1	Images and Transformations
2	Reading Forsyth & Ponce, chapter 7
3	What is an image? 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:
4	Images as functions
5	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
6	Image transformations 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?
7	Image processing Some operations preserve the range but change the domain of f : What kinds of operations can this perform? Many image transforms operate both on the domain and the range
8	Linear image transforms Let’s start with a 1-D image (a “signal”):
9	Linear image transforms Let’s start with a 1-D image (a “signal”):
10	Linear image transforms Let’s start with a 1-D image (a “signal”):
11	Linear image transforms Another example is the discrete Fourier transform Each row of M is a sinusoid (complex-valued) The frequency increases with the row number
12	Linear shift-invariant filters
13	2D linear transforms We can do the same thing for 2D images by concatenating all of the rows into one long vector:
14	2D filtering A 2D image f[i,j] can be filtered by a 2D kernel h[u,v] to produce an output image g[i,j]: This is called a cross-correlation operation and written: h is called the “filter,” “kernel,” or “mask.” The above allows negative filter indices. When you implement need to use: h[u+k,v+k] instead of h[u,v]
15	Noise Filtering 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
16	Mean filtering
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18	Mean filtering
19	Effect of mean filters
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:
22	Mean vs. Gaussian filtering
23	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?
24	Convolution theorems Let F be the 2D Fourier transform of f, H of h Define Convolution theorem: Convolution in the spatial (image) domain is equivalent to multiplication in the frequency (Fourier) domain. Symmetric theorem: Convolution in the frequency domain is equivalent to multiplication in the spatial domain. Why is this useful?
25	2D convolution theorem example
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