积分图像

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 andcolumn 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: (Bottomright + 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 parame
4000
ters 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;

    }

}