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Image Restoration CONTENT • Overview • Noise Models – – – – Gaussian Salt-and-pepper Uniform Rayleigh • Noise Removal using Spatial Filters – Order filters – Mean filters • Geometric Transforms Overview • Used to improve the appearance of an image by application of a restoration process that uses a mathematical model for image degradation • Types of degradation :– Blurring caused by motion or atmospheric disturbance – Geometric distortion caused by imperfect lenses – Superimposed interference patterns caused by mechanical systems – Noise from electronic sources • We see that sample degraded images and knowledge of the image acquisition process are inputs to development of a degradation model • After the model has been developed, the next step is the formulation of the inverse process • This inverse degradation process is then applied to the degraded image, d(r,c), which results in the output image, Î(r,c), • The output image Î(r,c), is the restored image which represents an estimate of original image, I(r,c) d̂ • Once the estimated image has been created, any knowledge gained by observation & analysis of this image is used as additional input for further development of degradation model • This process continues until satisfactory results are achieved • With this perspective, can define image restoration as the process of finding an approximation to the degradation process & finding the appropriate inverse process to estimate the original image System Model • Degradation process model consists of 2 parts, the degradation function & the noise function • General mode in spatial domain : d(r,c) = h(r,c) * I(r,c) + n(r,c) where d(r,c) = degraded image h(r,c) = degradation function I(r,c) = original image n(r,c) = additive noise function System Model • Frequency domain : D(u,v) = H(u,v) * I(u,v) + N(u,v) where D(u,v) = Fourier transform of the degraded image H(u,v) = Fourier transform of the degradation function I(u,v) = Fourier transform of the original image N(u,v) = Fourier transform of the additive noise function Noise Models • Any undesired information that contaminates an image • noise models is a random variable with a probability density function (PDF) that describes its shape and distribution • The actual distribution of noise in a specific image is the histogram of the noise • Noise can be modeled with Gaussian (“normal”), uniform, salt-and-pepper (“impulse”), or Rayleigh distribution • Gaussian model – occur from electronic noise in image acquisition system – Most problematic with poor lighting conditions or vary high temperatures – Also valid for film grain noise • Salt-and-pepper noise (also called impulse noise, shot noise or spike noise) typically caused by malfunctioning pixel element in camera sensors, faulty memory locations, or timing errors in digitization process • Uniform noise is useful - it can be used to generate any other type of noise distribution, and is often used to degrade images for the evaluation of image restoration algo since provides the most unbiased or neutral noise model Gaussian distribution 1 Hg 2 2 e ( g m ) 2 / 2 2 g gray level m mean (average) standard deviation var iance 2 • A bell-shapped • 70% of all values fall within the range from one standard deviation (σ) below the mean (m) to one above • About 95% fall within two standard deviations Uniform distribution • The gray level values of the noise are evenly distributed across a specific range H Uniform mean 1 b a 0 a b 2 (b - a) 2 variance 12 , for a g b elsewhere Salt-and-pepper A H salt& paper B for g a ("pepper") for g b ("salt") • There are only 2 possible values, a and b, and the probability of each is typically less than 0.2 – with numbers greater than this the noise will swamp out the image Rayleigh H Rayleigh 2g e g 2 / where : mean varia nce 4 (4 - ) 4 Original image without noise, and its histogram image with added Gaussian noise with mean = 0 and variance = 600, and its histogram image with added uniform noise with mean = 0 and variance = 600, and its histogram image with added salt-and-pepper noise with the probability of each 0.08, and its histogram Noise Removal Using Spatial Filters • Spatial filters can be effectively used to remove various types of noise • Operate on small neighborhoods, 3x3 to 11x11 • Will use the degradation model with the assumption that h(r,c) causes no degradation where the only corruption to the image is caused by additive noise d(r,c) = I(r,c) + n(r,c) where d(r,c) = degraded image I(r,c) = original image n(r,c) = additive noise function • Two primary categories; order filters and mean filters • Order filters – implemented by arranging the neighborhood pixels in order from smallest to largest gray level value, and using this ordering to select the “correct” value • Mean filters determine, in one sense or another, an average value • Mean filters work best with Gaussian or uniform noise • Order filters work best with salt-and-pepper, negative exponential, or Rayleigh noise • Mean filters have disad of blurring the image edges, or details • Order filters such as mean can be used to smooth images Order Filters • Operate on small subimages, windows, and replace the center pixel value (similar to convolution process) • Given an N x N wondow, W, the pixel values can be ordered as follows I1 , I 2 I 3 ...... I N 2 where I , I , I ,......., I are the Intensity 2 1 2 3 N (gray level) values 110 110 114 100 104 104 95 88 85 • (85, 88, 95, 100, 104, 104, 110, 110, 114) • Min = 85, Med = 104, max = 114 (will be replaced at the center value) • Median filter is most useful • Max & min filters can eliminate salt or pepper noise a) Image with added salt-and-pepper noise, the probability for salt = probability for pepper = 0.10, b) after median filtering with a 3x3 window, all the noise is not removed a) b) c) after median filtering with a 5x5 window, all the noise is removed, but the image is blurry acquiring the “painted” effect c) • Two order filters are midpoint and alphatrimmed mean filters – both order and mean filters since they rely on ordering the pixels values, but are then calculated by an averaging process • Midpoint filter – the average of max & min within the window; Ordered set I1 , I 2 I 3 ...... I N Midpoint 2 I1 I N 2 2 • Most useful for Gaussian & uniform noise • Alpha-trimmed mean is the average of pixel values within the window, but with some of the endpoint ranked excluded • Useful for images containing multiple types of noise, Gaussian and salt-and-pepper noise Ordered set I1 , I 2 I 3 ...... I N 1 Alpha - trimmed mean 2 N 2T 2 N 2 T I i T 1 i where T is the number of pixel values excluded at each end of the ordered set, and can range from 0 to (N2 – 1)/2 • Alpha-trimmed mean filter ranges from a mean to median filter, depending on the value selected for the T parameter Figure 9.3-5 Alpha-Trimmed Mean. This filter can vary between a mean filter and a median filter. a) Image with added noise: zero-mean Gaussian noise with a variance of 200, and saltand-pepper noise with probability of each = 0.03, b) result of alpha-trimmed mean filter, mask size = 3x3, T = 1, c) result of alpha-trimmed mean filter, mask size = 3x3, T = 2, d) result of alphatrimmed mean filter, mask size = 3x3, T = 4. As the T parameter increases the filter becomes more like a median filter, so becomes more effective at removing the salt-and pepper noise. a) b) c) d) Mean Filters • Function by finding some form of an average within the NxN window, using sliding window concept to process entire image • The most basic – arithmetic mean filter which finds the arithmetic average of pixel values ; 1 Arithmetic mean 2 N 2 d (r , c) r ,c )W where N = the number of pixels in (the NxN window, W • Smooths out local variations & work best with Gaussian, gamma and uniform noise • Contra-harmonic mean filter works well for images containing salt OR pepper type noise, depending on the filter order, R: d(r, c) d(r, c) R 1 Contra - harmonic mean ( r ,c )W R ( r ,c )W where W is the NxN window under consideration • Negative values of R, eliminates salt-type noise • Positive values, eliminates pepper-type noise • Geometric mean filter works best with Gaussian noise, & retains detail information better than an arithmetic mean filter • Defined as the product of pixel values within window, raised to the 1/N2 power: Geometric mean I(r, c) ( r ,c )W 1 N2 • Harmonic mean filter also fails with pepper noise but works well for salt noise; Purata H N2 ( r , c ) W 1 d(r, c) • Retaining detail information better than the arithmetic mean filter • Yp mean filter is defined as follows: 1/ p P d ( r , c) Yp mean 2 ( r ,c )W N Geometric Transforms • Images that have been spatially, or geometrically, distorted • Used to modify the location of pixel values within an image, typically to correct images that have been spatially warped • Often referred as rubber-sheet transforms image is modeled as a sheet of rubber and stretched and shrunk • Because of defective optics in image acquisition system, distortion in image display devices, or 2D imaging of 3D surfaces • This methods are used in map making, image registration, image morphing, and other applications requiring spatial modification • Simplest – translate, rotate, zoom & shrink • More sophisticated – 1) spatial transform & 2) gray level interpolation Input Image Spatial Transform Gray Level Interpolation Output Image Spatial Transforms • Used to map the input image location to a location in the output image; it defines how the pixel values in output image are to be arranged I(r,c) rˆ Rˆ ( r , c ) d ( rˆ, cˆ) cˆ Cˆ ( r , c ) • The original, undistorted image, I(r,c), and distorted (or degraded) image is d ( rˆ, cˆ) rˆ Rˆ (r , c), defines the row coordinate for the distorted image cˆ Cˆ (r , c), defines the column coordinate for the distorted image • Primary idea is to find a mathematical model for the geometric distortion process – Rˆ (r , c) and Cˆ (r , c) and apply the inverse process to find restored image • Different equations for different portions of the image • To determine the necessary equations, need to identify a set of points in the original image that match points in the distorted image • These sets of points are called tiepoints, used to define the equations Rˆ (r , c) and Cˆ (r , c) • Method to restore a geometrically distorted image consists of 3 steps: 1. Define quadriterals (4 sided polygons) with known, or best-guessed tiepoints for the entire image 2. Find the equations Rˆ (r , c) and Cˆ (r , c) for each set of tiepoints, 3. Remap all the pixels within each quadrilateral subimage using the equations corresponding to those tiepoints 1. 2 images are divided into subimages, defined by tiepoints (fig. 9.6.3 a&b) 2. Using bilinear model for the mapping equations, these 4 points to generate the ˆ ( r , c ) k r k c k rc k r ˆ R equations : Cˆ (r , c) k r k c k rc k cˆ 3. Involves application of the mapping equations, Rˆ (r , c) and Cˆ (r , c) , to all the (r,c) pairs 1 5 2 6 3 7 4 8 Exercise • Example 9.6.1 & 9.6.2 • The difficulty in above example arises when we try to determine the value of d(41.4, 20.6) – Since digital images are defined only at integer values for Î(r,c) as an estimate to the original image I(r,c) to represent the restored image Gray Level Interpolation • The simplest – nearest neighbor method, where the pixel is assigned the value of the closest pixel in the distorted image – Î(2,3) is set to the value of d(41,21), the row and column values determined by rounding – Easy to implement and computationally fast • More advance is to interpolate the value – More computationally extensive but more visually pleasing results – Easiest - neighborhood average. Provide smoother object edges but slightly blurry Gray Level Interpolation • Better results- uses bilinear interpolation with the equation: g (rˆ, cˆ) k1rˆ k2cˆ k3rˆcˆ k4 where g (rˆ, cˆ) = the gray level interpolating equation • Example 9.6.3 & 9.6.4