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Unformatted text preview: Lab # 3 - ACS I DATA COMPRESSION in IMAGE PROCESSING using SVD Goals. The goal of this lab is to demonstrate how the SVD can be used to remove redundancies in data; in this example we will be compressing image data. We will see that for a matrix of rank r , the SVD can give a rank p < r approximation to a matrix and that this approximation is the one that minimizes the Frobenius norm of the error. Introduction Any image from a digital camera, scanner or in a computer is a digital image . The “real world” color image is digitized by converting the images to numerical data. A pixel is the smallest element of the digital image. For example, a 3 megapixel camera has a grid of 2048 × 1536 = 3,145,828 pixels. Since the size of a digitized image is dimensioned in pixels of say m rows and n columns, it is easy for us to think of the image as an m × n matrix. However, each pixel of a color image has an RGB values (red, green, blue) which is represented by three numbers. The composite of the three RGB values creates the final color for the single pixel. So we can think of each entry in the m × n matrix as having three numerical values stored in that location, i.e., an m × n × 3 matrix. Now suppose we have a digital image taken with a 3 megapixel camera and each color pixel is determined by a 24-bit number (8 bits for intensity of red, blue and green). Then the information we have is roughly 3 × 10 6 × 24. However, when we print the picture suppose we only use 8-bit colors giving 2 8 = 256 colors. We are still using 3 million pixels but the information used to describe the image has been reduced to 3 × 10 6 × 8, i.e., a reduction of one-third. This is an example of image compression. In the figure below we give a grayscale image of the moon’s surface (the figure on the left) along with two different compressed images. Clearly the center image is not acceptable but the compressed image on the right has most of the critical information. We want to investigate using the SVD for doing data compression in image processing. Understanding the SVD Recall from the notes that the SVD is related to the familiar result that any n × n real symmetric matrix can be made orthogonally similar to a diagonal matrix which gives us the decomposition A = Q Λ Q T where Q is orthogonal and Λ is a diagonal matrix containing the eigenvalues of A . 1 Figure 1: The image on the left is the original image while the other two images represent the results of data compression....
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This note was uploaded on 01/15/2012 for the course ISC 5315 taught by Professor Staff during the Spring '11 term at FSU.

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