Calculate size of resulting array (matrix transformation) - matlab

Question

So I have this code to rotate and skew an image. it works well for the rotation, and the image will fit exactly in the canvas. However if I apply skewing the image doesn't fit anymore. Can someone explain how to calculate the proper array dimension for the resulting image rotated and skewed by specific angles? In particular, I'm using this 2 lines for the rotated image (and it works although I don't fully understand them). How should I modify them such as even when skewed, the final image will fit? Thanks!
rows_new = ceil(rows_init_img * abs(cos(rads)) + cols_init_img * abs(sin(rads)));
cols_new = ceil(rows_init_img * abs(sin(rads)) + cols_init_img * abs(cos(rads)));
full code
clc;
clear;
%% init values
%loading initial image
init_img = imread('name2.png');
% define rows/cols dimension of original image pixel matrix
[rows_init_img, cols_init_img,z]= size(init_img);
% skew angle in radians
sk_angle = 50;
sk_rads = 2*pi*sk_angle/360;
% rotation angle in radians
angle = 20;
rads = 2*pi*angle/360;
%% calculate size of final_image
orig_corners = [ 1, 1; 1, rows_init_img; 1, cols_init_img; rows_init_img, cols_init_img];
new_corners = uint8(zeros(size(orig_corners)));
for i = 1:size(orig_corners, 1)
for j = 1:size(orig_corners, 2)
% translate
a = i - final_origin_x;
b = j - final_origin_y;
% rotate
x = a * cos(rads) - b * sin(rads);
y = a * sin(rads) + b * cos(rads);
% skew along x axis (AFTER rotation)
x = x + sk_rads * y;
% translate
x = x + init_origin_x;
y = y + init_origin_y;
% round to turn values to positive integers
x = round(x);
y = round(y);
if (x >= 1 && y >= 1 && x <= size(orig_corners, 1) && y <= size(orig_corners, 2) )
new_corners(i, j) = init_img(x, y);
end
end
end
% calculating array dimesions such that rotated image gets fit in it exactly.
% rows_new = ceil(rows_init_img * abs(cos(rads)) + cols_init_img * abs(sin(rads)));
% cols_new = ceil(rows_init_img * abs(sin(rads)) + cols_init_img * abs(cos(rads)));
% define an array with calculated dimensions and fill the array with zeros ie.,black
% uint8 is important. without it will show noise WHY?
final_img = uint8(zeros([rows_new cols_new 3 ]));
%calculating center of original image
init_origin_x = ceil(rows_init_img/2);
init_origin_y = ceil(cols_init_img/2);
%calculating center of final image
final_origin_x = ceil( size(final_img, 1)/2 );
final_origin_y = ceil( size(final_img, 2)/2 );
%% main loop
% apply transformation to each pixel of the image
for i = 1:size(final_img ,1)
for j = 1:size(final_img, 2)
% translate
a = i - final_origin_x;
b = j - final_origin_y;
% rotate
x = a * cos(rads) - b * sin(rads);
y = a * sin(rads) + b * cos(rads);
% skew along x axis
x = x + sk_rads * y;
% translate
x = x + init_origin_x;
y = y + init_origin_y;
% round to turn values to positive integers
x = round(x);
y = round(y);
% make sure values exists (are part of the initial image) and copy
% them in the final image matrix
if (x >= 1 && y >= 1 && x <= size(init_img, 1) && y <= size(init_img, 2) )
final_img(i, j, :) = init_img(x, y, :);
end
end
end
%% display images
% original image
% figure('name','Original Image','numbertitle','off');
% imshow(init_img);
% result image
figure('name','Manipulated Image','numbertitle','off');
imshow(final_img);
updated code
clc;
clear;
%% init values
%loading initial image
init_img = imread('name2.png');
% define rows/cols dimension of original image pixel matrix
[rows_init_img, cols_init_img, z]= size(init_img);
% skew angle in radians
sk_angle = 50;
sk_rads = 2*pi*sk_angle/360;
% rotation angle in radians
angle = 20;
rads = 2*pi*angle/360;
%calculating center of original image
init_origin_x = ceil(rows_init_img/2);
init_origin_y = ceil(cols_init_img/2);
%% calculate size of final_image
orig_corners = [ 1, 1; 1, rows_init_img; 1, cols_init_img; rows_init_img, cols_init_img];
new_corners = uint8(zeros(size(orig_corners)));
for i = 1:size(orig_corners, 1)
for j = 1:size(orig_corners, 2)
% translate
a = i - final_origin_x; %at this point I don't have this variable because I can't create it yet
b = j - final_origin_y;
% rotate
x = a * cos(rads) - b * sin(rads);
y = a * sin(rads) + b * cos(rads);
% skew along x axis (AFTER rotation)
x = x + sk_rads * y;
% translate
x = x + init_origin_x;
y = y + init_origin_y;
% round to turn values to positive integers
x = round(x);
y = round(y);
if (x >= 1 && y >= 1 && x <= size(orig_corners, 1) && y <= size(orig_corners, 2) )
new_corners(i, j) = init_img(x, y);
end
end
end
% calculating array dimesions such that rotated image gets fit in it exactly.
rows_new = abs(max(new_corners(1, :)) - min(new_corners(1, :)));
cols_new = abs(max(new_corners(2, :)) - min(new_corners(2, :)));
% define an array with calculated dimensions and fill the array with zeros ie.,black
final_img = uint8(zeros([rows_new cols_new 3 ]));
%calculating center of final image
final_origin_x = ceil( size(final_img, 1)/2 );
final_origin_y = ceil( size(final_img, 2)/2 );
%% main loop
% apply transformation to each pixel of the image
for i = 1:size(final_img ,1)
for j = 1:size(final_img, 2)
% translate
a = i - final_origin_x;
b = j - final_origin_y;
% skew along x axis (BEFORE rotation)
a = a + sk_rads * b;
% rotate
x = a * cos(rads) - b * sin(rads);
y = a * sin(rads) + b * cos(rads);
% skew along x axis (AFTER rotation)
%x = x + sk_rads * y;
% translate
x = x + init_origin_x;
y = y + init_origin_y;
% round to turn values to positive integers
x = round(x);
y = round(y);
% make sure values exists (are part of the initial image) and copy
% them in the final image matrix
if (x >= 1 && y >= 1 && x <= size(init_img, 1) && y <= size(init_img, 2) )
final_img(i, j, :) = init_img(x, y, :);
end
end
end
%% display images
% original image
% figure('name','Original Image','numbertitle','off');
% imshow(init_img);
% result image
figure('name','Manipulated Image','numbertitle','off');
imshow(final_img);