1	Image filtering
2	Image filtering
3	Reading Szeliski, Chapter 3.1-3.2
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	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?
19	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.
20	More on filters… Cross-correlation/convolution is useful for, e.g., Blurring Sharpening Edge Detection Interpolation Convolution has a number of nice properties Commutative, associative Convolution corresponds to product in the Fourier domain More sophisticated filtering techniques can often yield superior results for these and other tasks: Polynomial (e.g., bicubic) filters Steerable filters Median filters Bilateral Filters … (see text, web for more details on these)