1	Image filtering
2	Image filtering
3	Reading Forsyth & Ponce, chapter 8 (in reader)
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? In computer vision we usually operate on 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	Image processing An image processing operation typically defines a new image g in terms of an existing image f. We can transform either the domain or the range of f. Range transformation: What’s kinds of operations can this perform?
9	Image processing Some operations preserve the range but change the domain of f : What kinds of operations can this perform?
10	Image processing Still other operations operate on both the domain and the range of f .
11	Noise Image processing is useful for noise reduction... Common types of noise: Salt and pepper noise: contains random occurrences of black and white pixels Impulse noise: contains random occurrences of white pixels Gaussian noise: variations in intensity drawn from a Gaussian normal distribution
12	Ideal noise reduction Given a camera and a still scene, how can you reduce noise?
13	Ideal noise reduction Given a camera and a still scene, how can you reduce noise?
14	Practical noise reduction How can we “smooth” away noise in a single image?
15	Mean filtering
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17	Mean filtering
18	Effect of mean filters
19	Cross-correlation filtering Let’s write this down as an equation. Assume the averaging window is (2k+1)x(2k+1):
20	Mean kernel What’s the kernel for a 3x3 mean filter?
21	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 ?
22	Mean vs. Gaussian filtering
23	Filtering an impulse
24	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?
25	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.
26	Median filters A Median Filter operates over a window by selecting the median intensity in the window. What advantage does a median filter have over a mean filter? Is a median filter a kind of convolution?
27	Comparison: salt and pepper noise
28	Comparison: Gaussian noise