Cascaded Gaussians Consider an extremely simple smoothing fi

Cascaded Gaussians

Consider an extremely simple smoothing filter G=[1 1]. Although this doesn\'t look much like a Gaussian filter, it is indeed a very simple approximation to a Gaussian, with sigma=1/2 (it also happens to be a simple box smoothing filter). Imagine convolving an input image by the filter G twice. This is the same as convolving the input image once with the filter G*G =[1 2 1], which has a larger size than G and thus operates over image pixel values in larger local neighborhoods. As discussed in class, it is also a Gaussian filter, and using the formula from class its standard deviation can be computed as 1/sqrt(2). The table below shows the set of Gaussians derived by cascading the [1 1] Gaussian filter. That is, each row is formed by convolving the previous row with a 1 times 2 filter having coefficients [1 1], retaining output values for all partial overlaps of the filter with the previous row. For example: [1 2 1]-[1 1] *[1 1] [1 3 3 1] = [1 1] * [1 2 1] [1 4 6 4 1] = [1 1] = [1 3 3 1] And so on, with each row(N) being the result of [1 1] * row(N-1). Note that this a rare example in this class of a filter with even size, leaving open the question of which pixel is to be considered the center pixel. If this bothers you, think of it as a 1 times 3 filter [0 1 1] or [1 1 0] and proceed as usual. (A) Fill in the blanks for rows 5 and 6. (B) Assume the standard deviation (sigma value) for the [1 1] Gaussian filter is 1/2. From this we can derive the effective sigma value of the other Gaussian filters formed by cascading the [1 1] filter. We have entered some of the sigma values in the table for you. Fill in the blanks for rows 3 and 4. (C) What is the effective sigma value for row 100 in the table?

Solution

Gaussian filtering is an important tool in image processing and computer vision. In this paper we discuss the background of Gaussian filtering and look at some methods for implementing it. Consideration of the central limit theorem suggests using a cascade of \"simple\" filters as a means of computing Gaussian filters. Among \"simple\" filters, uniform-coefficient finite-impulse-response digital filters are especially economical to implement. The idea of cascaded uniform filters has been around for a while [13], [16]. We show that this method is economical to implement, has good filtering characteristics, and is appropriate for hardware implementation. We point out an equivalence to one of Burt\'s methods [1], [3] under certain circumstances. As an extension, we describe an approach to implementing a Gaussian Pyramid which requires approximately two addition operations per pixel, per level, per dimension. We examine tradeoffs in choosing an algorithm for Gaussian filtering, and finally discuss an implementation.