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