# Image Processing Class (EGBE443) #4 — Filters

The center of it is the origin location and it is called the ‘hot spot’.Applying weight median filter to the image I, a hotspot location is at the orange shade (center of the filter matrix H)Applying the filterTo apply the filter to the image, please follow these stepMove the filter matrix over the image I and H(0,0) must go along with the current image position (u,v)Multiply each filter coefficient H(i,j) with the corresponding image element I(u+i,v+j)Average all result from the previous step and it is the result for the current location I(u,v)All steps can be described as equation belowType of linear filterSmoothing Filter (This filter has only positive integer.)Box filter..All members of this filter are the same.Gaussian filter..The weight of filter member depend on the location of the member..The center of the filter receive the maximum weigh and it decreases with distance from the center.2..Different FilterLaplace or Mexican hat filter..Some members of this filter are negative filter and it can calculate by summation of positive member and negative member.3D structure, 2D structure and example of filter (a) Box filter (b) Gaussian filter and (c) Laplace filterProperties of Linear FilterFirst, i will introduce an operation which associate with linear filter..This operation is call “Linear Convolution”..For 2-dimensional function I and H, the convolution operation is defined as the equationwhere * is the convolution operation filter, look at the equation you will see that this operation provide the similar result with the linear filter with the filter function which reflect in both horizontal and vertical axis..The convolution matrix H can be called kernel.Properties of Linear ConvolutionCommutativityLinearityAssociativitySeperability: the kernel H can be represented as the convolution of multiple kernels and can separated in a pair dimensional kernel x and y.Note that: In the linearity properties, adding scalar value b to the image I before perform convolution with the kernel dose not equal to adding scalar value b to convolution result between the image and the kernel.Non-Linear FiltersNoise removing with smoothing filter (a linear filter) was provide the result in burred of the image structure, line and edge..Non-Linear Filters were used to solve this problem and it works in non-linear manner.Type of non-linear filtersMinimum and Maximum Filters: The minimum and maximum value in the moving region R of the original image is the result of the minimum and maximum filter respectively..These filter were defined asThe equation of minimum and maximum filterMedian Filter: The result was calculated in the same way as the minimum and maximum filter..The median of all value in moving region R is the result of the median filter.. More details