Images and Transformations

Reading

	Forsyth & Ponce, chapter 7

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 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?

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

Linear image transforms

	Let’s start with a 1-D image (a “signal”):

Linear image transforms

	Let’s start with a 1-D image (a “signal”):

Linear image transforms

	Let’s start with a 1-D image (a “signal”):

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

Linear shift-invariant filters

2D linear transforms

	We can do the same thing for 2D images by concatenating all of the rows into one long vector:

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]

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

Mean filtering

Slide 17

Mean filtering

Effect of mean filters

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?

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?

2D convolution theorem example

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