bwdistsc 快速距离场计算函数解析

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之前的《距离场计算:维度诱导法(dimension-induction)的基本原理》,已经简述了快速距离场计算基本思路,bwdistsc函数是 Yuriy Mishchenko 对上述思路的实现。这里将函数源码做分析与解释。介意结合 Yuriy Mishchenko 论文阅读。

经测试,函数速度相当快,并且计算量与实际问题复杂度相关。若实际问题中物体较为规则,在相同网格数量下,算法执行速度将比复杂拓扑结构物体相比有明显提高。

文章为简明删去了部分原有注释,主要是权利声明等。需要使用函数的话可到上文链接处Mathworks网站下载最新版

声明与输入解析

matlabfunction D=bwdistsc(bw,aspect)

% parse inputs
if(nargin<2 || isempty(aspect)) aspect=[1 1 1]; end

% determine geometry of the data
if(iscell(bw)) shape=[size(bw{1}),length(bw)]; else shape=size(bw); end

% correct this for 2D data
if(length(shape)==2) shape=[shape,1]; end
if(length(aspect)==2) aspect=[aspect,1]; end

% allocate internal memory
D=cell(1,shape(3)); for k=1:shape(3) D{k}=zeros(shape(1:2)); end

以上都是预处理。二维还是三维,网格长宽比是否为1,等等。

计算距离场

matlab%%%%%%%%%%%%% scan along XY %%%%%%%%%%%%%%%%
for k=1:shape(3)    %循环,按第三个坐标(z)循环
    if(iscell(bw)) bwXY=bw{k}; else bwXY=bw(:,:,k); end
    %切片储存在bw中

    % initialize arrays
    DXY=zeros(shape(1:2));
    D1=zeros(shape(1:2));

    % if can, use 2D bwdist from image processing toolbox    
    if(exist('bwdist') && aspect(1)==aspect(2))
        D1=aspect(1)^2*bwdist(bwXY).^2;
    else    % if not, use full XY-scan

计算一维距离场

切成一条一条的计算。一行一行,从第一行向最后一行扫略(从上到下),再从最后一行向第一行扫略(从下到上),从而生成了两个记录矩阵。这两个矩阵中每一个网格点记录的值是当前列中,按当前方向扫略的情况下,离自己最近的物体网格点。

下文中"on"-pixel指的是输入bw中数值为1的网格,即认为是有物体的网格
虽然有很多“条”,但是都用了向量化计算,所以这里没有循环语句

matlab
% z的循环还在进行 %%%%%%%%%%%%%%% X-SCAN %%%%%%%%%%%%%%% % reference for nearest "on"-pixel in bw in x direction down % scan bottow-up (for all y), copy x-reference from previous row % unless there is "on"-pixel in that point in current row, then % that is the nearest pixel now xlower=repmat(Inf,shape(1:2)); xlower(1,find(bwXY(1,:)))=1; % fill in first row for i=2:shape(1) xlower(i,:)=xlower(i-1,:); % copy previous row xlower(i,find(bwXY(i,:)))=i;% unless there is pixel end % reference for nearest "on"-pixel in bw in x direction up xupper=repmat(Inf,shape(1:2)); xupper(end,find(bwXY(end,:)))=shape(1); for i=shape(1)-1:-1:1 xupper(i,:)=xupper(i+1,:); xupper(i,find(bwXY(i,:)))=i; end % build (X,Y) for points for which distance needs to be calculated idx=find(~bwXY); [x,y]=ind2sub(shape(1:2),idx); % update distances as shortest to "on" pixels up/down in the above % 将两个矩阵合并起来 DXY(idx)=aspect(1)^2*min((x-xlower(idx)).^2,(x-xupper(idx)).^2);

计算二维距离场

matlab
% z的循环还在继续 %%%%%%%%%%%%%%% Y-SCAN %%%%%%%%%%%%%%% % this will be the envelop of parabolas at different y D1=repmat(Inf,shape(1:2)); p=shape(2); for i=1:shape(2) % some auxiliary datasets % 取出平面上的一列。从左向右按列扫描 % d0中存放的是距离的平方 d0=DXY(:,i); % selecting starting point for x: % * if parabolas are incremented in increasing order of y, % then all below-envelop intervals are necessarily right- % open, which means starting point can always be chosen % at the right end of y-axis % * if starting point exists it should be below existing % current envelop at the right end of y-axis dtmp=d0+aspect(2)^2*(p-i)^2; %均匀网格下,aspect的三个分类分量均为1,可以忽略。写在这里真影响理解…… L=D1(:,p)>dtmp; % 比较D1中的一列与dtmp的大小 %一维数组L中储存的是大小比较的结果量(0或1,认为是bool) idx=find(L); D1(idx,p)=dtmp(L); % D1的最后一列被设置为dtmp的一部分(取最小值) % 上面这几句还可以精简 % these will keep track along which X should % keep updating distances map_lower=L; idx_lower=idx; %储存了dtmp比D1小的那些位置 % scan from starting points down in increments of 1 % 从尾巴开始向前扫描,重复类似上面的比较 % 但是只关心lower位置,下面有解释 for ii=p-1:-1:1 % new values for D dtmp=d0(idx_lower)+aspect(2)^2*(ii-i)^2; % these pixels are to be updated L=D1(idx_lower,ii)>dtmp; D1(idx_lower(L),ii)=dtmp(L); % other pixels are removed from scan map_lower(idx_lower)=L; idx_lower=idx_lower(L); if(isempty(idx_lower)) break; end end end end D{k}=D1; end %z的循环结束了

从尾巴开始向前扫描,重复类似上面的比较,但是只关心已经包含在lower中的位置。循环中不断缩小lower区域,lower若为空则可以提前结束循环

原因解释

参考论文中引理1,抛物线比下包络线低的位置只可能存在于一个区间,而不会断断续续的存在于多个区间。又由于所有的抛物线二次项系数相等,所以这一区间必定为无穷区间$(-\infty,x)$或者$(x,\infty)$或者$(-\infty,\infty)$(排除完全重合的情况)。如果是$(x,\infty)$的情况,则下面这个从右向左扫描,找到包络线与抛物线交点即不再继续的算法容易理解。

由于上面的i循环是从左往右扫描的,新加入的抛物线要么在包络线右半部分上升段与之相交(上述区间为$(x,\infty)$),或者整条都低于包络线(上述区间为$(-\infty,\infty)$)。$(-\infty,x)$的情况不存在,因为它已经被扫描方法排除了。可以画图观察。

计算三维距离场

matlab%%%%%%%%%%%%% scan along Z %%%%%%%%%%%%%%%%
% 新建一个跟D一样大的元胞数组D1,用Inf填充
D1=cell(size(D));
for k=1:shape(3) 
  D1{k}=repmat(Inf,shape(1:2)); 
end

% start building the envelope 
p=shape(3);
for k=1:shape(3)
    % if there are no objects in this slice, nothing to do
    if(isinf(D{k}(1,1)))    %有一个是Inf,即说明这一层里面一个物体点都没找到
      continue;
    end

下面的算法,跟由一维构建二维方式相同。由于每次在一个维度上做文章,所以无论扩展到多少维度,需要计算的总是只有那么一小撮抛物线,所以算法基本一样。需要的话都能写出递归来。

matlab    % selecting starting point for (x,y):
    % * if parabolas are incremented in increasing order of k, then all 
    %   intersections are necessarily at the right end of the envelop, 
    %   and so the starting point can be always chosen as the right end
    %   of the axis

    % check which points are valid starting points, & update the envelop
    dtmp=D{k}+aspect(3)^2*(p-k)^2;
    L=D1{p}>dtmp; 
    D1{p}(L)=dtmp(L);    

    % map_lower keeps track of which pixels can be yet updated with the 
    % new distance, i.e. all such XY that had been under the envelop for
    % all Deltak up to now, for Deltak<0
    map_lower=L;

    % these are maintained to keep fast track of whether map is empty
    idx_lower=find(map_lower);

    % scan away from the starting points in increments of -1
    for kk=p-1:-1:1
        % new values for D
        dtmp=D{k}(idx_lower)+aspect(3)^2*(kk-k)^2;

        % these pixels are to be updated
        L=D1{kk}(idx_lower)>dtmp;
        map_lower(idx_lower)=L;
        D1{kk}(idx_lower(L))=dtmp(L);

        % other pixels are removed from scan
        idx_lower=idx_lower(L);

        if(isempty(idx_lower)) break; end
    end
end

% prepare the answer
if(iscell(bw))
    D=cell(size(bw));
    for k=1:shape(3) D{k}=sqrt(D1{k}); end
else
    D=zeros(shape);
    for k=1:shape(3) D(:,:,k)=sqrt(D1{k}); end
end
end

计算完成

阅读 2.5k发布于 2015-08-04
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