ellipseDetectionByArcSupportLSs.m

function [ellipses, L, posi] = ellipseDetectionByArcSupportLSs(I, Tac, Tr, specified_polarity)
%input:
% I: input image
% Tac: elliptic angular coverage (completeness degree)
% Tni: ratio of support inliers on an ellipse
%output:
% ellipses: N by 5. (center_x, center_y, a, b, phi)
% reference:
% 1、von Gioi R Grompone, Jeremie Jakubowicz, Jean-
% Michel Morel, and Gregory Randall, “Lsd: a fast line
% segment detector with a false detection control.,” IEEE
% transactions on pattern analysis and machine intelligence,
% vol. 32, no. 4, pp. 722–732, 2010.
    angleCoverage = Tac;%default 165°
    Tmin = Tr;%default 0.6 
    unit_dis_tolerance = 2; %max([2, 0.005 * min([size(I, 1), size(I, 2)])]);%内点距离的容忍差小于max(2,0.5%*minsize)
    normal_tolerance = pi/9; %法线容忍角度20°= pi/9
    t0 = clock;
    if(size(I,3)>1)
        I = rgb2gray(I);
        [candidates, edge, normals, lsimg] = generateEllipseCandidates(I, 2, specified_polarity);%1,sobel; 2,canny
    else
        [candidates, edge, normals, lsimg] = generateEllipseCandidates(I, 2, specified_polarity);%1,sobel; 2,canny
    end
%    figure; imshow(edge);
%     return;
%    subplot(1,2,1);imshow(edge);%show edge image
%    subplot(1,2,2);imshow(lsimg);%show LS image
    t1 = clock;
    disp(['the time of generating ellipse candidates:',num2str(etime(t1,t0))]);
    candidates = candidates';%ellipse candidates matrix Transposition
    if(candidates(1) == 0)%表示没有找到候选圆
        candidates =  zeros(0, 5);
    end
    posi = candidates;
    normals    = normals';%norams matrix transposition
    [y, x]=find(edge);%找到非0元素的行(y)、列(x)的索引
%     ellipses = [];L=[];
%     return;
    [mylabels,labels, ellipses] = ellipseDetection(candidates ,[x, y], normals, unit_dis_tolerance, normal_tolerance, Tmin, angleCoverage, I);%后四个参数 0.5% 20° 0.6 180° 
    disp('-----------------------------------------------------------');
    disp(['running time:',num2str(etime(clock,t0)),'s']);
%     labels
%     size(labels)
%     size(y)
    warning('on', 'all');
     L = zeros(size(I, 1), size(I, 2));%创建与输入图像I一样大小的0矩阵L
     L(sub2ind(size(L), y, x)) = mylabels;%labels,长度等于edge_pixel_n x 1,如果第i个边缘点用于识别了第j个圆，则该行标记为j,否则为0。大小 edge_pixel_n x 1;现在转化存到图像中，在图像中标记
%     figure;imshow(L==2);%LLL
%     imwrite((L==2),'D:\Graduate Design\画图\edge_result.jpg');
end
%% ================================================================================================================================
%函数1
%输入
%candidates: ncandidates x 5
%points:     边缘像素点的坐标(x,y),nx2,n为总共的边缘点数
%lineLabels: 对相应的坐标(xi,yi)标记，对应靠近相应的线段，nx1,未标记则为0
%lines:      线段参数，-B,A,xmid,ymid，其中(xmid,ymid)对应相应的线段中点，mx4，m为总共m条线段
%输出
%labels：    长度等于n x 1,如果第i个边缘点用于识别了第j个圆，则该行标记为j,否则为0。大小 n x 1
%C：   识别出来的对称中心，长半轴，短半轴，和倾角，每一行格式是(x,y,a,b,phi)
function [mylabels,labels, ellipses] = ellipseDetection(candidates, points, normals, distance_tolerance, normal_tolerance, Tmin, angleCoverage, E)
    labels = zeros(size(points, 1), 1);
    mylabels = zeros(size(points, 1), 1);%测试
    ellipses = zeros(0, 5);
  
    %% 对于显著性很大的候选椭圆，且满足极其严格要求，直接检测出来，SE(salient ellipses)；同时对~SE按照goodness进行pseudo order
    goodness = zeros(size(candidates, 1), 1);%初始化时为0，当检测到显著椭圆时直接提取，相应位置的goodness(i) = -1标记。
    for i = 1 : size(candidates,1)
        %ellipse circumference is approximate pi * (1.5*sum(ellipseAxes)-sqrt(ellipseAxes(1)*ellipseAxes(2))
        ellipseCenter = candidates(i, 1 : 2);
        ellipseAxes   = candidates(i, 3:4);
        tbins = min( [ 180, floor( pi * (1.5*sum(ellipseAxes)-sqrt(ellipseAxes(1)*ellipseAxes(2)) ) * Tmin ) ] );%选分区
        %ellipse_normals = computePointAngle(candidates(i,:),points);
        %inliers = find( labels == 0 & dRosin_square(candidates(i,:),points) <= 1 );  % +-1个像素内找支持内点
        %加速计算，只挑出椭圆外接矩形内的边缘点(椭圆中的长轴a>b),s_dx存储的是相对points的索引
        s_dx = find( points(:,1) >= (ellipseCenter(1)-ellipseAxes(1)-1) & points(:,1) <= (ellipseCenter(1)+ellipseAxes(1)+1) & points(:,2) >= (ellipseCenter(2)-ellipseAxes(1)-1) & points(:,2) <= (ellipseCenter(2)+ellipseAxes(1)+1));
        inliers = s_dx(dRosin_square(candidates(i,:),points(s_dx,:)) <= 1);
        ellipse_normals = computePointAngle(candidates(i,:),points(inliers,:));
        p_dot_temp = dot(normals(inliers,:), ellipse_normals, 2); %加速后ellipse_normals(inliers,:)改为加速后ellipse_normals
        p_cnt = sum(p_dot_temp>0);%无奈之举，做一次极性统计，当改为C代码时注意拟合圆时内点极性的选取问题
        if(p_cnt > size(inliers,1)*0.5)
            %极性相异,也就是内黑外白    
            %ellipse_polarity = -1;
            inliers = inliers(p_dot_temp>0 & p_dot_temp >= 0.923879532511287 );%cos(pi/8) = 0.923879532511287, 夹角小于22.5°  
        else
            %极性相同,也就是内白外黑 
            %ellipse_polarity = 1;
            inliers = inliers(p_dot_temp<0 & (-p_dot_temp) >= 0.923879532511287 );
        end
        inliers = inliers(takeInliers(points(inliers, :), ellipseCenter, tbins)); 
        support_inliers_ratio = length(inliers)/floor( pi * (1.5*sum(ellipseAxes)-sqrt(ellipseAxes(1)*ellipseAxes(2)) ));
        completeness_ratio = calcuCompleteness(points(inliers,:),ellipseCenter,tbins)/360;
        goodness(i) = sqrt(support_inliers_ratio*completeness_ratio); %goodness = sqrt(r_i * r_c)
        %{
        if( support_inliers_ratio >= Tmin && completeness_ratio >= 0.75 ) %300/360 = 0.833333333333333 and ratio great than Tmin
            goodness(i) = -1;
            if (size(ellipses, 1) > 0)
                s_flag = false;
                for j = 1 : size(ellipses, 1)
                    %新识别出来的圆不能够与之前识别出来的圆重复，pi*0.1 = 0.314159265358979
                    if (sqrt((ellipses(j, 1) - candidates(i, 1)) .^ 2 + (ellipses(j, 2) - candidates(i, 2)) .^ 2) <= distance_tolerance ...
                            && sqrt((ellipses(j, 3) - candidates(i, 3)) .^ 2 + (ellipses(j, 4) - candidates(i, 4)) .^ 2 ) <= distance_tolerance ...
                            && abs( ellipses(j, 5) - candidates(i, 5) ) <= 0.314159265358979) %pi/10 = 18°
                        s_flag = true;
                        labels(inliers) = j;%如果重复了，就把该标签转移到之前的圆上面。
                        break;%打破内循环，继续下一个外循环
                    end
                end
                if (~s_flag)%如果不重复，则加入到识别的圆(circles)中
                    labels(inliers) = size(ellipses, 1) + 1;
                    ellipses = [ellipses; candidates(i, :)];
                    %drawEllipses(candidates(i, :)',E);
                end
            else
                labels(inliers) = size(ellipses, 1) + 1;
                ellipses = [ellipses; candidates(i, :)];%标记
                %drawEllipses(candidates(i, :)',E);
            end
        else
            goodness(i) = sqrt(support_inliers_ratio*completeness_ratio); %goodness = sqrt(r_i * r_c)
        end
        %}
    end
    %drawEllipses(ellipses',E);ellipses
    [goodness_descending, goodness_index] = sort(goodness,1,'descend');%here we can use pseudo order to speed up 
    candidates = candidates(goodness_index(goodness_descending>0),:);

    %%
%    t1 = clock;
    angles = [300; 210; 150; 90];%角度从大到小验证，列向量
    angles(angles < angleCoverage) = [];%只保留大于angleCoverage的部分
    if (isempty(angles) || angles(end) ~= angleCoverage)%如果angels为空了，或者angles最小的~=angleCoverage，则把angleCoverage加入进来
        angles = [angles; angleCoverage];
    end
%    disp('开始对一组圆的完整角度进行验证，开始angleLoop，在每次循环里设置一个angleCoverage，对候选圆进行验证，包括圆周上的内点的连通性分析，数量分析，完整度分析，从而找到有效圆，同时剔除无效圆');
    for angleLoop = 1 : length(angles)
        idx = find(labels == 0);%labels大小为边缘像素总数edge_nx1，初始化时labels全为0，找到labels中等于0的索引
        if (length(idx) < 2 * pi * (6 * distance_tolerance) * Tmin)%当idx数量小于一定值时
            break;
        end
        [L2, L, C, validCandidates] = subEllipseDetection( candidates, points(idx, :), normals(idx, :), distance_tolerance, normal_tolerance, Tmin, angles(angleLoop), E, angleLoop);
        candidates = candidates(validCandidates, :);%根据logical向量validCandidates进行剔除掉不成立的圆，剩下的圆继续用于下一个angleloop验证
      % size(candidates)
      % disp(angleLoop)
        if (size(C, 1) > 0)
            for i = 1 : size(C, 1)
                flag = false;
                for j = 1 : size(ellipses, 1)
                    %新识别出来的圆不能够与之前识别出来的圆重复，pi*0.1 = 0.314159265358979
                    if (sqrt((C(i, 1) - ellipses(j, 1)) .^ 2 + (C(i, 2) - ellipses(j, 2)) .^ 2) <= distance_tolerance ...
                        && sqrt((C(i, 3) - ellipses(j, 3)) .^ 2 + (C(i, 4) - ellipses(j, 4)) .^ 2) <= distance_tolerance ...
                        && abs(C(i, 5) - ellipses(j, 5)) <= 0.314159265358979) %pi/10 = 18°
                        flag = true;
                        labels(idx(L == i)) = j;%如果重复了，就把该标签转移到之前的圆上面。注意注意：idx存的是索引，label是标记该边缘点用在了第j个圆上，idx、labels都是一维类向量(n x 1)，labels与边缘点points(n x 2)总数一样,而idx则不一定
                        %==================================================
                        mylabels(idx(L2 == i)) = j;
                        %==================================================
                        break;%打破内循环，继续下一个外循环
                    end 
                end
                if (~flag)%如果不重复，则加入到识别的圆(circles)中
                    labels(idx(L == i)) = size(ellipses, 1) + 1;
                    %=================================================================
                    %%显示拟合出圆时所用的内点  my code 
                    mylabels(idx(L2 == i)) = size(ellipses, 1) + 1;%测试
                    %=================================================================
                    ellipses = [ellipses; C(i, :)];
                end
            end
        end
    end
%    t2 = clock;
%    disp(['聚类和验证时间：',num2str(etime(t2,t1))]);
end

%% ================================================================================================================================
%函数2
%输入
%list：      聚类候选的圆心和半径组合，(x,y,a,b,r)，大小 candidate_n x 5.
%points:     边缘像素点的坐标(x,y),nx2,n为总共的边缘点数
%normals:    每一个边缘点对应的梯度向量，normals大小为nx2，格式为(xi,yi)
 
%输出
%labels：    如果第i个边缘点用于检测到了第j个圆，则labels第i行赋值为j，否则为0.长度与points一致，n x 1
%circles:    此次检测到的圆,(x,y,z),若检测到detectnum个，则大小为detectnum x 3
%validCandidates: list的候选圆中，如果第i个圆被检测到了或者不满足圆条件(圆周上内点数量不足)，则第i个位置为false(初始化时为true)，这样在下一个angleloop轮次验证时可以剔除掉，不必要重复验证。
%                 validCandidates的大小为 candidate_n x 1.
function [mylabels,labels, ellipses, validCandidates] = subEllipseDetection( list, points, normals, distance_tolerance, normal_tolerance, Tmin, angleCoverage,E,angleLoop)
    labels = zeros(size(points, 1), 1);%边缘像素点的总数量n,n x 1
    mylabels = zeros(size(points, 1), 1);%测试
    ellipses = zeros(0, 5);
    ellipse_polarity = 0; %椭圆极性
    max_dis = max(points) - min(points);
    maxSemiMajor = max(max_dis);%最大的可能半径(此处可改为/2)
    maxSemiMinor = min(max_dis);
    distance_tolerance_square = distance_tolerance*distance_tolerance;
    validCandidates = true(size(list, 1), 1);%logical向量，大小 candidate_n x 1
    convergence = list;%候选椭圆副本
    for i = 1 : size(list, 1)
        ellipseCenter = list(i, 1 : 2);
        ellipseAxes = list(i, 3:4);
        ellipsePhi  = list(i,5);
        %ellipse circumference is approximate pi * (1.5*sum(ellipseAxes)-sqrt(ellipseAxes(1)*ellipseAxes(2))
        tbins = min( [ 180, floor( pi * (1.5*sum(ellipseAxes)-sqrt(ellipseAxes(1)*ellipseAxes(2)) ) * Tmin ) ] );%选分区
       
        %找到这个圆list(i,:)的圆周上的内点,find里面判断的结果为logical向量，是内点则对应在points中的行对应位置为1，否则为0，大小为n x 1
        %通过find找出非0元素的索引后，inliers则是保存着相应为内点的行的值，长度 inlier_n x 1
       
        %下行代码有问题，未进行极性分析，导致邻近的2 * distance_tolerance的极性相反的错误内点被纳入，从而拟合错误.
%           inliers = find(labels == 0 & abs(sqrt((points(:, 1) - circleCenter(1)) .^ 2 + (points(:, 2) - circleCenter(2)) .^ 2) - circleRadius) <= 2 * distance_tolerance & radtodeg(real(acos(abs(dot(normals, circle_normals, 2))))) <= normal_tolerance);
        %===============================================================================================================================================================
        %上行代码改为如下代码,my code
        
%         size(labels)
%         size(dRosin_square(list(i,:),points) )
%         ppp = (dRosin_square(list(i,:),points) <= 4*distance_tolerance_square)
%           if( i == 11 && angleCoverage == 165)
%          drawEllipses(list(i,:)',E);
%           end
        %此处非常重要，距离应该要比之前的更狭小的范围寻找内点
        %ellipse_normals = computePointAngle(list(i,:),points);
        %inliers = find(labels == 0 & (dRosin_square(list(i,:),points) <= distance_tolerance_square) );  % 2.25 * distance_tolerance_square , 4*distance_tolerance_square.
        %加速计算，只挑出椭圆外接矩形内的边缘点(椭圆中的长轴a>b),i_dx存储的是相对在points中的索引.
        i_dx = find( points(:,1) >= (ellipseCenter(1)-ellipseAxes(1)-distance_tolerance) & points(:,1) <= (ellipseCenter(1)+ellipseAxes(1)+distance_tolerance) & points(:,2) >= (ellipseCenter(2)-ellipseAxes(1)-distance_tolerance) & points(:,2) <= (ellipseCenter(2)+ellipseAxes(1)+distance_tolerance));
        inliers = i_dx(labels(i_dx) == 0 & (dRosin_square(list(i,:),points(i_dx,:)) <= distance_tolerance_square) );
        ellipse_normals = computePointAngle(list(i,:),points(inliers,:));%ellipse_normals长度与inliers长度一致
        
%         if( i == 11 && angleCoverage == 165)
%         testim = zeros(size(E,1),size(E,2));
%         testim(sub2ind(size(E),points(inliers,2),points(inliers,1))) = 1;
%         figure;imshow(testim);
%         end 
        
        p_dot_temp = dot(normals(inliers,:), ellipse_normals, 2);
        p_cnt = sum(p_dot_temp>0);%无奈之举，做一次极性统计，当改为C代码时注意拟合圆时内点极性的选取问题
        if(p_cnt > size(inliers,1)*0.5)
            %极性相异,也就是内黑外白    
            ellipse_polarity = -1;
            inliers = inliers(p_dot_temp>0 & p_dot_temp >= 0.923879532511287 );%cos(pi/8) = 0.923879532511287, 夹角小于22.5°  
        else
            %极性相同,也就是内白外黑 
            ellipse_polarity = 1;
            inliers = inliers(p_dot_temp<0 & (-p_dot_temp) >= 0.923879532511287 );
        end
        
%         if( i == 11 && angleCoverage == 165)
%         testim = zeros(size(E,1),size(E,2));
%         testim(sub2ind(size(E),points(inliers,2),points(inliers,1))) = 1;
%         figure;imshow(testim);
%         end
        
        inliers2 = inliers;
        inliers3 = 0;
        %=================================================================================================================================================================
        %连通域分析，inliers为存的是在边缘点的行下标
%         size(points)
%          size(inliers)
%         size(points(inliers, :))
%         size(takeInliers(points(inliers, :), circleCenter, tbins))
        %连通域分析，得到有效的内点，内点提纯，也就是inliers中进一步产出有效的inliers,个数会减少，大小inlier_n2 x 1。注意注意：inliers中存的是在points中的行下标
        inliers = inliers(takeInliers(points(inliers, :), ellipseCenter, tbins));
        
%         if( i == 11 && angleCoverage == 165)
%         testim = zeros(size(E,1),size(E,2));
%         testim(sub2ind(size(E),points(inliers,2),points(inliers,1))) = 1;
%         figure;imshow(testim);
%         end

         [new_ellipse,new_info] = fitEllipse(points(inliers,1),points(inliers,2));
         
%           if( i == 11 && angleCoverage == 165)
%          drawEllipses(new_ellipse',E);
%           end

%         if angleLoop == 2   %mycode
%         dispimg = zeros(size(E,1),size(E,2),3);
%         dispimg(:,:,1) = E.*255;%边缘提取出来的是0-1图像
%         dispimg(:,:,2) = E.*255;
%         dispimg(:,:,3) = E.*255;
%         for i = 1:length(inliers)
%         dispimg(points(inliers(i),2),points(inliers(i),1),:)=[0 0 255];
%         end
%         dispimg = drawCircle(dispimg,[newa newb],newr);
%         figure;
%         imshow(uint8(dispimg));
%         end

        if (new_info == 1)%如果是用最小二乘法拟合的而得出的结果
            %新对称中心和老对称中心的距离小于4*distance_tolerance, (a,b)的距离也是小于4*distance_tolerance,倾角phi小于0.314159265358979 = 0.1pi = 18°,因为新拟合出来的不能和原来的椭圆中心差很多,
            if ( (((new_ellipse(1) - ellipseCenter(1))^2 + (new_ellipse(2) - ellipseCenter(2))^2 ) <= 16 * distance_tolerance_square) ...
                && (((new_ellipse(3) - ellipseAxes(1))^2 + (new_ellipse(4) - ellipseAxes(2))^2 ) <= 16 * distance_tolerance_square) ...
                && (abs(new_ellipse(5) - ellipsePhi) <= 0.314159265358979) )
                ellipse_normals = computePointAngle(new_ellipse,points);
                %重新做一次找内点，连通性分析的内点提纯,这次的新的内点会用于后面的完整度分析
                %newinliers = find( (labels == 0) & (dRosin_square(new_ellipse,points) <= distance_tolerance_square) ...
                %    & ((dot(normals, ellipse_normals, 2)*(-ellipse_polarity)) >= 0.923879532511287) ); % (2*distance_tolerance)^2, cos(pi/8) = 0.923879532511287, 夹角小于22.5°
                %加速计算，只挑出椭圆外接矩形内的边缘点(椭圆中的长轴a>b),i_dx存储的是相对在points中的索引
                i_dx = find( points(:,1) >= (new_ellipse(1)-new_ellipse(3)-distance_tolerance) & points(:,1) <= (new_ellipse(1)+new_ellipse(3)+distance_tolerance) & points(:,2) >= (new_ellipse(2)-new_ellipse(3)-distance_tolerance) & points(:,2) <= (new_ellipse(2)+new_ellipse(3)+distance_tolerance));
                ellipse_normals = computePointAngle(new_ellipse,points(i_dx,:));%ellipse_normals长度与i_dx长度一致
                newinliers = i_dx(labels(i_dx) == 0 & (dRosin_square(new_ellipse,points(i_dx,:)) <= distance_tolerance_square & ((dot(normals(i_dx,:), ellipse_normals, 2)*(-ellipse_polarity)) >= 0.923879532511287) ) );
                newinliers = newinliers(takeInliers(points(newinliers, :), new_ellipse(1:2), tbins));
                if (length(newinliers) >= length(inliers))
                    %a = newa; b = newb; r = newr; cnd = newcnd;
                    inliers = newinliers;
                    inliers3 = newinliers;%my code，just test
                    %======================================================================
                    %二次拟合
                    %[newa, newb, newr, newcnd] = fitCircle(points(inliers, :));
                    [new_new_ellipse,new_new_info] = fitEllipse(points(inliers,1),points(inliers,2));
                    if(new_new_info == 1)
                       new_ellipse = new_new_ellipse;
                    end
                    %=======================================================================
                end
            end
        else
            new_ellipse = list(i,:);  %candidates
        end
        
        %内点数量大于圆周上的一定比例，Tmin为比例阈值
%         length(inliers)
%         floor( pi * (1.5*sum(new_ellipse(3:4))-sqrt(new_ellipse(3)*new_ellipse(4))) * Tmin )
        if (length(inliers) >= floor( pi * (1.5*sum(new_ellipse(3:4))-sqrt(new_ellipse(3)*new_ellipse(4))) * Tmin ))
            convergence(i, :) = new_ellipse;
            %与之前的圆心和半径参数几乎一致，重复了，因此把这个圆淘汰(排在最开头的和它重复的圆不一定会被淘汰)
            if (any( (sqrt(sum((convergence(1 : i - 1, 1 : 2) - repmat(new_ellipse(1:2), i - 1, 1)) .^ 2, 2)) <= distance_tolerance) ...
                & (sqrt(sum((convergence(1 : i - 1, 3 : 4) - repmat(new_ellipse(3:4), i - 1, 1)) .^ 2, 2)) <= distance_tolerance) ...
                & (abs(convergence(1 : i - 1, 5) - repmat(new_ellipse(5), i - 1, 1)) <= 0.314159265358979) ))
                validCandidates(i) = false;
            end
            %如果内点在圆周上满足angleCoverage的完整度
            %completeOrNot =  isComplete(points(inliers, :), new_ellipse(1:2), tbins, angleCoverage);
            completeOrNot = calcuCompleteness(points(inliers,:),new_ellipse(1:2),tbins) >= angleCoverage;
            if (new_info == 1 && new_ellipse(3) < maxSemiMajor && new_ellipse(4) < maxSemiMinor && completeOrNot )
                %且满足和其它圆参数大于distance_tolerance，也就是指和其它圆是不同的
                if (all( (sqrt(sum((ellipses(:, 1 : 2) - repmat(new_ellipse(1:2), size(ellipses, 1), 1)) .^ 2, 2)) > distance_tolerance) ...
                   | (sqrt(sum((ellipses(:, 3 : 4) - repmat(new_ellipse(3:4), size(ellipses, 1), 1)) .^ 2, 2)) > distance_tolerance) ...
                   | (abs(ellipses(:, 5) - repmat(new_ellipse(5), size(ellipses, 1), 1)) >= 0.314159265358979 ) )) %0.1 * pi = 0.314159265358979 = 18°
                    %size(inliers)
                    %line_normal = pca(points(inliers, :));%得到2x2的pca变换矩阵，因此第二列便是由内点统计出的梯度
                    %line_normal = line_normal(:, 2)';%取出第二列并且变为1 x 2 的行向量
                    %line_point = mean(points(inliers, :));%内点取平均
                    %防止数据点过于集中
                    %if (sum(abs(dot(points(inliers, :) - repmat(line_point, length(inliers), 1), repmat(line_normal, length(inliers), 1), 2)) <= distance_tolerance & radtodeg(real(acos(abs(dot(normals(inliers, :), repmat(line_normal, length(inliers), 1), 2))))) <= normal_tolerance) / length(inliers) < 0.8)
                         labels(inliers) = size(ellipses, 1) + 1;%标记，这些内点已经用过了，构成了新检测到圆周
                         %==================================================================
                         if(all(inliers3) == 1)
                         mylabels(inliers3) = size(ellipses,1) + 1; %显示拟合出圆时所用的内点  SSS
                         end
                         %==================================================================
                        ellipses = [ellipses; new_ellipse];%将该圆参数加入进去
                        validCandidates(i) = false;%第i个候选圆检测完毕
                        %disp([angleCoverage,i]);
                        %drawEllipses(new_ellipse',E);
                    %end
                end
            end
        else
            validCandidates(i) = false;%其它情况，淘汰该候选圆
        end
        
    end %for
end%fun
%% ================================================================================================================================
%函数4
%圆的最小二乘法拟合(此处可以改用快速圆拟合方法)
%输入：
%points: 联通性分析后的提纯后的内点，设大小为 fpn x 2,格式(xi,yi)
%输出：
%a   ：拟合后的圆心横坐标x
%b   ：拟合后的圆心纵坐标y
%c   ：拟合后的圆心半径r
%cnd ：1表示数据代入方程后是奇异的，直接用平均值估计；0表示数据是用最小二乘法拟合的
function [a, b, r, cnd] = fitCircle(points)
%{
    A = [sum(points(:, 1)), sum(points(:, 2)), size(points, 1); sum(points(:, 1) .* points(:, 2)), sum(points(:, 2) .* points(:, 2)), sum(points(:, 2)); sum(points(:, 1) .* points(:, 1)), sum(points(:, 1) .* points(:, 2)), sum(points(:, 1))];
    %用最小二乘法时，A'A正则矩阵如果接近0，则意味着方程组线性，求平均值即可
    if (abs(det(A)) < 1e-9)
        cnd = 1;
        a = mean(points(:, 1));
        b = mean(points(:, 2));
        r = min(max(points) - min(points));
        return;
    end
    cnd = 0;
    B = [-sum(points(:, 1) .* points(:, 1) + points(:, 2) .* points(:, 2)); -sum(points(:, 1) .* points(:, 1) .* points(:, 2) + points(:, 2) .* points(:, 2) .* points(:, 2)); -sum(points(:, 1) .* points(:, 1) .* points(:, 1) + points(:, 1) .* points(:, 2) .* points(:, 2))];
    t = A \ B;
    a = -0.5 * t(1);
    b = -0.5 * t(2);
    r = sqrt((t(1) .^ 2 + t(2) .^ 2) / 4 - t(3));
 %}
    A = [sum(points(:, 1) .* points(:, 1)),sum(points(:, 1) .* points(:, 2)),sum(points(:, 1)); sum(points(:, 1) .* points(:, 2)),sum(points(:, 2) .* points(:, 2)),sum(points(:, 2)); sum(points(:, 1)),sum(points(:, 2)),size(points, 1)]; 
    %用最小二乘法时，A'A正则矩阵如果接近0，则意味着方程组线性，求平均值即可
    if (abs(det(A)) < 1e-9)
        cnd = 1;
        a = mean(points(:, 1));
        b = mean(points(:, 2));
        r = min(max(points) - min(points));
        return;
    end
    cnd = 0;
    B = [sum(-points(:, 1) .* points(:, 1) .* points(:, 1) - points(:, 1) .* points(:, 2) .* points(:, 2));sum(-points(:, 1) .* points(:, 1) .* points(:, 2) - points(:, 2) .* points(:, 2) .* points(:, 2)); sum(-points(:, 1) .* points(:, 1) - points(:, 2) .* points(:, 2))];
    t = A \ B;
    a = -0.5 * t(1);
    b = -0.5 * t(2);
    r = sqrt((t(1) .^ 2 + t(2) .^ 2) / 4 - t(3));
end
%% ================================================================================================================================
%函数5
%输入
%x     : 连通性分析后，满足数量2piRT的提纯后的内点(x,y)，将参与到完整度分析环节.num x 2
%center: 圆心(x,y)  1 x 2
%tbins ：分区总数
%angleCoverage: 需要达到的圆完整度
%输出
%result： true or false，表示该圆完整与不完整
%longest_inliers:
function [result, longest_inliers] = isComplete(x, center, tbins, angleCoverage)
    [theta, ~] = cart2pol(x(:, 1) - center(1), x(:, 2) - center(2));%theta为(-pi,pi)的角度，num x 1
    tmin = -pi; tmax = pi;
    tt = round((theta - tmin) / (tmax - tmin) * tbins + 0.5);%theta的第i个元素落在第j个bin，则tt第i行标记为j，大小num x 1
    tt(tt < 1) = 1; tt(tt > tbins) = tbins;
    h = histc(tt, 1 : tbins);
    longest_run = 0;
    start_idx = 1;
    end_idx = 1;
    while (start_idx <= tbins)
        if (h(start_idx) > 0)%找到bin中vote第一个大于0的
            end_idx = start_idx;
            while (start_idx <= tbins && h(start_idx) > 0)%直到bin第一个小于0的
                start_idx = start_idx + 1;
            end
            inliers = [end_idx, start_idx - 1];%此区间为连通区域
            inliers = find(tt >= inliers(1) & tt <= inliers(2));%在tt中找到落在此区间的内点的索引
            run = max(theta(inliers)) - min(theta(inliers));%角度差
            if (longest_run < run)%此举是为了找到最大的完整的且连通的跨度
                longest_run = run;
                longest_inliers = inliers;
            end
        end
        start_idx = start_idx + 1;
    end
    if (h(1) > 0 && h(tbins) > 0)%如果第一个bin和最后一个bin都大于0，有可能最大连通区域是头尾相连的这种情况
        start_idx = 1;
        while (start_idx < tbins && h(start_idx) > 0)%找到bin中vote第一个大于0的
            start_idx = start_idx + 1;
        end
        end_idx = tbins;%end_idx直接从最尾部开始往回找
        while (end_idx > 1 && end_idx > start_idx && h(end_idx) > 0)
            end_idx = end_idx - 1;
        end
        inliers = [start_idx - 1, end_idx + 1];
        run = max(theta(tt <= inliers(1)) + 2 * pi) - min(theta(tt >= inliers(2)));
        inliers = find(tt <= inliers(1) | tt >= inliers(2));
        if (longest_run < run)
            longest_run = run;
            longest_inliers = inliers;
        end
    end
    %最大的连通的跨度大于了angleCoverage，或者虽然最大连通跨度小于，但完整度足够了
    longest_run_deg = radtodeg(longest_run);
    h_greatthanzero_num = sum(h>0);
    result =  longest_run_deg >= angleCoverage || h_greatthanzero_num * (360 / tbins) >= min([360, 1.2*angleCoverage]);  %1.2 * angleCoverage
end
function [completeness] = calcuCompleteness(x, center, tbins)
    [theta, ~] = cart2pol(x(:, 1) - center(1), x(:, 2) - center(2));%theta为(-pi,pi)的角度，num x 1
    tmin = -pi; tmax = pi;
    tt = round((theta - tmin) / (tmax - tmin) * tbins + 0.5);%theta的第i个元素落在第j个bin，则tt第i行标记为j，大小num x 1
    tt(tt < 1) = 1; tt(tt > tbins) = tbins;
    h = histc(tt, 1 : tbins);
    h_greatthanzero_num = sum(h>0);
    completeness = h_greatthanzero_num*(360 / tbins);
end
%% ================================================================================================================================
%函数6
%连通性分析，对圆周上的内点进行提纯
%输入
%x：椭圆周上的内点(x,y),设为inlier_n x 2 
%center：一个椭圆的中心(x,y) 1x2
%tbins: 分区 = min( 180 , pi*(1.5*(a+b)-sqrt(a*b)) ) 
%输出
%idx：为与x一样长的，inlier_n x 1的logical向量，返回有效的满足一定连通长度的内点，对应位置有效则为1，否则为0
function idx = takeInliers(x, center, tbins)
   [theta, ~] = cart2pol(x(:, 1) - center(1), x(:, 2) - center(2));%得到[-pi,pi]的方位角，等价于 theta = atan2(x(:, 2) - center(2) , x(:, 1) - center(1)); 
    tmin = -pi; tmax = pi;
    tt = round((theta - tmin) / (tmax - tmin) * tbins + 0.5);%将内点分区到[1 tbins]
    tt(tt < 1) = 1; tt(tt > tbins) = tbins;
    h = histc(tt, 1 : tbins);%h为直方图[1 tbins]的统计结果
    mark = zeros(tbins, 1);
    compSize = zeros(tbins, 1);
    nComps = 0;
    queue = zeros(tbins, 1);
    du = [-1, 1];
    for i = 1 : tbins
        if (h(i) > 0 && mark(i) == 0)%如果落在第i个分区内的值大于0，且mark(i)为0
            nComps = nComps + 1;
            mark(i) = nComps;%标记第nComps个连通区域
            front = 1; rear = 1;
            queue(front) = i;%将该分区加入队列，并以此开始任务
            while (front <= rear)
                u = queue(front);
                front = front + 1;
                for j = 1 : 2
                    v = u + du(j);
                    if (v == 0)
                        v = tbins;
                    end
                    if (v > tbins)
                        v = 1;
                    end
                    if (mark(v) == 0 && h(v) > 0)
                        rear = rear + 1;
                        queue(rear) = v;
                        mark(v) = nComps;%标记第nComps个连通区域
                    end
                end
            end
            compSize(nComps) = sum(ismember(tt, find(mark == nComps)));%得到构成连通域为nComps的内点数量
        end
    end
    compSize(nComps + 1 : end) = [];
    maxCompSize = max(compSize);
    validComps = find(compSize >= maxCompSize * 0.1 & compSize > 10);%大于等于最大连通长度的0.1倍的连通区域是有效的
    validBins = find(ismember(mark, validComps));%有效的分区
    idx = ismember(tt, validBins);%有效的内点
end
%% compute the points' normals belong to an ellipse, the normals have been already normalized. 
%param: [x0 y0 a b phi].
%points: [xi yi], n x 2
function [ellipse_normals] = computePointAngle(ellipse, points)
%convert [x0 y0 a b phi] to Ax^2+Bxy+Cy^2+Dx+Ey+F = 0
a_square = ellipse(3)^2;
b_square = ellipse(4)^2;
sin_phi = sin(ellipse(5));
cos_phi = cos(ellipse(5)); 
sin_square = sin_phi^2;
cos_square = cos_phi^2;
A = b_square*cos_square+a_square*sin_square;
B = (b_square-a_square)*sin_phi*cos_phi*2;
C = b_square*sin_square+a_square*cos_square;
D = -2*A*ellipse(1)-B*ellipse(2);
E = -2*C*ellipse(2)-B*ellipse(1);
% F = A*ellipse(1)^2+C*ellipse(2)^2+B*ellipse(1)*ellipse(2)-(ellipse(3)*ellipse(4)).^2;
% A = A/F;
% B = B/F;
% C = C/F;
% D = D/F;
% E = E/F;
% F = 1;
%calculate points' normals to ellipse
angles = atan2(C*points(:,2)+B/2*points(:,1)+E/2, A*points(:,1)+B/2*points(:,2)+D/2);
ellipse_normals = [cos(angles),sin(angles)];
end

%% param为[x0 y0 a b Phi],1 x 5 或者 5 x 1
%points为待计算rosin distance的点，每一行为(xi,yi),size是 n x 2
%dmin为输出估计距离的平方.
%调用注意，当a = b时，也就是椭圆退化成圆时，dmin会变成无穷大NAN，不能用此函数
function [dmin]= dRosin_square(param,points)
ae2 = param(3).*param(3);
be2 = param(4).*param(4);
x = points(:,1) - param(1);
y = points(:,2) - param(2);
xp = x*cos(-param(5))-y*sin(-param(5));
yp = x*sin(-param(5))+y*cos(-param(5));
fe2 = ae2-be2;
X = xp.*xp;
Y = yp.*yp;
delta = (X+Y+fe2).^2-4*fe2*X;
A = (X+Y+fe2-sqrt(delta))/2;
ah = sqrt(A);
bh2 = fe2-A;
term = A*be2+ae2*bh2;
xi = ah.*sqrt(ae2*(be2+bh2)./term);
yi = param(4)*sqrt(bh2.*(ae2-A)./term);
d = zeros(size(points,1),4);%n x 4
d(:,1) = (xp-xi).^2+(yp-yi).^2;
d(:,2) = (xp-xi).^2+(yp+yi).^2;
d(:,3) = (xp+xi).^2+(yp-yi).^2;
d(:,4) = (xp+xi).^2+(yp+yi).^2;
dmin = min(d,[],2); %返回距离的平方
%[dmin, ii] = min(d,[],2); %返回距离的平方
% for jj = 1:length(dmin)
%     if(ii(jj) == 1)
%         xi(jj) = xi(jj);
%         yi(jj) = yi(jj);
%     elseif (ii(jj) == 2)
%         xi(jj) = xi(jj);
%         yi(jj) = -yi(jj);
%     elseif (ii(jj) == 3)
%         xi(jj) = -xi(jj);
%         yi(jj) = yi(jj);
%     elseif(ii(jj) == 4)
%          xi(jj) = -xi(jj);
%         yi(jj) = -yi(jj);
%     end
% end
% 
% xi =  xi*cos(param(5))-yi*sin(param(5));
% yi =  xi*sin(param(5))+yi*cos(param(5));
% 
% testim = zeros(300,300);
% testim(sub2ind([300 300],uint16(yi+param(2)),uint16(xi+param(1)))) = 1;
% figure;imshow(uint8(testim).*255);
end