Image filtering

Reading

Forsyth & Ponce, chapter 7

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:

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?

Image processing

Still other operations operate on both the domain and the range of f .

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

Ideal noise reduction

Practical noise reduction

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

Mean filtering

Slide 16

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):

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:

What happens if you increase s ?

Mean vs. Gaussian filtering

Filtering an impulse

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?

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.

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


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:


	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


	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?


	Some operations preserve the range but change the domain of f :

	What kinds of operations can this perform?


	Still other operations operate on both the domain and the range of f .


	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


	Let’s write this down as an equation. Assume the averaging window is (2k+1)x(2k+1):


	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 ?


	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?


	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.


	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?