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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.

这里$u$理解为到达点的时间,

[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:

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
   
    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
    
     1 && D( ii-1,jj )
     
      1 && D( ii,jj-1 )
      
       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
      
     
    
   

运行结果: