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?

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

Filtering noise

How can we “smooth” away noise in an image?

Mean filtering

Slide 10

Mean filtering

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?

Mean vs. Gaussian filtering

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

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)

Image gradient

Scale space (Witkin 83)

Scale space: insights

As the scale is increased

edge position can change

edges can disappear

new edges are not created

Bottom line

need to consider edges at different scales
(or else know what scale you care about)

Slide 20

Slide 21

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.


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:


	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


	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 ? Photoshop demo


	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)


	As the scale is increased
		edge position can change
		edges can disappear
		new edges are not created







	Bottom line
		need to consider edges at different scales (or else know what scale you care about)


	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.