Interpolation. Applications to image processing
Interpolation. Applications to image
processing
Contents
Rigid transformations
Nonlinear transformations
Rigid transformations
One important application interpolation is the rigid transformation of images. Let (i; j )
denote the pixels of an image and I(i; j ) their corresponding intensities. By rigid
transformation we mean a linear transformation of the pixel coordinates, i.e.
Ò Ó
Ò Ó
p
i
=A
q
j
where A is a 2 Â 2 matrix.
Among rigid transformations, we find the subclass of affine transformations. Here we use
matlab for resizing, rotation and shear. To do this, we define a 3 Â 3 matrix of the form
1
a11 a12 0
A
@a
21 a22 0
0
0 1
0
and use the commands maketform and imtransform.
I=imread('cameraman.tif');
I=double(I);
s=[2,3];
tform1 = maketform('affine',[s(1) 0 0; 0 s(2) 0; 0 0 1]);
I1 = imtransform(I,tform1);
% scaling
sh=[0.5 0.2];
tform2 = maketform('affine',[1 sh(1) 0; sh(2) 1 0; 0 0 1]);
I2 = imtransform(I,tform2);
% shear
theta=3*pi/4;
A=[cos(theta) sin(theta) 0; -sin(theta) cos(theta) 0; 0 0 1];
tform3 = maketform('affine',A);
I3 = imtransform(I,tform3);
% rotation
figure
subplot(2,2,1),imagesc(I),axis image
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title('Original','FontSize',18)
subplot(2,2,2),imagesc(I1),axis image
title('Scaled','FontSize',18)
subplot(2,2,3),imagesc(I2),axis image
title('Shear','FontSize',18)
subplot(2,2,4),imagesc(I3),axis image
title('Rotation','FontSize',18)
colormap(gray)
Exercise 5.1 We want to straighten the tower of the picture tower_bw.jpg. To do that we
have to rotate it a suitable angle. But once we rotate it, a non-defined region appears
(black in the figure). So then, we have to crop (manually) the image to eliminate that
region. Make a script to perform this task. Show the initial and final images and the
correction applied (in degrees).
Hint: Use a vertical line to decide if the tower is straight. For instance, if your rotated
image is I , then set I(:; 9 : 12) = 255 to draw a vertical white line in columns 9 to 12.
%The result is:
Exercise5_1
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correction (degrees) -> -7.2
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Interpolation. Applications to image processing
Nonlinear transformations
This is the case when, in the transformation
Ò Ó
Ò Ó
p
i
=A
q
j
the matrix A may depend on the particular pixel in which it is acting, i.e. in (i; j ) , in the
above notation.
For these transformations, we use the Matlab command interp2, which interpolates
2
2
functions defined in a 2D domain. For instance, we consider f (x; y ) = eÀx Ày given in a
coarser grid and then compute its values in a finer grid. The plots are
clear
[x,y] = meshgrid(-2:0.25:2);
z = exp(-x.^2-y.^2);
[p,q] = meshgrid(-2:0.125:2);
zfinal = interp2(x,y,z,p,q);
mesh(x,y,z), hold on
mesh(p,q,zfinal+2)
colormap(jet)
view([34,10])
% coarser grid
% finer grid
% interpolation
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Command interp2 may be used for affine transformations too. For instance, we rotate Ò
radians the tower image with center at (x0 ; y0 ) :
Ò Ó Ò
Ó Ò Ó
ÓÒ
cos(Ò) sen(Ò)
p
x0
x À x0
+
=
y À y0
y0
q
Àsen(Ò) cos(Ò)
clear all
I=imread('tower_bw.jpg');
I=im2double(I);
m=size(I,1);n=size(I,2);
[x,y] = meshgrid(1:n,1:m);
% grid inherited from the original image
theta=pi/4;
x0=fix(n/2);y0=fix(m/2);
p=(x-x0).*cos(theta)+(y-y0).*sin(theta)+x0;
q=-(x-x0).*sin(theta)+(y-y0).*cos(theta)+y0;
% angle of rotation
% center of rotation
% transformed coordinates (new pixel locations)
Ifinal=interp2(x,y,I,p,q,'bicubic');
% intensity values interpolation at new locations
figure
subplot(1,2,1),imagesc(I),axis image
title('Original','FontSize',18)
subplot(1,2,2),imagesc(Ifinal),axis image
title('Transformed','FontSize',18)
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colormap(gray)
Exercise5_2 The goal of this exercise is to introduce an eddy in the swimingpool, as
shown in the figure (final result of the exercise)
Exercise5_2
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To complete this exercise, use the rotation script above, introducing the following changes:
Define an online funtion r=@(x,y)... representing the radious of a circumference
centered at (x0 ; y0 ) .
Define the angle as Ò = 10 Ã exp(À0:1 Ã r (x; y )) .
Define the new pixel locations in terms of this angle, as in the script above.
If the picture were a intensity image, then we would just use interp2 as above. Since our
picture is a rgb image, we have to use the interpolation for each color channel. To do this,
we use a loop in this way
% Ifinal=zeros(size(Ic));
% initialize the final matrix
% for k=1:3 % colors
% select first channel
% interpolate and produce Ichannel;
% Ifinal(:,:,k)= Ichannel;
% add k channel
% end
% Ifinal=im2uint8(Ifinal);
% convert to unsigned integer
Exercise5_3 The goal of this exercise is to introduce a non uniform scaling of an image.
We also want to introduce automatic cropping.
In this case, we define the transformation as
p = d à s(x; y ): à (x À x0) + x
q = d à s(x; y ): à (y À y 0) + y;
for d = 0:01 . After interpolation, we get a small picture full of non-defined intensity values.
But the final result should be
Exercise5_3
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By using the function find, locate the pixels at which the interpolated image is positive.
Then, define another image restricted to this set of indices.
Exercise 5_4 Modify the script of Exercise5_3 to make a function with
Input: an intensity image, a center point (x0 ; y0 ) , a deformation parameter d.
Output: The transformed cropped image matrix.
Use it. For instance
I=imread('einstein_bw.jpg');
A=Exercise5_4(I,361,593,0.01);
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The result is funnier!!
Published with MATLAB® 7.14
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