关于积分图（Integral Image）

在学习opencv的surf的时候，又一次看到了积分图这个概念，便努力回忆上次是怎么用到它的。记起来了，是在学习人脸检测时，Viola-Jones的文章中提到的。积分图概念的创始人不记得了，但第一次让积分图推广使用的应该就是Viola&Jones了。

在网上看到有人对从SIFT转而想到SURF而感到惊叹的时候，我却觉得根本没什么大不了。俺第一次看到SIFT的时候，就一下子想到可以借鉴到adaboost的人脸检测的。

回忆积分图概念的过程中，搜到一篇很好的介绍积分图的帖子，遂转贴出并向原作者致敬。目前为止看到的结果是积分图用到了Haar小波，LBP，HOG和SURF的 特征提取中，此种手法也是用空间换取时间的算法优化，值得学习。

How it works

Suppose an image is w pixels wide and h pixels high. Then the integral of this will be w+1 pixels wide and h+1 pixels high. The first row and column of the integral image are all zeros.

All other pixels have a value equal to the sum of all pixels before it.

See the integral in the above image? Every pixel is the summation of the pixels before it (above and to the left).

Now, to calculate the summation of the pixels in the black box, you take the corresponding box in the integral. You sum as follows: (Bottom right + top left – top right – bottom left).

So for the 3,5,4,1 box, the calculations would go like this: (30+0-17-0 = 13). For the 4,1 box, it would be (0+15-10-0 = 5).

This way, you can calculate summations in rectangular regions rapidly.

More than just summations!

With the basic idea in mind, you can extend it to more types of summations. You can calculate the sum of squares. You can rotate the image by 45 degrees and then do the summations. Then, you can calculate the totals in any arbitrary rectangular region that is upright or tilted at 45 degrees.

You can calculate summations on irregular areas too (only those with 90 degree corners though). Not just that, you can do super fast blurs, approximate gradients and compute means and standard deviations very fast.

Calculating Integral Images in OpenCV

OpenCV comes with a predefined function to calculate an integral image.

void cvIntegral(const CvArr* image, CvArr* sum, CvArr* sqsum=NULL, CvArr* tilted_sum=NULL);

The parameters are, as always, self explanatory:

• image: the source image

• sum: the sum summation integral image

• sqsum: the square sum integral image

• tiled_sum: image is rotated by 45 degrees and then its integral is calculated

Summary

Calculating integral images is trivial. But they let you do more complex stuff (like blurring, HAAR wavelets, etc) super fast. And cvIntegral in OpenCV calculates integral images for you.

也贴出只针对8位灰度图的代码，相当精简，看起来会比openCV的稍稍舒服一些

 

for( y = 0; y < image->height; y++, src += image->width, sum += sum_width )

{

    int s = sum[-1] = 0;

    for( x = 0; x < image->width; x ++ )

    {

        s += src[x];

        sum[x] = sum[x - sum_width] + s;

    }

}

原博客地址：http://blog.sina.com.cn/s/blog_5584da9601018f5x.html