Raffaella Lanzarotti

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• Image Enhancement: – Color values modified– Pixel positions unchanged

• Geometric transformation– Pixel positions modifed– Colors unchanged

• Goal:– To create special effects– To register two images taken of the same scene at different times or different positions

– To morph one image to another

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• Let be an image• is a warping image of if

• and are called the warpingfunctions

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B(k, l) = A(x(k), y(l))

A(i, j)A(i, j)B(k, l)

x(...) y(...)

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x(k) = k + 100

y(l) = l + 20

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x = k+ t

x =

x(k)y(l)

, k =

k

l

, t =

t

x

t

y

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• T = maketform('affine',[1 0 0;0 1 0; 100 20 1]);

• J = imtransform(img,T,'XData',[1,size(img,2)],'YData',[1 size(img,1)]);

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The translationvector occupies the last row in the transformation matrixin MATLAB!!!

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x(k) = k · sky(l) = l · sl

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x = S · k

x =

x(k)y(l)

, k =

k

l

, S =

s

x

00 s

y

,

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T = affine2d([1.5 0 0; 0 .5 0; 0 0 1]);J = imwarp(img,T);

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x(k, l) = (k x0) cos() + (l y0) sin() + x0

y(k, l) = (k x0) sin() + (l y0) cos() + y0

(x0, y0) = the center of the image

Where:

= rotation angle

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x = Rk, where:

R =

cos sin sin cos

,

x =

x(k)y(l)

, k =

k x0

l y0

Theta= pi/6;T = affine2d([cos(Theta) sin(Theta) 0;

-­‐sin(Theta) cos(Theta) 0;0 0 1]);

J = imwarp(img, T);

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StR

• A generalization of geometric transformations:

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Q =

C d0T 1

C = any non singular 2 2 matrix

d = any vector in R2

• Maps straight lines into strainght lines• Affine mapping coefficients uniquelydetermined from displacements of 3 vertices

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A =

a1 a2b1 b2

, d =

a0b0

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x = a0 + a1k + a2l

y = b0 + b1k + b2l

• Construct the transformation matrix T, givencouples of control points:

T = fitgeotrans(movingPts, fixedPts, ‘affine’); %takes the 3 pairs of control points, movingPts and fixedPts, and uses them to infer the affine transformation

• Example-­-­-­-­-­-­-­Create an affine transformation that maps the triangle with vertices(0,0), (6,3), (-­2,5) to the triangle with vertices (-­1,-­1), (0,-­10), (4,4):X= [ 0 0;;

6 3;;-­2 5];;

Y = [-­‐1 -­‐1;0 -­‐10;4 4];

T =fitgeotrans(X,Y, 'affine');26

• Construct the struct corresponding to the affine transformation matrix T:

tform = affine2d(T) % builds a TFORM struct for an Ndimensional affine transformation. T defines a forward transformation such thatTFORMFWD(U,T), where U is a 1-­‐by-­‐N vector, returns a 1-­‐by-­‐N vector X such that X = U * T(1:N,1:N) + T(N+1,1:N).T hasboth forward and inverse transformations. N=2 for 2D image transformation

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tform =

C

T0

d

T 1

T

B = IMWARP(A,TFORM, INTERP) %transforms the image A according to the 2-­‐D spatialtransformation defined by TFORM; INTERP specifies the interpolation filter

• Example 1-­-­-­-­-­-­-­-­-­Apply a horizontal shear to an intensity image.

I = imread('cameraman.tif');tform = affine2d([1 0 0; .5 1 0; 0 0 1]);J = imwarp(I,tform);figure, imshow(I), figure, imshow(J)

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Goal: map a square (or more generally a quadrangle) to a quadrangle

It requires 8 parameters:

x = a0 + a1k + a2l + a3kl

y = b0 + b1k + b2l + b3kl

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• Includes all deformations that can be modeled by polynomial transformations(affine and bilinear are special cases)

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y = b0 + b1k + b2l + b3kl + b4k2 + b5l

2 + ...x = a0 + a1k + a2l + a3kl + a4k

2 + a5l2 + ...

• Polynolial:

tform = fitgeotrans(movingPts,fixedPts,'polynomial',degree)

• OR more in general:

tform = fitgeotrans(movingPts,fixedPts,transformationType)

takes the pairs of controlpoints, ”movingPts” and ”fixedPts”, and uses them to infer the geometric transformation specifiedby transformationType.

Where “transformationType”…

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Wave1: x(k, l) = k + 20 sin

2

128l

; y(k, l) = l

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Wave2: x(k, l) = k + 20 sin

2

30k

; y(k, l) = l

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Warp: x(k, l) =sign(k x0)

x0 (k x0)

2 + x0; y(k, l) = l

(x0, y0) = the center of the image

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(x0, y0) = the center of the image