用扩展盒模糊逼近高斯模糊 [英] Approximating Gaussian Blur Using Extended Box Blur

问题描述

问题如下，如何使用Box Blur / Extended Box Blur对给定STD的高斯模糊滤波器进行近似。

The problem is a s following, how to approximate a Gaussian Blur Filter with a given STD using Box Blur / Extended Box Blur.

更具体地说，我知道这是Photoshop应用其高斯模糊的方式。

More specifically, I know this is the way Photoshop applies its Gaussian Blur.

首先，一篇关于扩展盒模糊可以在这里看到 - 扩展盒高斯卷积的理论基础过滤。

First, an article about "Extended Box Blur can be seen here - Theoretical Foundations of Gaussian Convolution by Extended Box Filtering.

我遇到的问题与文章中的图2有关。

解释这个问题的最佳方法是使用一个例子。

The problem I'm having is with Figure 2 in the article.
The best way to explain this would be using an example.

假设我们需要近似高斯模糊，STD为15.4 - > Var = 237.16。

为了得到一个很好的近似我们将用6次迭代的Box Blur来做。

Let's say we need to approximate a Gaussian Blur with STD of 15.4 -> Var = 237.16.
In order to have a good approximation we'll do that with 6 iterations of a Box Blur.

现在，我如何选择Box Blur的长度（我们将在可分离的方式，即在1D工作）？

我应该选择不同的长度（看来我必须）？

目标是匹配GB的Blur级别（这是它的STD / VAR）。

Now, How do I choose the length of the Box Blur (We'll do it in a separable manner, namely, working in 1D)?
Should I chose different lengths (It seems I have to)?
The target is matching the GB Level of Blur (Which is its STD / VAR).

谢谢。

PS

我正在使用MATLAB，所以代码很简单： - ）。

P.S.
I'm working on MATLAB, so code is easy :-).

推荐答案

<这是我的文章的MATLAB实现：

This is my MATLAB implementation of the article:

```

function [ vBoxBlurKernel ] = GenerateBoxBlurKernel( boxBlurVar, numIterations )
% ----------------------------------------------------------------------------------------------- %
% [ boxBlurKernel ] = GenerateBoxBlurKernel( boxBlurVar, numIterations )
%   Approximates 1D Gaussian Kernel by iterative convolutions of "Extended Box Filter".
% Input:
%   - boxBlurVar        -   BoxFilter Varaiance.
%                           The variance of the output Box Filter.
%                           Scalar, Floating Point (0, inf).
%   - numIterations     -   Number of Iterations.
%                           The number of convolution iterations in order
%                           to produce the output Box Filter.
%                           Scalar, Floating Point [1, inf), Integer.
% Output:
%   - vBoxBlurKernel    -   Output Box Filter.
%                           The Box Filter with 'boxBlurVar' Variance.
%                           Vector, Floating Point, (0, 1).
% Remarks:
%   1.  The output Box Filter has a variance of '' as if it is treated as
%       Discrete Probability Function.
%   2.  References: "Theoretical Foundations of Gaussian Convolution by Extended Box Filtering"
%   3.  Prefixes:
%       -   'm' - Matrix.
%       -   'v' - Vector.
% TODO:
%   1.  F
%   Release Notes:
%   -   1.0.001     07/05/2014  xxxx xxxxxx
%       *   Accurate calculation of the "Extended Box Filter" length as in
%           the reference.
%   -   1.0.000     06/05/2014  xxxx xxxxxx
%       *   First release version.
% ----------------------------------------------------------------------------------------------- %

boxBlurLength = sqrt(((12 * boxBlurVar) / numIterations) + 1);
boxBlurRadius = (boxBlurLength - 1) / 2;

% 'boxBlurRadiusInt' -> 'l' in the reference
boxBlurRadiusInt    = floor(boxBlurRadius);
% boxBlurRadiusFrac   = boxBlurRadius - boxBlurRadiusInt;

% The length of the "Integer" part of the filter.
% 'boxBlurLengthInt' -> 'L' in the reference
boxBlurLengthInt = 2 * boxBlurRadiusInt + 1;

a1 = ((2 * boxBlurRadiusInt) + 1);
a2 = (boxBlurRadiusInt * (boxBlurRadiusInt + 1)) - ((3 * boxBlurVar) / numIterations);
a3 = (6 * ((boxBlurVar / numIterations) - ((boxBlurRadiusInt + 1) ^ 2)));

alpha = a1 * (a2 / a3);
ww = alpha / ((2 * boxBlurRadiusInt) + 1 + (2 * alpha));

% The length of the "Extended Box Filter".
% 'boxBlurLength' -> '\Gamma' in the reference.
boxBlurLength = (2 * (alpha + boxBlurRadiusInt)) + 1;

% The "Single Box Filter" with Varaince - boxBlurVar / numIterations
% It is normalized by definition.
vSingleBoxBlurKernel = [ww, (ones(1, boxBlurLengthInt) / boxBlurLength), ww];
% vBoxBlurKernel = vBoxBlurKernel / sum(vBoxBlurKernel);

vBoxBlurKernel = vSingleBoxBlurKernel;

% singleBoxKernelVar = sum(([-(boxBlurRadiusInt + 1):(boxBlurRadiusInt + 1)] .^ 2) .* boxBlurKernel)
% boxKernelVar = numIterations * singleBoxKernelVar


for iIter = 2:numIterations
    vBoxBlurKernel = conv2(vBoxBlurKernel, vSingleBoxBlurKernel, 'full');
end


end

这是一个演示尝试一下：

Here's a demo to try it:

% Box Blur Demo

gaussianKernelStd = 9.6;
gaussianKernelVar = gaussianKernelStd * gaussianKernelStd;
gaussianKernelRadius = ceil(6 * gaussianKernelStd);
gaussianKernel = exp(-([-gaussianKernelRadius:gaussianKernelRadius] .^ 2) / (2 * gaussianKernelVar));
gaussianKernel = gaussianKernel / sum(gaussianKernel);

boxBlurKernel = GenerateBoxBlurKernel(gaussianKernelVar, 6);
boxBlurKernelRadius = (length(boxBlurKernel) - 1) / 2;

figure();
plot([-gaussianKernelRadius:gaussianKernelRadius], gaussianKernel, [-boxBlurKernelRadius:boxBlurKernelRadius], boxBlurKernel);

sum(([-boxBlurKernelRadius:boxBlurKernelRadius] .^ 2) .* boxBlurKernel)
sum(([-gaussianKernelRadius:gaussianKernelRadius] .^ 2) .* gaussianKernel)

棘手的部分是计算扩展盒过滤器的有效长度。

使用常规Box Filter的方差计算长度不是长度。

The tricky part is the calculation of the effective length of the "Extended Box Filter".
It's not the length by the calculation of the length using the variance of a regular "Box Filter".

文章很棒，这种方法很棒。

The article is great and this method is amazing.

这篇关于用扩展盒模糊逼近高斯模糊的文章就介绍到这了，希望我们推荐的答案对大家有所帮助，也希望大家多多支持IT屋！