本文介绍了使用三次三次插值旋转图像而不会旋转的处理方法，对大家解决问题具有一定的参考价值，需要的朋友们下面随着小编来一起学习吧！
问题描述
我已经使用
下面的示例使用" CachedBicubicInterpolator "代码版本，并且还支持RGB图像:
函数BicubicInterpolationTest2()关闭所有;％全部清除；img ='peppers.png';input_image = double(imread(img))./255;H =大小(input_image，1);％ 高度W =大小(input_image，2);％ 宽度th = 120 * pi/180；％旋转120度s0 = 0.8；s1 = 0.8；x0 = -W/2;x1 = -H/2;T = [1 0 x0;...0 1 x1;...0 0 1];RST = [(s0 * cos(th))(-s1 * sin(th))((s0 * x0 * cos(th))-(s1 * x1 * sin(th)));...(s0 * sin(th))(s1 * cos(th))((s0 * x0 * sin(th))+(s1 * x1 * cos(th)));...0 0 1];M = inv(T)* RST;N = inv(M);output_image = zeros(H，W，size(input_image，3));对于我= 1:W对于j = 1:Hx = [i;j;1];y = N * x;a = y(1)/y(3);b = y(2)/y(3);x1 =楼板面积(a);y1 =底数(b);％双三次插值(应用灰度图像)如果((x1> = 2)&&(y1> = 2)&&(x1< = W-2)&&(y1< = H-2))％加载4x4像素P = input_image(y1-1:y1 + 2，x1-1:x1 + 2，:);内插法dx = a-x1;dy = b-y1;％双三次插值output_image(j，i，:) = bicubicInterpolate(P，dx，dy);结尾结尾结尾imshow(输出图像);%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%通过与Matalb内置函数imwarp比较来验证实现:tform = affine2d(M');ref_image = imwarp(input_image，tform，'OutputView'，imref2d(size(input_image))，'Interp'，'cubic');图; imshow(ref_image)图; imshow(abs(output_image-ref_image)，[]); impixelinfomax_diff = max(abs(output_image(:)-ref_image(:)));disp(['与imwarp的最大差='，num2str(max_diff)]);%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%函数[p0，p1，p2，p3] = list4(P)P =挤压(P);p0 = P(1，:);p1 = P(2，:);p2 = P(3，:);p3 = P(4，:);％http://www.paulinternet.nl/?page = bicubic％public void updateCoefficients(double [] [] p){％a00 = p [1] [1];％a01 = -.5 * p [1] [0] + .5 * p [1] [2];％a02 = p [1] [0]-2.5 * p [1] [1] + 2 * p [1] [2]-.5 * p [1] [3]；％a03 = -.5 * p [1] [0] + 1.5 * p [1] [1]-1.5 * p [1] [2] + .5 * p [1] [3]；％a10 = -.5 * p [0] [1] + .5 * p [2] [1]；％a11 = .25 * p [0] [0]-.25 * p [0] [2]-.25 * p [2] [0] + .25 * p [2] [2]；％a12 = -.5 * p [0] [0] + 1.25 * p [0] [1]-p [0] [2] + .25 * p [0] [3] + .5 * p [2] [0]-1.25 * p [2] [1] + p [2] [2]-.25 * p [2] [3];％a13 = .25 * p [0] [0]-.75 * p [0] [1] + .75 * p [0] [2]-.25 * p [0] [3]-.25 *p [2] [0] + .75 * p [2] [1]-.75 * p [2] [2] + .25 * p [2] [3];％a20 = p [0] [1]-2.5 * p [1] [1] + 2 * p [2] [1]-.5 * p [3] [1]；％a21 = -.5 * p [0] [0] + .5 * p [0] [2] + 1.25 * p [1] [0]-1.25 * p [1] [2]-p [2][0] + p [2] [2] + .25 * p [3] [0]-.25 * p [3] [2];％a22 = p [0] [0]-2.5 * p [0] [1] + 2 * p [0] [2]-.5 * p [0] [3]-2.5 * p [1] [0] + 6.25 * p [1] [1]-5 * p [1] [2] + 1.25 * p [1] [3] + 2 * p [2] [0]-5 * p [2] [1] + 4 * p [2] [2]-p [2] [3]-.5 * p [3] [0] + 1.25 * p [3] [1]-p [3] [2] +.25 * p [3] [3];％a23 = -.5 * p [0] [0] + 1.5 * p [0] [1]-1.5 * p [0] [2] + .5 * p [0] [3] + 1.25 * p [1] [0]-3.75 * p [1] [1] + 3.75 * p [1] [2]-1.25 * p [1] [3]-p [2] [0] + 3 * p [2][1]-3 * p [2] [2] + p [2] [3] + .25 * p [3] [0]-.75 * p [3] [1] + .75 * p [3] [2]-.25 * p [3] [3];％a30 = -.5 * p [0] [1] + 1.5 * p [1] [1]-1.5 * p [2] [1] + .5 * p [3] [1]；％a31 = .25 * p [0] [0]-.25 * p [0] [2]-.75 * p [1] [0] + .75 * p [1] [2] + .75 *p [2] [0]-.75 * p [2] [2]-.25 * p [3] [0] + .25 * p [3] [2];％a32 = -.5 * p [0] [0] + 1.25 * p [0] [1]-p [0] [2] + .25 * p [0] [3] + 1.5 * p [1][0]-3.75 * p [1] [1] + 3 * p [1] [2]-.75 * p [1] [3]-1.5 * p [2] [0] + 3.75 * p [2] [1]-3 * p [2] [2] + .75 * p [2] [3] + .5 * p [3] [0]-1.25 * p [3] [1] + p [3] [2]-.25 * p [3] [3];％a33 = .25 * p [0] [0]-.75 * p [0] [1] + .75 * p [0] [2]-.25 * p [0] [3]-.75 *p [1] [0] + 2.25 * p [1] [1]-2.25 * p [1] [2] + .75 * p [1] [3] + .75 * p [2] [0]-2.25 * p [2] [1] + 2.25 * p [2] [2]-.75 * p [2] [3]-.25 * p [3] [0] + .75 * p [3] [1]-.75 * p [3] [2] + .25 * p [3] [3];％}％public double getValue(double x，double y){％double x2 = x * x;％double x3 = x2 * x;％double y2 = y * y;％double y3 = y2 * y;％回报百分比(a00 + a01 * y + a02 * y2 + a03 * y3)+％(a10 + a11 * y + a12 * y2 + a13 * y3)* x +％(a20 + a21 * y + a22 * y2 + a23 * y3)* x2 +％(a30 + a31 * y + a32 * y2 + a33 * y3)* x3;％}函数q = bicubicInterpolate(P，x，y)[p00，p01，p02，p03] ＝ list4(P(1，:，:))；[p10，p11，p12，p13] = list4(P(2，:，:));[p20，p21，p22，p23] = list4(P(3，:，:));[p30，p31，p32，p33] = list4(P(4，:，:));a00 = p11;a01 = -.5 * p10 + .5 * p12;a02 = p10-2.5 * p11 + 2 * p12-.5 * p13;a03 = -.5 * p10 + 1.5 * p11-1.5 * p12 + .5 * p13;a10 = -.5 * p01 + .5 * p21;a11 = .25 * p00-.25 * p02-.25 * p20 + .25 * p22;a12 = -.5 * p00 + 1.25 * p01-p02 + .25 * p03 + .5 * p20-1.25 * p21 + p22-.25 * p23;a13 = .25 * p00-.75 * p01 + .75 * p02-.25 * p03-.25 * p20 + .75 * p21-.75 * p22 + .25 * p23;a20 = p01-2.5 * p11 + 2 * p21-.5 * p31;a21 = -.5 * p00 + .5 * p02 + 1.25 * p10-1.25 * p12-p20 + p22 + .25 * p30-.25 * p32;a22 = p00-2.5 * p01 + 2 * p02-.5 * p03-2.5 * p10 + 6.25 * p11-5 * p12 + 1.25 * p13 + 2 * p20-5 * p21 + 4 * p22-p23-.5 *p30 + 1.25 * p31-p32 + .25 * p33;a23 = -.5 * p00 + 1.5 * p01-1.5 * p02 + .5 * p03 + 1.25 * p10-3.75 * p11 + 3.75 * p12-1.25 * p13-p20 + 3 * p21-3 * p22 + p23 +.25 * p30-.75 * p31 + .75 * p32-.25 * p33;a30 = -.5 * p01 + 1.5 * p11-1.5 * p21 + .5 * p31;a31 = .25 * p00-.25 * p02-.75 * p10 + .75 * p12 + .75 * p20-.75 * p22-.25 * p30 + .25 * p32;a32 = -.5 * p00 + 1.25 * p01-p02 + .25 * p03 + 1.5 * p10-3.75 * p11 + 3 * p12-.75 * p13-1.5 * p20 + 3.75 * p21-3 * p22 + .75* p23 + .5 * p30-1.25 * p31 + p32-.25 * p33;a33 = .25 * p00-.75 * p01 + .75 * p02-.25 * p03-.75 * p10 + 2.25 * p11-2.25 * p12 + .75 * p13 + .75 * p20-2.25 * p21 + 2.25* p22-.75 * p23-.25 * p30 + .75 * p31-.75 * p32 + .25 * p33;x2 = x * x;x3 = x2 * x;y2 = y * y;y3 = y2 * y;％q =(a00 + a01 * y + a02 * y2 + a03 * y3)+ ...％(a10 + a11 * y + a12 * y2 + a13 * y3)* x + ...％(a20 + a21 * y + a22 * y2 + a23 * y3)* x2 + ...％(a30 + a31 * y + a32 * y2 + a33 * y3)* x3;q =(a00 + a01 * x + a02 * x2 + a03 * x3)+ ...(a10 + a11 * x + a12 * x2 + a13 * x3)* y + ...(a20 + a21 * x + a22 * x2 + a23 * x3)* y2 + ...(a30 + a31 * x + a32 * x2 + a33 * x3)* y3; 结果:
I have implemented a code for image warping using bilinear interpolation:
Matlab image rotation
I would like to improve the code by using bicubic interpolation to rotate the image WITHOUT using the built-in functions like imrotate or imwarp and interp functions in MATLAB.
解决方案
I successfully managed to implement a full working example.
Code is based on Anna1994's code: Matlab image rotation
Biqubic code is also based on Java (and C++) implementation posted here: http://www.paulinternet.nl/?page=bicubic
The following code applies image rotation example using biqubic interpolation:
function BicubicInterpolationTest()
close all;
% clear all;
img = 'cameraman.tif';
input_image =double(imread(img))./255;
H=size(input_image,1); % height
W=size(input_image,2); % width
th=120*pi/180; %Rotate 120 degrees
s0 = 2;
s1 = 2;
x0 = -W/2;
x1 = -H/2;
T=[1 0 x0 ; ...
0 1 x1 ; ...
0 0 1];
RST = [ (s0*cos(th)) (-s1*sin(th)) ((s0*x0*cos(th))-(s1*x1*sin(th))); ...
(s0*sin(th)) (s1*cos(th)) ((s0*x0*sin(th))+(s1*x1*cos(th))); ...
0 0 1];
M=inv(T)*RST;
N = inv(M);
output_image=zeros(H,W,size(input_image,3));
for i=1:W
for j=1:H
x = [i ; j ; 1];
y = N * x;
a = y(1)/y(3);
b = y(2)/y(3);
%Nearest neighbor
% a = round(a);
% b = round(b);
%Bilinear interpolation (applies RGB image):
% x1 = floor(a);
% y1 = floor(b);
% x2 = x1 + 1;
% y2 = y1 + 1;
% if ((x1 >= 1) && (y1 >= 1) && (x2 <= W) && (y2 <= H))
% %Load 2x2 pixels
% i11 = input_image(y1, x1, :); %Top left pixel
% i21 = input_image(y2, x1, :); %Bottom left pixel
% i12 = input_image(y1, x2, :); %Top right pixel
% i22 = input_image(y2, x2, :); %Bottom right pixel
%
% %Interpolation wieghts
% dx = x2 - a;
% dy = y2 - b;
%
% %Bi-lienar interpolation
% output_image(j, i, :) = i11*dx*dy + i21*dx*(1-dy) + i12*(1-dx)*dy + i22*(1-dx)*(1-dy);
% end
x1 = floor(a);
y1 = floor(b);
%Bicubic interpolation (applies grayscale image)
if ((x1 >= 2) && (y1 >= 2) && (x1 <= W-2) && (y1 <= H-2))
%Load 4x4 pixels
P = input_image(y1-1:y1+2, x1-1:x1+2);
%Interpolation wieghts
dx = a - x1;
dy = b - y1;
%Bi-bicubic interpolation
output_image(j, i) = bicubicInterpolate(P, dx, dy);
end
end
end
imshow(output_image);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Verify implementation by comparing with Matalb build in function imwarp:
tform = affine2d(M');
ref_image = imwarp(input_image, tform, 'OutputView', imref2d(size(input_image)), 'Interp', 'cubic');
figure;imshow(ref_image)
figure;imshow(output_image - ref_image)
max_diff = max(abs(output_image(:) - ref_image(:)));
disp(['Maximum difference from imwarp = ', num2str(max_diff)]);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%http://www.paulinternet.nl/?page=bicubic
%double cubicInterpolate (double p[4], double x) {
% return p[1] + 0.5 * x*(p[2] - p[0] + x*(2.0*p[0] - 5.0*p[1] + 4.0*p[2] - p[3] + x*(3.0*(p[1] - p[2]) + p[3] - p[0])));
%}
function q = cubicInterpolate(p, x)
q = p(2) + 0.5 * x*(p(3) - p(1) + x*(2.0*p(1) - 5.0*p(2) + 4.0*p(3) - p(4) + x*(3.0*(p(2) - p(3)) + p(4) - p(1))));
%http://www.paulinternet.nl/?page=bicubic
% double bicubicInterpolate (double p[4][4], double x, double y) {
% double arr[4];
% arr[0] = cubicInterpolate(p[0], y);
% arr[1] = cubicInterpolate(p[1], y);
% arr[2] = cubicInterpolate(p[2], y);
% arr[3] = cubicInterpolate(p[3], y);
% return cubicInterpolate(arr, x);
% }
function q = bicubicInterpolate(p, x, y)
q1 = cubicInterpolate(p(1,:), x);
q2 = cubicInterpolate(p(2,:), x);
q3 = cubicInterpolate(p(3,:), x);
q4 = cubicInterpolate(p(4,:), x);
q = cubicInterpolate([q1, q2, q3, q4], y);
I verified implementation by comparing to Matalb build in function imwarp
Result:
The following example uses the "CachedBicubicInterpolator" code version, and also supports RGB image:
function BicubicInterpolationTest2()
close all;
% clear all;
img = 'peppers.png';
input_image = double(imread(img))./255;
H=size(input_image,1); % height
W=size(input_image,2); % width
th=120*pi/180; %Rotate 120 degrees
s0 = 0.8;
s1 = 0.8;
x0 = -W/2;
x1 = -H/2;
T=[1 0 x0 ; ...
0 1 x1 ; ...
0 0 1];
RST = [ (s0*cos(th)) (-s1*sin(th)) ((s0*x0*cos(th))-(s1*x1*sin(th))); ...
(s0*sin(th)) (s1*cos(th)) ((s0*x0*sin(th))+(s1*x1*cos(th))); ...
0 0 1];
M=inv(T)*RST;
N = inv(M);
output_image=zeros(H,W,size(input_image,3));
for i=1:W
for j=1:H
x = [i ; j ; 1];
y = N * x;
a = y(1)/y(3);
b = y(2)/y(3);
x1 = floor(a);
y1 = floor(b);
%Bicubic interpolation (applies grayscale image)
if ((x1 >= 2) && (y1 >= 2) && (x1 <= W-2) && (y1 <= H-2))
%Load 4x4 pixels
P = input_image(y1-1:y1+2, x1-1:x1+2, :);
%Interpolation wieghts
dx = a - x1;
dy = b - y1;
%Bi-bicubic interpolation
output_image(j, i, :) = bicubicInterpolate(P, dx, dy);
end
end
end
imshow(output_image);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Verify implementation by comparing with Matalb build in function imwarp:
tform = affine2d(M');
ref_image = imwarp(input_image, tform, 'OutputView', imref2d(size(input_image)), 'Interp', 'cubic');
figure;imshow(ref_image)
figure;imshow(abs(output_image - ref_image), []);impixelinfo
max_diff = max(abs(output_image(:) - ref_image(:)));
disp(['Maximum difference from imwarp = ', num2str(max_diff)]);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [p0, p1, p2, p3] = list4(P)
P = squeeze(P);
p0 = P(1, :);
p1 = P(2, :);
p2 = P(3, :);
p3 = P(4, :);
%http://www.paulinternet.nl/?page=bicubic
% public void updateCoefficients (double[][] p) {
% a00 = p[1][1];
% a01 = -.5*p[1][0] + .5*p[1][2];
% a02 = p[1][0] - 2.5*p[1][1] + 2*p[1][2] - .5*p[1][3];
% a03 = -.5*p[1][0] + 1.5*p[1][1] - 1.5*p[1][2] + .5*p[1][3];
% a10 = -.5*p[0][1] + .5*p[2][1];
% a11 = .25*p[0][0] - .25*p[0][2] - .25*p[2][0] + .25*p[2][2];
% a12 = -.5*p[0][0] + 1.25*p[0][1] - p[0][2] + .25*p[0][3] + .5*p[2][0] - 1.25*p[2][1] + p[2][2] - .25*p[2][3];
% a13 = .25*p[0][0] - .75*p[0][1] + .75*p[0][2] - .25*p[0][3] - .25*p[2][0] + .75*p[2][1] - .75*p[2][2] + .25*p[2][3];
% a20 = p[0][1] - 2.5*p[1][1] + 2*p[2][1] - .5*p[3][1];
% a21 = -.5*p[0][0] + .5*p[0][2] + 1.25*p[1][0] - 1.25*p[1][2] - p[2][0] + p[2][2] + .25*p[3][0] - .25*p[3][2];
% a22 = p[0][0] - 2.5*p[0][1] + 2*p[0][2] - .5*p[0][3] - 2.5*p[1][0] + 6.25*p[1][1] - 5*p[1][2] + 1.25*p[1][3] + 2*p[2][0] - 5*p[2][1] + 4*p[2][2] - p[2][3] - .5*p[3][0] + 1.25*p[3][1] - p[3][2] + .25*p[3][3];
% a23 = -.5*p[0][0] + 1.5*p[0][1] - 1.5*p[0][2] + .5*p[0][3] + 1.25*p[1][0] - 3.75*p[1][1] + 3.75*p[1][2] - 1.25*p[1][3] - p[2][0] + 3*p[2][1] - 3*p[2][2] + p[2][3] + .25*p[3][0] - .75*p[3][1] + .75*p[3][2] - .25*p[3][3];
% a30 = -.5*p[0][1] + 1.5*p[1][1] - 1.5*p[2][1] + .5*p[3][1];
% a31 = .25*p[0][0] - .25*p[0][2] - .75*p[1][0] + .75*p[1][2] + .75*p[2][0] - .75*p[2][2] - .25*p[3][0] + .25*p[3][2];
% a32 = -.5*p[0][0] + 1.25*p[0][1] - p[0][2] + .25*p[0][3] + 1.5*p[1][0] - 3.75*p[1][1] + 3*p[1][2] - .75*p[1][3] - 1.5*p[2][0] + 3.75*p[2][1] - 3*p[2][2] + .75*p[2][3] + .5*p[3][0] - 1.25*p[3][1] + p[3][2] - .25*p[3][3];
% a33 = .25*p[0][0] - .75*p[0][1] + .75*p[0][2] - .25*p[0][3] - .75*p[1][0] + 2.25*p[1][1] - 2.25*p[1][2] + .75*p[1][3] + .75*p[2][0] - 2.25*p[2][1] + 2.25*p[2][2] - .75*p[2][3] - .25*p[3][0] + .75*p[3][1] - .75*p[3][2] + .25*p[3][3];
% }
% public double getValue (double x, double y) {
% double x2 = x * x;
% double x3 = x2 * x;
% double y2 = y * y;
% double y3 = y2 * y;
%
% return (a00 + a01 * y + a02 * y2 + a03 * y3) +
% (a10 + a11 * y + a12 * y2 + a13 * y3) * x +
% (a20 + a21 * y + a22 * y2 + a23 * y3) * x2 +
% (a30 + a31 * y + a32 * y2 + a33 * y3) * x3;
% }
function q = bicubicInterpolate(P, x, y)
[p00, p01, p02, p03] = list4(P(1, :, :));
[p10, p11, p12, p13] = list4(P(2, :, :));
[p20, p21, p22, p23] = list4(P(3, :, :));
[p30, p31, p32, p33] = list4(P(4, :, :));
a00 = p11;
a01 = -.5*p10 + .5*p12;
a02 = p10 - 2.5*p11 + 2*p12 - .5*p13;
a03 = -.5*p10 + 1.5*p11 - 1.5*p12 + .5*p13;
a10 = -.5*p01 + .5*p21;
a11 = .25*p00 - .25*p02 - .25*p20 + .25*p22;
a12 = -.5*p00 + 1.25*p01 - p02 + .25*p03 + .5*p20 - 1.25*p21 + p22 - .25*p23;
a13 = .25*p00 - .75*p01 + .75*p02 - .25*p03 - .25*p20 + .75*p21 - .75*p22 + .25*p23;
a20 = p01 - 2.5*p11 + 2*p21 - .5*p31;
a21 = -.5*p00 + .5*p02 + 1.25*p10 - 1.25*p12 - p20 + p22 + .25*p30 - .25*p32;
a22 = p00 - 2.5*p01 + 2*p02 - .5*p03 - 2.5*p10 + 6.25*p11 - 5*p12 + 1.25*p13 + 2*p20 - 5*p21 + 4*p22 - p23 - .5*p30 + 1.25*p31 - p32 + .25*p33;
a23 = -.5*p00 + 1.5*p01 - 1.5*p02 + .5*p03 + 1.25*p10 - 3.75*p11 + 3.75*p12 - 1.25*p13 - p20 + 3*p21 - 3*p22 + p23 + .25*p30 - .75*p31 + .75*p32 - .25*p33;
a30 = -.5*p01 + 1.5*p11 - 1.5*p21 + .5*p31;
a31 = .25*p00 - .25*p02 - .75*p10 + .75*p12 + .75*p20 - .75*p22 - .25*p30 + .25*p32;
a32 = -.5*p00 + 1.25*p01 - p02 + .25*p03 + 1.5*p10 - 3.75*p11 + 3*p12 - .75*p13 - 1.5*p20 + 3.75*p21 - 3*p22 + .75*p23 + .5*p30 - 1.25*p31 + p32 - .25*p33;
a33 = .25*p00 - .75*p01 + .75*p02 - .25*p03 - .75*p10 + 2.25*p11 - 2.25*p12 + .75*p13 + .75*p20 - 2.25*p21 + 2.25*p22 - .75*p23 - .25*p30 + .75*p31 - .75*p32 + .25*p33;
x2 = x * x;
x3 = x2 * x;
y2 = y * y;
y3 = y2 * y;
% q = (a00 + a01 * y + a02 * y2 + a03 * y3) +...
% (a10 + a11 * y + a12 * y2 + a13 * y3) * x +...
% (a20 + a21 * y + a22 * y2 + a23 * y3) * x2 +...
% (a30 + a31 * y + a32 * y2 + a33 * y3) * x3;
q = (a00 + a01 * x + a02 * x2 + a03 * x3) +...
(a10 + a11 * x + a12 * x2 + a13 * x3) * y +...
(a20 + a21 * x + a22 * x2 + a23 * x3) * y2 +...
(a30 + a31 * x + a32 * x2 + a33 * x3) * y3;
Result:
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