A12 - Geometric Distortion Correction

August 19, 2008

We have an image of a Capiz window, an analog to a grid, whose left side is slightly tilted. We aim to correct this distortion by creating an ideal grid, then mapping into it the grayscale values of the corresponding pixels in the distorted image.


First, we choose an eminently distorted area in the image. This area is covered by a rectangle in figure 1. We designate the rectangle to be the ideal polygon, how the shape of the set of small squares in the window appear. The ideal polygon vertices are labeled as [xi, yi] , while the vertices of the distorted area are tagged as [x^i, y^i].

The transformation of [x, y] to [x^, y^] can be expressed by a function r and s.

x^ = r(x,y)
y ^ = s(x,y) (1)

If the polygon is not too large, we could assume that the transformations for both directions happen linearly. Thus, we guess.

x^ = c1x + c2y + c3xy + c4
y^ = c5x + c6y + c7xy + c8 (2)

From equation (2), we could obtain the transformation vectors C14 = [c1, c2, c3, c4]t and
C58 = [c5, c6, c7, c8] t, where t stands for transposed via

C14 = (inv(T) ) X
C58 = (inv(T) ) Y

where inv means inv(T) means the inverse of T, T is a 4 by 4 matrix with row i given by [xi, yi, xiyi, 1], X = [[x^1,x^2, x^3, x^4] t and Y = [y^1, y^2, y^3, y^4] t.

In the ideal rectangle, we find the corresponding pixel in the distorted image. If the corresponding coordinates are integer-valued, we simply duplicate the point intensity value into the ideal space. Otherwise, we employ the bilinear interpolation wherein if x^ or y^ is not an element of the set of integers, the value at (x^, y^) is given by

v(x^,y^) = ax^ + by^ + cx^y^ + d (3)

To find the unknowns a, b, c and d, we use the nearest neighbors of (x^,y^).

We, see that the image has been quite corrected. The sides of the squares without using bilinear interpolation are crooked. Meanwhile, the sides of the squares when bilinear interpolation was employed were straight, though the resulting image had less contrast,

//1 Call the image
M = imread('D:\ap187\distorted_capiz_window.jpg');
M = im2gray(M);
imshow(M,[]);
[m,k] = size(M);

//2 Select large subgrid vertices
g = locate(4,1);
g = [46.723164 238.70056;
99.265537 239.83051;
100.96045 41.525424;
50.112994 40.960452 ];
gprime(:,1) = g(:,2);
gprime(:,2) = g(:,1);
gprime(:,1) = abs(gprime(:,1) - m - 1);
//3 Generate ideal grid
x_1 = gprime(1,1);
x_2 = gprime(1,1);
x_3 = gprime(3,1);
x_4 = x_3;
y_1 = gprime(1,2);
y_2 = gprime(2,2);
y_3 = gprime(2,2);
y_4 = gprime(1,2);

//4 Obtain transformation matrix C
T = [x_1 y_1 x_1*y_1 1;
x_2 y_2 x_2*y_2 1;
x_3 y_3 x_3*y_3 1;
x_4 y_4 x_4*y_4 1];

X = gprime(:,1);
Y = gprime(:,2);

C14 = (inv(T))*X;
C58 = (inv(T))*Y;

//5 Map the distorted image into the ideal space (w/ bilinear interpolation)
v = zeros(m,k);

for x = 5:m-5;
for y = 5:k-5;
t = [x y x*y 1];
xhat = t*C14;
yhat = t*C58;
xhat_integer = int(xhat);
yhat_integer = int(yhat);

if xhat_integer == xhat & yhat_integer == yhat then
if xhat_integer == 0 then
xhat_integer = 1;
end
if yhat_integer == 0 then
yhat_integer = 1;
end

v(x,y) = M(xhat_integer, yhat_integer);

else
xplus = xhat_integer + 1;
xminus = xhat_integer;
yplus = yhat_integer + 1;
yminus = yhat_integer;

nearestneighbors = [xplus yplus xplus*yplus 1;
xplus yminus xplus*yminus 1;
xminus yminus xminus*yminus 1;
xminus yplus xminus*yminus 1];

vhat = [M(xplus,yplus); M(xplus,yminus); M(xminus,yminus); M(xminus,yplus)];
a_b_c_d = inv(nearestneighbors)*vhat;
nu = [x y y*x 1];
v(x,y) = nu*a_b_c_d;
end
end
end

//6 Mapping without bilinear interpolation
v2 = zeros(m,k);

for x = 5:m-5;
for y = 5:k-5;
t = [x y x*y 1];
xhat = t*C14;
yhat = t*C58;
xhat_integer = int(xhat);
yhat_integer = int(yhat);
v2(x,y) = M(xhat_integer, yhat_integer);
end
end

//7 Showing the images, M - original, v - corrected with interpolation, v2 - corrected without interpolation
subplot(221)
imshow(M,[]);
subplot(222)
imshow(v2,[]);
subplot(223)
imshow(v3,[]);

I give myself 7/10 because it took me this long to solve the problem.

Thank you to Mark Leo for some explanations.