Question

我使用bilinear interpolation实现了图像变形代码：
Matlab image rotation。

我希望使用bicubic interpolation来旋转图像，而无需使用内置函数（如imrotate或imwarp以及MATLAB中的interp函数来改进代码。

Answer 1

我成功地设法实现了一个完整的工作示例。

代码基于Anna1994的代码：Matlab image rotation
Biqubic代码也基于此处发布的Java（和C ++）实现：http://www.paulinternet.nl/?page=bicubic

以下代码使用biqubic插值应用图像旋转示例：

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);

我通过比较函数imwarp

中的Matalb构建来验证实现

结果：

以下示例使用＆＃34; CachedBicubicInterpolator＆＃34;代码版本，也支持RGB图像：

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;

结果：

使用双三次插值旋转图像而不进行旋转

1 个答案: