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