3 Spatial FilteringThe word “filtering” has been borrowed from the frequency domain.Filters are classified as:Low-pass (i.e., preserve low frequencies)High-pass (i.e., preserve high frequencies)Band-pass (i.e., preserve frequencies within a band)Band-reject (i.e., reject frequencies within a band)
4 Spatial Filtering (cont’d) Spatial filtering is defined by:(1) A neighborhood(2) An operation that is performed on the pixels inside the neighborhoodoutput image
5 Spatial Filtering - Neighborhood Typically, the neighborhood is rectangular and its sizeis much smaller than that of f(x,y)- e.g., 3x3 or 5x5
6 Spatial filtering - Operation Assume the origin of themask is the center of themask.for a 3 x 3 mask:for a K x K mask:
7 Spatial filtering - Operation A filtered image is generated as the center of the mask moves to every pixel in the input image.output image
8 Handling Pixels Close to Boundaries pad with zeroes0 0 0 ……………………….0or0 0 0 ……………………….0
9 Linear vs Non-Linear Spatial Filtering Methods A filtering method is linear when the output is a weighted sum of the input pixels.Methods that do not satisfy the above property are called non-linear.e.g.,
10 Linear Spatial Filtering Methods Two main linear spatial filtering methods:CorrelationConvolution
12 Correlation (cont’d)Often used in applications where we need to measure the similarity between images or parts of images(e.g., pattern matching).
13 ConvolutionSimilar to correlation except that the mask is first flipped both horizontally and vertically.Note: if w(x,y) is symmetric, that is w(x,y)=w(-x,-y), then convolution is equivalent to correlation!
15 How do we choose the elements of a mask? Typically, by sampling certain functions.Gaussian1st derivativeof Gaussian2nd derivative
16 Filters Smoothing (i.e., low-pass filters) Reduce noise and eliminate small details.The elements of the mask must be positive.Sum of mask elements is 1 (after normalization)Gaussian
17 Filters (cont’d) Sharpening (i.e., high-pass filters) Highlight fine detail or enhance detail that has been blurred.The elements of the mask contain both positive and negative weights.Sum of the mask weights is 0 (after normalization)1st derivativeof Gaussian2nd derivative
25 Smoothing Filters: Median Filtering (non-linear) Very effective for removing “salt and pepper” noise (i.e., random occurrences of black and white pixels).medianfilteringaveraging
26 Smoothing Filters: Median Filtering (cont’d) Replace each pixel by the median in a neighborhood around the pixel.
27 Sharpening Filters (High Pass filtering) Useful for emphasizing transitions in image intensity (e.g., edges).
28 Sharpening Filters (cont’d) Note that the response of high-pass filtering might be negative.Values must be re-mapped to [0, 255]sharpened imagesoriginal image
29 Sharpening Filters: Unsharp Masking Obtain a sharp image by subtracting a lowpass filtered (i.e., smoothed) image from the original image:-=
30 Sharpening Filters: High Boost Image sharpening emphasizes edges but details (i.e., low frequency components) might be lost.High boost filter: amplify input image, then subtract a lowpass image.(A-1)+=
31 Sharpening Filters: High Boost (cont’d) If A=1, we get a high pass filterIf A>1, part of the original image is added back to the high pass filtered image.
39 Sharpening Filters: Gradient Computation (cont’d) We can implement and using masks:(x+1/2,y)good approximationat (x+1/2,y)(x,y+1/2)**good approximationat (x,y+1/2)Example: approximate gradient at z5
40 Sharpening Filters: Gradient Computation (cont’d) A different approximation of the gradient:good approximation(x+1/2,y+1/2)*We can implement and using the following masks: