Notes

Outline

Image filtering

Reading Forsyth & Ponce, chapters 8.1-8.2 http://www.cs.washington.edu/education/courses/490cv/02wi/readings/book-7-revised-a-indx.pdf

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:

Images as functions

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

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?

Image processing Some operations preserve the range but change the domain of f : What kinds of operations can this perform?

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

Practical noise reduction How can we “smooth” away noise in a single image?

Mean filtering

Slide 11

Mean filtering

Effect of mean filters

Cross-correlation filtering Let’s write this down as an equation.  Assume the averaging window is (2k+1)x(2k+1): We can generalize this idea by allowing different weights for different neighboring pixels: 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]

Mean kernel What’s the kernel for a 3x3 mean filter?

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:

Mean vs. Gaussian filtering

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?

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?

Comparison: salt and pepper noise

Comparison: Gaussian noise