{"id":3198,"date":"2019-04-02T07:00:18","date_gmt":"2019-04-02T11:00:18","guid":{"rendered":"https:\/\/blogs.mathworks.com\/steve\/?p=3198"},"modified":"2019-11-01T22:36:45","modified_gmt":"2019-11-02T02:36:45","slug":"multiresolution-image-pyramids-and-impyramid-part-1","status":"publish","type":"post","link":"https:\/\/blogs.mathworks.com\/steve\/2019\/04\/02\/multiresolution-image-pyramids-and-impyramid-part-1\/","title":{"rendered":"Multiresolution image pyramids and impyramid – part 1"},"content":{"rendered":"

There's a function in the Image Processing Toolbox that I'm not especially fond of: impyramid<\/tt>.<\/p>

I shouldn't tell you who designed this function, but I'll give you a hint: his initials are \"Steve Eddins.\"<\/p>

Years ago, we had occasional user requests for creating multiresolution image pyramids, such as Gaussian pyramids and Laplacian pyramids. I thought these requests would be easy to satisfy by impyramid<\/tt>, but I was wrong. I was reminded of the problems by some code written by one our Pilot engineers, Steve Kuznicki. Steve made a nice of use impyramid<\/tt> to implement a multiresolution image blending method, but weaknesses in the design of impyramid<\/tt> made some of the code awkward. Today I want to discuss those weaknesses. In a follow-up post later, I'll demonstrate some possible improvements.<\/p>

I'll start by explaining multiresolution pyramids. Below an example of one. This form is sometimes called a lowpass pyramid.<\/p>

\"\" <\/p>

The original image is shown in the upper left. The image is lowpass filtered and then subsampled by a factor of 2 in each direction to form the image in the upper right. That process is repeated several times, reaching the tiny version of the original image at the lower right.<\/p>

So, I think \"multiresolution pyramid\" and \"lowpass pyramid\" are both sensible terms, but why is this frequently called a \"Gaussian pyramid\"? The answer comes from an implementation technique described in Burt, \"Fast Filter Transforms for Image Processing,\" Computer Graphics and Image Processing<\/i>, vol. 16, pp. 20-51, 1981. Burt described a procedure for forming a multiresolution pyramid by filtering at each step with the kernel $h_1(x)$ in Fig. 2 from the paper:<\/p>

\"\" <\/p>

The coefficients of $h_1(x)$ are $1\/4 - a\/2$, 0.25, and $a$, with typical values for $a$ around 0.4.<\/p>

As you successively apply this filter at each stage of a multiresolution pyramid, the overall effect becomes similar to filtering with a continuous Gaussian-shaped kernel, and this is where the name Gaussian pyramid<\/i> comes from.<\/p>

At the time this paper was written, computers were vastly slower than they are now, especially with floating-point computation. The floating-point convolution of a 512x512 image (considered large then, and considered quite small now) with a filter of modest size could take minutes.<\/p>

That explains part of the significance of Burt's implementation method. Not only is the filter applied at each pyramid stage very small, but when $a = 0.375$, all the filter coefficients are exact binary fractions and so can be implemented by direct processor bit-shift instructions on the integer pixel values. That allowed for reasonably fast implementations on the computers of that time.<\/p>

When I try to put impyramid<\/tt> to good use, or when I have seen other people try to use it, the following problems stand out to me:<\/p>