1	Image filtering
2	Image filtering
3	Reading Forsyth & Ponce, chapter 7
4	What is an image?
5	Images as functions We can think of an image as a function, f, from R² to R: f( x, y ) gives the intensity at position ( x, y ) Realistically, we expect the image only to be defined over a rectangle, with a finite range: f: [a,b]x[c,d] à [0,1] A color image is just three functions pasted together. We can write this as a “vector-valued” function:
6	Images as functions
7	What is a digital image? We usually work with digital (discrete) images: Sample the 2D space on a regular grid Quantize each sample (round to nearest integer) If our samples are D apart, we can write this as: f[i ,j] = Quantize{ f(i D, j D) } The image can now be represented as a matrix of integer values
8	Filtering noise How can we “smooth” away noise in an image?
9	Mean filtering
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11	Mean filtering
12	Cross-correlation filtering Let’s write this down as an equation. Assume the averaging window is (2k+1)x(2k+1):
13	Mean kernel What’s the kernel for a 3x3 mean filter?
14	Mean vs. Gaussian filtering
15	Gaussian filtering A Gaussian kernel gives less weight to pixels further from the center of the window This kernel is an approximation of a Gaussian function: What happens if you increase s ? Photoshop demo
16	Image gradient How can we differentiate a digital image F[x,y]? Option 1: reconstruct a continuous image, f, then take gradient Option 2: take discrete derivative (finite difference)
17	Image gradient
18	Scale space (Witkin 83)
19	Scale space: insights As the scale is increased edge position can change edges can disappear new edges are not created Bottom line need to consider edges at different scales (or else know what scale you care about)
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22	Convolution A convolution operation is a cross-correlation where the filter is flipped both horizontally and vertically before being applied to the image: It is written: Suppose H is a Gaussian or mean kernel. How does convolution differ from cross-correlation? Suppose F is an impulse function (previous slide) What will G look like?
23	Continuous Filters We can also apply filters to continuous images. In the case of cross correlation: In the case of convolution: Note that the image and filter are infinite.