The mean and variance are given by:  This noise is “one-sided” and the density function is skewed 21/11/15 Duong Anh Duc - Digital Image Processing 10 Example 21/11/15 Duong Anh Duc - Digital Image Processing 67 Inverse Filter  The simplest approach to restoration is direct inverse filtering This is obtained as follows: where 21/11/15 Duong Anh Duc - Digital Image Processing 68 Inverse Filter  We can rewrite this in the spatial domain as follows:  In practice, we actually use a slightly modified filter: where is a small value This avoids numerical problems when |H(u,v)| is small 21/11/15 Duong Anh Duc - Digital Image Processing 69 Inverse Filter  The inverse filter works fine provided there is no noise This is illustrated in the following example  Let us now analyze the performance of the inverse filter in the presence of noise Indeed, in this case: G(u,v) = H(u,v)F(u,v) + N(u,v) which gives 21/11/15 Duong Anh Duc - Digital Image Processing 70 Inverse Filter  Hence noise actually gets amplified at frequencies where |H(u,v)| is zero or very small In fact, the contribution from the noise term dominates at these frequencies  As illustrated by an example, the inverse filter fails miserably in the presence of noise It is therefore, seldom used in practice, in the presence of noise 21/11/15 Duong Anh Duc - Digital Image Processing 71 Inverse Filtering example (no noise) 21/11/15 Duong Anh Duc - Digital Image Processing 72 Inverse Filtering example (no noise) g(m,n) 21/11/15 Duong Anh Duc - Digital Image Processing 73 Inverse Filtering example (no noise) fˆ m, n 21/11/15 Duong Anh Duc - Digital Image Processing 74 Inverse Filtering example (no noise) f(m,n) 21/11/15 Duong Anh Duc - Digital Image Processing 75 Inverse Filtering example (no noise) fˆ m, n g(m,n) r0= 11 MSE = 0.02 21/11/15 MSE = 0.008 Duong Anh Duc - Digital Image Processing 76 Inverse Filtering example (no noise) fˆ m, n g(m,n) r0= 15 MSE = 0.017 21/11/15 MSE = 0.005 Duong Anh Duc - Digital Image Processing 77 Inverse Filtering example (no noise) fˆ m, n g(m,n) r0= 23 MSE = 0.013 21/11/15 MSE = 0.0016 Duong Anh Duc - Digital Image Processing 78 Inverse Filtering example (with noise) We will add the Zero­mean  Gaussian noise with variance  21/11/15 f(m,n) Duong Anh Duc - Digital Image Processing 79 Inverse Filtering example (with noise)  = 0.03 MSE = 0.008 21/11/15 g(m,n)  = 0.01 MSE = 0.007 Duong Anh Duc - Digital Image Processing  = 0.02 MSE = 0.0075 80 Inverse Filtering example (with noise) fˆ m, n MSE = 0.09 21/11/15 MSE = 0.09 Duong Anh Duc - Digital Image Processing MSE = 0.047 81 ... “!” represents factorial 21/11/15 Duong Anh Duc - Digital Image Processing 13 Plot of density function of different noise models 21/11/15 Duong Anh Duc - Digital Image Processing 14 Plot of density... different noise models 21/11/15 Duong Anh Duc - Digital Image Processing 15 Plot of density function of different noise models 21/11/15 Duong Anh Duc - Digital Image Processing 16 Test pattern... types of noise 21/11/15 Duong Anh Duc - Digital Image Processing 17 Test pattern and illustration of the effect of different types of noise 21/11/15 Duong Anh Duc - Digital Image Processing 18 Test