C306 | |
Fall 2001 | |
Martin Jagersand |
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 are some of the most important methods in image processing | ||
Will be used later in: | ||
Filtering | ||
Restoration | ||
Enhancement | ||
Compression | ||
Image analysis |
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): | ||
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:
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]; |
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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(:)'] |
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Flatten the test area: CRf = CR(:) |
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Compute coordinate change: | |
NewCoord = B*CRf % = (10,0,0,0)’ |
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Test2: Step edge: SE = [ -1 1 -1 1] SEf = SE(:) NewCoordSE = B*SEf % =(0,1.41,-1.41,0)’ |
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Verify that we can transform
back: SEtestf =B'*NewCoordSE reshape(SEtestf, 2, 2) % SEtest = % -1.0000 1.0000 % -1.0000 1.0000 |
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e1 e2 e3 | |
e8 e9 |
t = 0:pi/50:10*pi I = sin(t) subplot(221) plot(t,I) |
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Image(I) ??? | |
subplot(222) image(64*(I+1)/2) colormap gray |
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What does the F-t express? |