歸一化交叉相關Normalization cross correlation (NCC)

相關系數，圖像匹配

NCC正如其名字，是用來描述兩個目標的相關程度的，也就是說可以用來刻畫目標間的相似性。一般NCC也會被用來進行圖像匹配，即在一個圖像中搜索與一小塊已知區域的NCC最高的區域作為對應匹配，然后對准整幅圖像。

假設兩幅圖像,那么其NCC可如下計算：

其中表示點乘運算。

比如下面兩幅圖像，我們想對齊

part1.jpg

part2.jpg

首先在一幅圖像中選取兩個圖像都有的區域作為templete

templete.jpg

然后在另一幅圖像中選取包含templete的區域

region.jpg

最終對齊的結果如下

result.jpg

1. function NCC 

   ---

2. close all 

   ---

3. part2 =imread('part1.png'); 

   ---

4. part1 = imread('part2.png'); 

   ---

5.  

   ---

6. imshow(part2);title('part2') 

   ---

7. figure, imshow(part1);title('part1'); 

   ---

8. % 選取template和搜索區域region. Note:template的大小必須小於region 

   ---

9. figure, 

   ---

10. [sub_part2,rect_part2]=imcrop(part2);% template 

    ---

11. [sub_part1,rect_part1]=imcrop(part1);% region 

    ---

12.  

    ---

13. figure, imshow(sub_part2) 

    ---

14. figure, imshow(sub_part1); 

    ---

15.  

    ---

16. [x,y,z] = normcorr2(double(sub_part2(:,:,1)),double(sub_part1(:,:,1))); 

    ---

17. figure,surf(x,y,z),shading flat 

    ---

18. %% 尋找偏置 

    ---

19. [~,imax]=max(z(:));% 

    ---

20. [yoffset,xoffset]=ind2sub(size(z),imax(1)); 

    ---

21. %% 圖像對齊 

    ---

22. % 配對點在第一幅圖像中到邊界的距離 

    ---

23.  

    ---

24. rect_part1=floor(rect_part1); 

    ---

25. left1=rect_part1(1)+xoffset; 

    ---

26. right1=size(part1,2)-left1; 

    ---

27. up1=rect_part1(2)+yoffset; 

    ---

28. down1 = size(part1,1)-left1; 

    ---

29. % 配對點在第二幅圖像中到邊界的距離 

    ---

30. rect_part2=floor(rect_part2); 

    ---

31. left2=rect_part2(1); 

    ---

32. right2=size(part2,2)-left2; 

    ---

33. up2=rect_part2(2); 

    ---

34. down2 = size(part2,1)-up2; 

    ---

35.  

    ---

36. img=zeros(max(up1,up2)+max(down1,down2)+1,max(left1,left2)+max(right1,right2)+1,size(part1,3)); 

    ---

37. sx=max(left1,left2)+1; 

    ---

38. sy=max(up1,up2)+1; 

    ---

39. img(sy-up1+1:sy-up1+size(part1,1),sx-left1+1:sx-left1+size(part1,2),:)=part1; 

    ---

40. img(sy-up2+1:sy-up2+size(part2,1),sx-left2+1:sx-left2+size(part2,2),:)=part2; 

    ---

41. imshow(uint8(img)); 

    ---

42. end 

    ---

43.  

    ---

44.  

    ---

45.  

    ---

46. function [x,y,z]=normcorr2(I1,I2) 

    ---

47. % I1,I2是單通道圖像，即矩陣 

    ---

48. % I1 is template 

    ---

49. % I2 is the region where match pairs are searched 

    ---

50. [m1,n1]=size(I1); 

    ---

51. [m2,n2]=size(I2); 

    ---

52. z=conv2(I2,rot90(I1,2),'valid'); 

    ---

53. im2=cumsum(I2.^2,1); 

    ---

54. im2=cumsum(im2,2); 

    ---

55. sum1=sum(sum(I1.^2,1),2); 

    ---

56. % nz=zeros(m2-m1+1,n2-n1+1); 

    ---

57. wrapIm2=zeros(1+m2,1+n2); 

    ---

58. wrapIm2(2:end,2:end)=im2; 

    ---

59. nz=(wrapIm2(m1+1:m2+1,n1+1:n2+1)+wrapIm2(1:m2-m1+1,1:n2-n1+1)-wrapIm2(1:m2-m1+1,n1+1:n2+1)-wrapIm2(m1+1:m2+1,1:n2-n1+1))*sum1; 

    ---

60. nz=sqrt(nz); 

    ---

61. z=z./nz; 

    ---

62. [x,y]=meshgrid(1:n2-n1+1,1:m2-m1+1); 

    ---

63. end 

    ---

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有些文獻在計算卷積的時候，總說要把矩陣逆排序之后再對應相乘，包括conv2這個函數也是這么計算的。但實際上我們有一個templete和region，直接就是對應位置乘積之后作為卷積結果，也只有這樣才能保證在匹配的位置處NCC系數最大，所以在代碼里，我先對templete進行了逆序操作，在使用conv2卷積。

另外matlab2016添加了image registration 工具箱，里面可以直接計算NCC系數了，具體參見
http://cn.mathworks.com/help/images/examples/registering-an-image-using-normalized-cross-correlation.html.