More on image transforms

	C306
	Fall 2001
	Martin Jagersand

Assignment

	Heard concerns about long time between turn in and demo; matlab problems
	Consider:
		New due date Tue Oct 9, 17:00 at start of lab
		Everyone has to demo in Tue or Thu section or make individual appointment  with TA

Image operations:
Point op v.s. Transforms

	So far we have considered operations on one image point, ie contrast stretching, histogram eq. Etc
	By contrast, image transforms are defined over regions of an image or the whole image

Image transforms

	Image transforms are some of the most important methods in image processing
	Will be used later in:
		Filtering
		Restoration
		Enhancement
		Compression
		Image analysis

Vector spaces

	We will start with the concepts of
		vector spaces and
		coordinate changes,
	just as used in the camera models:
	Examples:
	Euclidean world space
	Image space,
	(q by q image):

Basis (of a vector space)

	A member can be described as a linear comb
	Example: 3D vectors
	Example: Image           = a          +b         +c

Vector spaces
Important properties from lin alg

	Product
		With scalar
		“dot” product
		Vector product (for 3-vectors)
	vector sum, difference
	Norm
	Orthogonality
	Basis
	See Shapiro-Stockman p. 178-181

Finding local image properties:

Roberts basis in matlab

	Basis:
	W1 = 1/2*[ 1 1   1 1]; W2 = 1/sqrt(2)*[ 0 1 -1 0]; W3 = 1/sqrt(2)*[ 1 0   0 -1]; W4 = 1/2*[ -1 1    1 -1];
	Test region:
	CR = [ 5 5   5 5]; % Coefficients: c1 = sum(sum(W1.*CR)) % c1 = 10 c2 = sum(sum(W2.*CR)) % c2 = 0 % Do same for W3 and W4

Better way to implement:
Use matrix algebra!

	Construct basis matrix B = [ W1(:)' W2(:)' W3(:)' W4(:)']
	Flatten the test area: CRf = CR(:)
	Compute coordinate change:
	NewCoord = B*CRf % = (10,0,0,0)’
	Test2: Step edge: SE = [ -1 1 -1 1] SEf = SE(:) NewCoordSE = B*SEf % =(0,1.41,-1.41,0)’
	Verify that we can transform back: SEtestf =B'*NewCoordSE reshape(SEtestf, 2, 2) % SEtest = %   -1.0000    1.0000 %   -1.0000    1.0000

Standard basis

	e1               e2               e3
	e8               e9

Frei-Chen basis

Frei-Chen example:

Sine function

	t = 0:pi/50:10*pi I = sin(t) subplot(221) plot(t,I)
	Image(I) ???
	subplot(222) image(64*(I+1)/2) colormap gray

Adding sines

Slide 16

Analyzing an image

Manmade object images:

	What does the F-t express?

Cropping in Fourier space

Slide 20