Đang chuẩn bị nút TẢI XUỐNG, xin hãy chờ
Tải xuống
Image Transformations What is this chapter about? This chapter is concerned with the development of some of the most important tools of linear Image Processing, namely the ways by which we express an image as the linear superposition of some elementary images. How can we define an elementary image? | Image Processing The Fundamentals. Maria Petrou and Panagiota Bosdogianni Copyright 1999 John Wiley Sons Ltd Print ISBN 0-471-99883-4 Electronic ISBN 0-470-84190-7 Chapter 2 Image Transformations What is this chapter about This chapter is concerned with the development of some of the most important tools of linear Image Processing namely the ways by which we express an image as the linear superposition of some elementary images. How can we define an elementary image We can define an elementary image as the outer product of two vectors. What is the outer product of two vectors Consider two vectors N x 1 UiN Their outer product is defined as un Ui2 UuVji Vjl Vj2 .VjN U il VjN Ui2Vj2 Ui2VjN 2.1 UiNVji Therefore the outer product of these two vectors is an N x N matrix which can be thought of as an image. How can we expand an image in terms of vector outer products We saw in the previous chapter that a general separable linear transformation of an image matrix f can be written as 9 h fhr 2-2 22 Image Processing The Fundamentals where g is the output image and hc and hr are the transforming matrices. We can use the inverse matrices of h and hr to solve this expression for f in terms of g as follows Multiply both sides of the equation with if 1 on the left and h 1 on the right ftf 1ghr 1 if 1h fhrhr 1 f 2-3 Thus we write f f 1ghr 1 2-4 Suppose that we partition matrices if 1 vectors respectively and hr 1 in their column and row 2-5 Then 2-6 We may also write matrix g as a sum of N2 N x N matrices each one having only one non-zero element 9 911 0 0 . 0 . . 0 0 912 0 . . 0 . 0 . . 0 0 0 . 0 . 0 . 0 2-7 k 0 0 . . o kO 0 . . o 0 0 . 9nn Then equation 2.6 can be written as N N f ÎL Ei uivî i i j i 2-8 This is an expansion of image f in terms of vector outer products. The outer product Uivf may be interpreted as an image so that the sum over all combinations of the outer products appropriately weighted by the gij coefficients represents the original image . Image .
