Image Processing

Today’s readings

Forsyth & Ponce, chapters 8.1-8.2

http://www.cs.washington.edu/education/courses/490cv/02wi/readings/book-7-revised-a-indx.pdf

For Monday

Watt, 10.3-10.4 (handout)

Cipolla and Gee (handout)

supplemental: Forsyth, chapter 9

Announcements

HW0 due now

Project 0 is out today. Li will give a help-session in the last 10 minutes of class.

If you don’t have a CSE account

turn in your account forms right after class!

If you want to add

go ahead and register in EE400B, SLN 8590

If you don’t have access to 228

talk to me after class today

All announcements are posted via the class webpage

You may also use the mailing list (cse490cv@cs) for questions

What’s an image?

We can think of an image as a function, f, from R² to R:

gives the intensity of a channel at position

typically only defined over a rectangle, with a finite range

A color image is just three functions pasted together

We can write this as a “vector-valued” function:

What’s a digital image?

We usually operate on digital (discrete) images:

Sample the space on a regular grid

Quantize each sample (round to nearest integer)

If our samples are d apart, we can write this as:

Image processing

An image processing operation 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

Noise

Common types of noise:

Salt and pepper noise: random black and white pixels

Impulse noise: random white pixels

Gaussian noise: variations in intensity drawn from a Gaussian (normal) distribution

Reducing Noise

Mean filtering

Replace each pixel with an average of the pixels in the k´k box around it

3´3 case:

Mean filtering

What about border pixels?

Effect of filter size

What happens if we

use a larger mean filter?

5´5? 7´7?

Weighted average filtering

Gaussian filter

Comparison of mean vs. gaussian filter

Linear shift-invariant filters

What happens when a filter is applied to an impulse?

helps explain why mean filters can give “blocky” results

Cross correlation

For a k´k filter, the equation becomes (assume k is odd):

If we choose the origin of h, F, and G to be the center, we can simplify:

This filtering operation is known as a cross-correlation, written

For continuous images, cross-correlation is defined as:

Similar to convolution:

Convolution flips the filter horizontally and vertically before applying it

how does convolution with a Gaussian filter differ from cross-correlation?

Linear shift-invariant filters

Both cross-correlation and convolution filters have the following properties

linear:

how about scale? Is the following true?

shift-invariant: same operation applied for every x,y in f

Any filter with these properties (LSI) may be written as a cross-correlation

Is this also true for convolution?

Convolution has nice properties in the Fourier Domain

we won’t cover this, but see Forsyth Chap. 8.3 for a nice discussion

Symmetry in Convolutions

Convolutions have a symmetry property:

Try it out:

Median filter

A median filter operates over a k´k window by returning the median pixel value in that window

What advantage might a median filter have over a mean filter?

Can a median filter be implemented by a cross-correlation?

Slide 21

Slide 22

Image Scaling

Image sub-sampling

Even worse for synthetic images

Sampling and the Nyquist rate

Aliasing can arise when you sample a continuous signal or image

Demo applet http://www.cs.brown.edu/exploratory/research/applets/repository/nyquist/nyquist.html

occurs when your sampling rate is not high enough to capture the amount of detail in your image

formally, the image contains structure at different scales

called “frequencies” in the Fourier domain

the sampling rate must be high enough to capture the highest frequency in the image

To avoid aliasing:

sampling rate > 2 * max frequency in the image

i.e., need more than two samples per period

This minimum sampling rate is called the Nyquist rate

Subsampling with Gaussian pre-filtering

Some times we want many resolutions

Gaussian pyramid construction

Image resampling

So far, we considered only power-of-two subsampling

What about arbitrary scale reduction?

How can we increase the size of the image?

Image resampling

So what to do if we don’t know

Image resampling

What does the 2D version of this hat function look like?

Subsampling with bilinear pre-filtering

Things to take away from this lecture

Things to take away from this lecture

An image as a function

Digital vs. continuous images

Image transformation: range vs. domain

Types of noise

LSI filters

cross-correlation and convolution

properties of LSI filters

mean, Gaussian, bilinear filters

Median filtering

Image scaling

Image resampling

Aliasing

Gaussian pyramids


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

For Monday
	Watt, 10.3-10.4 (handout)
	Cipolla and Gee (handout)
		supplemental: Forsyth, chapter 9


	HW0 due now
	Project 0 is out today. Li will give a help-session in the last 10 minutes of class.
	If you don’t have a CSE account
		turn in your account forms right after class!
	If you want to add
		go ahead and register in EE400B, SLN 8590
	If you don’t have access to 228
		talk to me after class today

	All announcements are posted via the class webpage
	You may also use the mailing list (cse490cv@cs) for questions


	We can think of an image as a function, f, from R² to R:
		gives the intensity of a channel at position
		typically only defined over a rectangle, with a finite range


	A color image is just three functions pasted together
		We can write this as a “vector-valued” function:


	We usually operate on digital (discrete) images:
		Sample the space on a regular grid
		Quantize each sample (round to nearest integer)
	If our samples are d apart, we can write this as:


	An image processing operation 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


	Common types of noise:
		Salt and pepper noise: random black and white pixels
		Impulse noise: random white pixels
		Gaussian noise: variations in intensity drawn from a Gaussian (normal) distribution


	Replace each pixel with an average of the pixels in the k´k box around it

		3´3 case:


	What happens when a filter is applied to an impulse?










		helps explain why mean filters can give “blocky” results


For a k´k filter, the equation becomes (assume k is odd):


If we choose the origin of h, F, and G to be the center, we can simplify:


This filtering operation is known as a cross-correlation, written

For continuous images, cross-correlation is defined as:


Similar to convolution:


	Convolution flips the filter horizontally and vertically before applying it
		how does convolution with a Gaussian filter differ from cross-correlation?


Both cross-correlation and convolution filters have the following properties
	linear:
		how about scale? Is the following true?

	shift-invariant: same operation applied for every x,y in f
Any filter with these properties (LSI) may be written as a cross-correlation
	Is this also true for convolution?

Convolution has nice properties in the Fourier Domain
	we won’t cover this, but see Forsyth Chap. 8.3 for a nice discussion


	A median filter operates over a k´k window by returning the median pixel value in that window

	What advantage might a median filter have over a mean filter?


	Can a median filter be implemented by a cross-correlation?


Aliasing can arise when you sample a continuous signal or image
	Demo applet http://www.cs.brown.edu/exploratory/research/applets/repository/nyquist/nyquist.html
	occurs when your sampling rate is not high enough to capture the amount of detail in your image
	formally, the image contains structure at different scales
		called “frequencies” in the Fourier domain
	the sampling rate must be high enough to capture the highest frequency in the image
To avoid aliasing:
	sampling rate > 2 * max frequency in the image
		i.e., need more than two samples per period
	This minimum sampling rate is called the Nyquist rate


	So far, we considered only power-of-two subsampling
		What about arbitrary scale reduction?
		How can we increase the size of the image?


Things to take away from this lecture
	An image as a function
	Digital vs. continuous images
	Image transformation: range vs. domain
	Types of noise
	LSI filters
		cross-correlation and convolution
		properties of LSI filters
		mean, Gaussian, bilinear filters
	Median filtering
	Image scaling
	Image resampling
	Aliasing
	Gaussian pyramids