Keren配准算法
Keren算法是一种梯度算法，它在平移变换模型的基础上扩展到旋转-平移模型。Keren算法采用了泰勒展开及近似的思想，通过连续使用两次泰勒展开并取近似后得到闭式解公式。原始配准公式与近似结果分别如公式（1）和（2）所示
其中f(x,y)和g(x,y)分别表示两幅图像的灰度分布，x_0为水平偏移，y_0为竖直偏移，θ_0为旋转角度。
将公式（3）在(x,y)处做泰勒级数展开，并取一阶近似后：
计算近似表达式（3）与原式的误差平方和并对x_0 、y_0和θ_0求取偏导数，忽略非线性项后联立求解得到线性方程组，如公式（4）：
其中，
。解该线性方程可以获得x_0 、y_0和θ_0的估计值。
由于是基于泰勒展开及近似求取得到的各个量的估计值，所以只有在小角度偏转及平移情况下Keren算法才有较好的配准效果。
主函数
clc;
close all;
clear all;
num = 2;
S = cell(num,1);
S0 = cell(num,1);
path1 = 'E:\';
for i = 1:num
S{i} = imread([path1,num2str(i),'.jpg']);
OO = size(S{i},3);
if OO == 3
S{i} = im2double(rgb2gray(S{i}));
else
S{i} = im2double((S{i}));
end
end
[delta_est, phi_est] = keren(S);
keren函数
function [delta_est, phi_est] = keren(im)
for imnr = 2:length(im)
lp = fspecial('ga',5,1);
im0{1} = im{1};
im1{1} = im{imnr};
for i=2:3
im0{i} = imresize(conv2(im0{i-1},lp,'same'),0.5,'bicubic');
im1{i} = imresize(conv2(im1{i-1},lp,'same'),0.5,'bicubic');
end
stot = zeros(1,3);
% do actual registration, based on pyramid
for pyrlevel=3:-1:1
f0 = im0{pyrlevel};
f1 = im1{pyrlevel};
[y0,x0]=size(f0);
xmean=x0/2; ymean=y0/2;
x=kron([-xmean:xmean-1],ones(y0,1));
y=kron(ones(1,x0),[-ymean:ymean-1]');
sigma=1;
g1 = zeros(y0,x0); g2 = g1; g3 = g1;
for i=1:y0
for j=1:x0
g1(i,j)=-exp(-((i-ymean)^2+(j-xmean)^2)/(2*sigma^2))*(i-ymean)/2/pi/sigma^2; % d/dy
g2(i,j)=-exp(-((i-ymean)^2+(j-xmean)^2)/(2*sigma^2))*(j-xmean)/2/pi/sigma^2; % d/dx
g3(i,j)= exp(-((i-ymean)^2+(j-xmean)^2)/(2*sigma^2))/2/pi/sigma^2;
end
end
a=real(ifft2(fft2(f1).*fft2(g2)));
c=real(ifft2(fft2(f1).*fft2(g1)));
b=real(ifft2(fft2(f1).*fft2(g3)))-real(ifft2(fft2(f0).*fft2(g3)));
R=c.*x-a.*y; % df1/dy*x-df1/dx*y
a11 = sum(sum(a.*a)); a12 = sum(sum(a.*c)); a13 = sum(sum(R.*a));
a21 = sum(sum(a.*c)); a22 = sum(sum(c.*c)); a23 = sum(sum(R.*c));
a31 = sum(sum(R.*a)); a32 = sum(sum(R.*c)); a33 = sum(sum(R.*R));
b1 = sum(sum(a.*b)); b2 = sum(sum(c.*b)); b3 = sum(sum(R.*b));
Ainv = [a11 a12 a13; a21 a22 a23; a31 a32 a33]^(-1);
s = Ainv*[b1; b2; b3];
st = s;
it=1;
while ((abs(s(1))+abs(s(2))+abs(s(3))*180/pi/20>0.1)&it<25)
% first shift and then rotate, because we treat the reference image
f0_ = shift(f0,-st(1),-st(2));
f0_ = imrotate(f0_,-st(3)*180/pi,'bicubic','crop');
b = real(ifft2(fft2(f1).*fft2(g3)))-real(ifft2(fft2(f0_).*fft2(g3)));
s = Ainv*[sum(sum(a.*b)); sum(sum(c.*b)); sum(sum(R.*b))];
st = st+s;
it = it+1;
end
% it
st(3)=-st(3)*180/pi;
st = st';
st(1:2) = st(2:-1:1);
stot = [2*stot(1:2)+st(1:2) stot(3)+st(3)];
if pyrlevel>1
% first rotate and then shift, because this is cancelling the
% motion on the image to be registered
im1{pyrlevel-1} = imrotate(im1{pyrlevel-1},-stot(3),'bicubic','crop');
im1{pyrlevel-1} = shift(im1{pyrlevel-1},2*stot(2),2*stot(1)); % twice the parameters found at larger scale
end
end
delta_est(imnr,:) = stot(1:2)
phi_est(imnr) = stot(3);
end
shift函数
function im2 = shift(im1,x1,y1)
[y0,x0,z0]=size(im1);
x1int=floor(x1); x1dec=x1-x1int;
y1int=floor(y1); y1dec=y1-y1int;
im2=im1;
for z=1:z0
if y1>=0
for y=-y0:-y1int-2
im2(-y,:,z)=(1-y1dec)*im2(-y1int-y,:,z)+y1dec*im2(-y1int-y-1,:,z);
end
if y1int<y0
im2(y1int+1,:,z)=(1-y1dec)*im2(1,:,z);
end
for y=max(-y1int,-y0):-1
im2(-y,:,z)=zeros(1,x0);
end
else
if y1dec==0
y1dec=y1dec+1;
y1int=y1int-1;
end
for y=1:y0+y1int
im2(y,:,z)=y1dec*im2(-y1int+y-1,:,z)+(1-y1dec)*im2(-y1int+y,:,z);
end
if -y1int<=y0
im2(y0+y1int+1,:,z)=y1dec*im2(y0,:,z);
end
for y=max(1,y0+y1int+2):y0
im2(y,:,z)=zeros(1,x0);
end
end
if x1>=0
for x=-x0:-x1int-2
im2(:,-x,z)=(1-x1dec)*im2(:,-x1int-x,z)+x1dec*im2(:,-x1int-x-1,z);
end
if x1int<x0
im2(:,x1int+1,z)=(1-x1dec)*im2(:,1,z);
end
for x=max(-x1int,-x0):-1
im2(:,-x,z)=zeros(y0,1);
end
else
if x1dec==0
x1dec=x1dec+1;
x1int=x1int-1;
end
for x=1:x0+x1int
im2(:,x,z)=x1dec*im2(:,-x1int+x-1,z)+(1-x1dec)*im2(:,-x1int+x,z);
end
if -x1int<=x0
im2(:,x0+x1int+1,z)=x1dec*im2(:,x0,z);
end
for x=max(1,x0+x1int+2):x0
im2(:,x,z)=zeros(y0,1);
end
end
end
****直接运行主函数 ，同文件夹放两张图片，命名1 和2。2为需要配准的图片。****