2D Fast Marching Computations

Fast Marching method 跟 dijkstra 方法类似,只不过dijkstra方法的路径只能沿网格,而Fast Marching method的方法可以沿斜线.
[Level Set Methods and Fast Marching Methods p94-95

]

这里u理解为到达点的时间, Fijk理解为在点ijk的流速.
然后就可以跟Boundary Value Formulation对应起来了.

[Level Set Methods and Fast Marching Methods p7

]

本例,首先加载一张灰度图片, 将里面像素的值W作为该点的流速F. 然后算到达该点的时间,也就是程序里面的距离D.

matlab部分源代码如下:
example1.m

n = 400;[M,W] = load_potential_map('road2', n);%start_point = [16;219];%end_point = [394;192];% You can use instead the function[start_point,end_point] = pick_start_end_point(W);clear options;options.nb_iter_max = Inf;options.end_points = end_point; % stop propagation when end point reached[D,S] = perform_fast_marching(W, start_point, options);% nicde color for displayA = convert_distance_color(D);imageplot({W A}, {'Potential map' 'Distance to starting point'}); colormap gray(256);

perform_front_propagation_2d_slow.m

function [D,S,father] = perform_front_propagation_2d_slow(W,start_points,end_points,nb_iter_max,H)%   [D,S] = perform_front_propagation_2d_slow(W,start_points,end_points,nb_iter_max,H);%%   [The mex function is perform_front_propagation_2d]%%   'D' is a 2D array containing the value of the distance function to seed.%   'S' is a 2D array containing the state of each point : %       -1 : dead, distance have been computed.%        0 : open, distance is being computed but not set.%        1 : far, distance not already computed.%   'W' is the weight matrix (inverse of the speed).%   'start_points' is a 2 x num_start_points matrix where k is the number of starting points.%   'H' is an heuristic (distance that remains to goal). This is a 2D matrix.%   %   Copyright (c) 2004 Gabriel Peyr?data.D = W.*0 + Inf; % action 先把所有点的距离标为Infstart_ind = sub2ind(size(W), start_points(1,:), start_points(2,:));data.D( start_ind ) = 0; %将起点的距离设置为0data.O = start_points; % 将起点加入Open list % S=1 : far,  S=0 : open,  S=-1 : closedata.S = ones(size(W));% 将所点的状态设为Fardata.S( start_ind ) = 0; % 将起点的状态设为open(trial)data.W = W;data.father = zeros( [size(W),2] );% father维度400*400*2,父节点有两个,因为走斜线verbose = 1;if nargin<3    end_points = [];endif nargin<4    nb_iter_max = round( 1.2*size(W,1)^2 );endif nargin<5    data.H = W.*0;else    if isempty(H)        data.H = W.*0;    else        data.H = H;    endendif ~isempty(end_points)    end_ind = sub2ind(size(W), end_points(1,:), end_points(2,:));else    end_ind = [];endstr = 'Performing Fast Marching algorithm.';if verbose    h = waitbar(0,str);endi = 0; while i<nb_iter_max && ~isempty(data.O) && isempty( find( data.S(end_ind)==-1 ) )    i = i+1;    data = perform_fast_marching_step(data);    if verbose        waitbar(i/nb_iter_max, h, sprintf('Performing Fast Marching algorithm, iteration %d.', i) );    endendif verbose    close(h);endD = data.D;S = data.S;father = data.father;function data1 = perform_fast_marching_step(data)%有多个起点也是一样的,只需将他们的距离都设为0即可% perform_fast_marching_step - perform one step in the Fast Marching algorithm%%   data1 = perform_fast_marching_step(data);%%   Data is a structure that records the state before/after a step %   of the FM algorithm.%%   Copyright (c) 2004 Gabriel Peyr?% some constantkClose = -1;kOpen = 0;kFar = 1;D = data.D; % action, a 2D matrixO = data.O; % open listS = data.S; % state, either 'O' or 'C', a 2D matrixH = data.H; % HeuristicW = data.W; % weight matrix, a 2D array (speed function)father = data.father;[n,p] = size(D);  % size of the grid,   n,p都为400% step sizeh = 1/n;if isempty(O)%看开集是否为空    data1 = data;    return;endind_O = sub2ind(size(D), O(1,:), O(2,:));%获取开集里面的顶点的索引[m,I] = min( D(ind_O)+H(ind_O) ); %m里面是最小值,I里面是该最小值在被检测矩阵里面的索引I = I(1);%取第一个索引% selected vertex% 取开集中的第I个点的索引i = O(1,I);j = O(2,I);O(:,I) = [];  % pop from open ,将此点从开集中移除S(i,j) = kClose; % now its close, 将此点加入闭集(known set)中% its neighbor% 准备遍历他的右,上,左,下的邻近点nei = [1,0; 0,1; -1,0; 0,-1 ];for k = 1:4    ii = i+nei(k,1);    jj = j+nei(k,2);    if ii>0 && jj>0 && ii<=n && jj<=p        f = [0 0];  % current father        %%% update the action using Upwind resolution        P = h/W(ii,jj);        % neighbors values        a1 = Inf;        if ii<n            a1 = D( ii+1,jj );            f(1) = sub2ind(size(W), ii+1,jj);        end        if ii>1 && D( ii-1,jj )<a1            a1 = D( ii-1,jj );            f(1) = sub2ind(size(W), ii-1,jj);        end        a2 = Inf;        if jj<n            a2 = D( ii,jj+1 );            f(2) = sub2ind(size(W), ii,jj+1);        end        if jj>1 && D( ii,jj-1 )<a2            a2 = D( ii,jj-1 );            f(2) = sub2ind(size(W), ii,jj-1);        end        if a1>a2    % swap to reorder            tmp = a1; a1 = a2; a2 = tmp;            f = f([2 1]);        end        % now the equation is   (a-a1)^2+(a-a2)^2 = P^2, with a >= a2 >= a1.        % 书上95页公式为:(ux^2 + uy^2)^(1/2)=1/Fijk        % u理解为到达点的时间,Fijk理解为在点ijk处的流速        % 那么 ux = (a-a1)/(1/400), uy = (a-a2)/(1/400)        % 所以方程变为:((a-a1)^2/(1/400))^2+((a-a2)^2/(1/400))^2 = (1/Wij)^2        % 把1/400移到右边,则得P        if P^2 > (a2-a1)^2%delta 大于0            delta = 2*P^2-(a2-a1)^2;            A1 = (a1+a2+sqrt(delta))/2;        else%否则用dijkstra方法,沿着格子走,公式为:max|ux,uy|=1/Fijk            % (a-a1) / (1/400) = 1 / Wij            A1 = a1 + P;            f(2) = 0;%将第2个父节点设为0        end        switch S(ii,jj)            case kClose%闭集不用更新                % check if action has change. Should not appen for FM                if A1<D(ii,jj)                    % warning('FastMarching:NonMonotone', 'The update is not monotone');                    % pop from Close and add to Open                    if 0        % don't reopen close points                        O(:,end+1) = [ii;jj];                        S(ii,jj) = kOpen;                        D(ii,jj) = A1;                    end                end            case kOpen%开集才更新                % check if action has change.                if A1<D(ii,jj)                    D(ii,jj) = A1;                    father(ii,jj,:) = f;                end            case kFar%远集不仅更新,而且加入开集                if D(ii,jj)~=Inf                    warning('FastMarching:BadInit', 'Action must be initialized to Inf');                  end                    % add to open                O(:,end+1) = [ii;jj];                S(ii,jj) = kOpen;                % action must have change.                D(ii,jj) = A1;                father(ii,jj,:) = f;            otherwise                error('Unknown state');        end    endenddata1.D = D;data1.O = O;data1.S = S;data1.W = W;data1.H = H;data1.father = father;

运行结果:

matlab完整源代码