face alignment by 3000 fps系列学习总结（三）

训练

我们主要以3000fps matlab实现为叙述主体。

总体目标

我们需要为68个特征点的每一个特征点训练5棵随机树，每棵树4层深，即为所谓的随机森林。

开始训练

分配样本
事实上，对于每个特征点，要训练随机森林，我们需要从现有的样本和特征中抽取一部分，训练成若干个树。
现在，我们有N（此处N=1622）个样本（图片和shape）和无数个像素差特征。训练时，对于每棵树，我们从N个样本采取有放回抽样的方法随机选取若干样本，再随机选取M个特征点。然后使用这些素材加以训练。这是一般的方法。不过为了简化，我们将N个样本平均分成5份，且允许彼此之间有重叠。然后分配好的样本用来作为68个特征点的共同素材。

示意图：
代码：

    dbsize = length(Tr_Data);% rf = cell(1, params.max_numtrees);overlap_ratio = params.bagging_overlap;%重叠比例Q = floor(double(dbsize)/((1-params.bagging_overlap)*(params.max_numtrees))); %每颗树分配的样本个数Data = cell(1, params.max_numtrees); %为训练每棵树准备的样本数据
for t = 1:params.max_numtrees% calculate the number of samples for each random tree% train t-th random treeis = max(floor((t-1)*Q - (t-1)*Q*overlap_ratio + 1), 1); ie = min(is + Q, dbsize);Data{t} = Tr_Data(is:ie);
end

2.随机森林训练全程

代码：

% divide local region into grid
params.radius = ([0:1/30:1]');
params.angles = 2*pi*[0:1/36:1]';rfs = cell(length(params.meanshape), params.max_numtrees); %随机森林的大小为68*5%parfor i = 1:length(params.meanshape)
for i = 1:length(params.meanshape)rf = cell(1, params.max_numtrees);disp(strcat(num2str(i), 'th landmark is processing...'));for t = 1:params.max_numtrees% disp(strcat('training', {''}, num2str(t), '-th tree for', {''}, num2str(lmarkID), '-th landmark'));% calculate the number of samples for each random tree% train t-th random treeis = max(floor((t-1)*Q - (t-1)*Q*overlap_ratio + 1), 1); %样本的序号ie = min(is + Q, dbsize);max_numnodes = 2^params.max_depth - 1; %最大的节点数自然是满二叉树的节点个数rf{t}.ind_samples = cell(max_numnodes, 1); %节点包含的样本序号rf{t}.issplit     = zeros(max_numnodes, 1);%是否分割rf{t}.pnode       = zeros(max_numnodes, 1);rf{t}.depth       = zeros(max_numnodes, 1);%当前深度rf{t}.cnodes      = zeros(max_numnodes, 2);%当前节点的左右子节点序号rf{t}.isleafnode  = zeros(max_numnodes, 1); %判断节点是否是叶子节点rf{t}.feat        = zeros(max_numnodes, 4); %围绕特征点随机选取的2个点的坐标（r1,a1,r2,a2）rf{t}.thresh      = zeros(max_numnodes, 1); %分割节点的阈值rf{t}.ind_samples{1} = 1:(ie - is + 1)*(params.augnumber); %第t棵树的样本序号，也是根节点包含的样本序号rf{t}.issplit(1)     = 0;rf{t}.pnode(1)       = 0;rf{t}.depth(1)       = 1;rf{t}.cnodes(1, 1:2) = [0 0];rf{t}.isleafnode(1)  = 1;rf{t}.feat(1, :)     = zeros(1, 4);rf{t}.thresh(1)      = 0;num_nodes = 1; %num_nodes为现有的节点个数num_leafnodes = 1;%num_leafnodes为现有的叶子节点个数stop = 0;while(~stop)  %这个循环用于产生随机树，直到没有再可以分割的点num_nodes_iter = num_nodes; %num_nodes为现有的节点个数num_split = 0; %分割节点的个数for n = 1:num_nodes_iterif ~rf{t}.issplit(n) %如果第t棵树第n个节点已经分过，就跳过去if rf{t}.depth(n) == params.max_depth % || length(rf{t}.ind_samples{n}) < 20if rf{t}.depth(n) == 1  %应该去掉吧????????????????rf{t}.depth(n) = 1;endrf{t}.issplit(n) = 1; else% separate the samples into left and right path[thresh, feat, lcind, rcind, isvalid] = splitnode(i, rf{t}.ind_samples{n}, Data{t}, params, stage);%{if ~isvalidrf{t}.feat(n, :)   = [0 0 0 0];rf{t}.thresh(n)    = 0;rf{t}.issplit(n)   = 1;rf{t}.cnodes(n, :) = [0 0];rf{t}.isleafnode(n)   = 1;continue;end%}% set the threshold and featture for current noderf{t}.feat(n, :)   = feat;rf{t}.thresh(n)    = thresh;rf{t}.issplit(n)   = 1;rf{t}.cnodes(n, :) = [num_nodes+1 num_nodes+2]; %当前节点的左右子节点序号rf{t}.isleafnode(n)   = 0;% add left and right child nodes into the random treerf{t}.ind_samples{num_nodes+1} = lcind;rf{t}.issplit(num_nodes+1)     = 0;rf{t}.pnode(num_nodes+1)       = n;rf{t}.depth(num_nodes+1)       = rf{t}.depth(n) + 1;rf{t}.cnodes(num_nodes+1, :)   = [0 0];rf{t}.isleafnode(num_nodes+1)     = 1;rf{t}.ind_samples{num_nodes+2} = rcind;rf{t}.issplit(num_nodes+2)     = 0;rf{t}.pnode(num_nodes+2)       = n;rf{t}.depth(num_nodes+2)       = rf{t}.depth(n) + 1;rf{t}.cnodes(num_nodes+2, :)   = [0 0];rf{t}.isleafnode(num_nodes+2)  = 1;num_split = num_split + 1; %分割节点的次数,实际上一层分割节点的个数num_leafnodes = num_leafnodes + 1;num_nodes     = num_nodes + 2;endendendif num_split == 0stop = 1;elserf{t}.num_leafnodes = num_leafnodes;rf{t}.num_nodes     = num_nodes;           rf{t}.id_leafnodes = find(rf{t}.isleafnode == 1); end            endend% disp(strcat(num2str(i), 'th landmark is over'));rfs(i, :) = rf;
end

3.分裂节点全程
流程图:

代码：

function [thresh, feat, lcind, rcind, isvalid] = splitnode(lmarkID, ind_samples, Tr_Data, params, stage)if isempty(ind_samples)thresh = 0;feat = [0 0 0 0];rcind = [];lcind = [];isvalid = 1;return;
end% generate params.max_rand cndidate feature
% anglepairs = samplerandfeat(params.max_numfeat);
% radiuspairs = [rand([params.max_numfeat, 1]) rand([params.max_numfeat, 1])];
[radiuspairs, anglepairs] = getproposals(params.max_numfeats(stage), params.radius, params.angles);angles_cos = cos(anglepairs);
angles_sin = sin(anglepairs);% extract pixel difference features from pairspdfeats = zeros(params.max_numfeats(stage), length(ind_samples)); %所有的样本均要提取相应阶段的像素差特征，即比如说1000*541shapes_residual = zeros(length(ind_samples), 2);for i = 1:length(ind_samples)s = floor((ind_samples(i)-1)/(params.augnumber)) + 1; %共用样本的序号k = mod(ind_samples(i)-1, (params.augnumber)) + 1; %不能共用盒子，而是对于同一张图片的不同shape使用各自的盒子，使用余运算，显然小于params.augnumber，又加1，所以答案从1：params.augnumber% calculate the relative location under the coordinate of meanshape %x1=angles_cos(:, 1)).*radiuspairs(:, 1)pixel_a_x_imgcoord = (angles_cos(:, 1)).*radiuspairs(:, 1)*params.max_raio_radius(stage)*Tr_Data{s}.intermediate_bboxes{stage}(k, 3);pixel_a_y_imgcoord = (angles_sin(:, 1)).*radiuspairs(:, 1)*params.max_raio_radius(stage)*Tr_Data{s}.intermediate_bboxes{stage}(k, 4);pixel_b_x_imgcoord = (angles_cos(:, 2)).*radiuspairs(:, 2)*params.max_raio_radius(stage)*Tr_Data{s}.intermediate_bboxes{stage}(k, 3);pixel_b_y_imgcoord = (angles_sin(:, 2)).*radiuspairs(:, 2)*params.max_raio_radius(stage)*Tr_Data{s}.intermediate_bboxes{stage}(k, 4);% no transformation%{pixel_a_x_lmcoord = pixel_a_x_imgcoord;pixel_a_y_lmcoord = pixel_a_y_imgcoord;pixel_b_x_lmcoord = pixel_b_x_imgcoord;pixel_b_y_lmcoord = pixel_b_y_imgcoord;%}% transform the pixels from image coordinate (meanshape) to coordinate of current shape%以下计算出的都是中心化的坐标[pixel_a_x_lmcoord, pixel_a_y_lmcoord] = transformPointsForward(Tr_Data{s}.meanshape2tf{k}, pixel_a_x_imgcoord', pixel_a_y_imgcoord');    pixel_a_x_lmcoord = pixel_a_x_lmcoord';pixel_a_y_lmcoord = pixel_a_y_lmcoord';[pixel_b_x_lmcoord, pixel_b_y_lmcoord] = transformPointsForward(Tr_Data{s}.meanshape2tf{k}, pixel_b_x_imgcoord', pixel_b_y_imgcoord');pixel_b_x_lmcoord = pixel_b_x_lmcoord';pixel_b_y_lmcoord = pixel_b_y_lmcoord';     %转化为绝对坐标pixel_a_x = int32(bsxfun(@plus, pixel_a_x_lmcoord, Tr_Data{s}.intermediate_shapes{stage}(lmarkID, 1, k)));pixel_a_y = int32(bsxfun(@plus, pixel_a_y_lmcoord, Tr_Data{s}.intermediate_shapes{stage}(lmarkID, 2, k)));pixel_b_x = int32(bsxfun(@plus, pixel_b_x_lmcoord, Tr_Data{s}.intermediate_shapes{stage}(lmarkID, 1, k)));pixel_b_y = int32(bsxfun(@plus, pixel_b_y_lmcoord, Tr_Data{s}.intermediate_shapes{stage}(lmarkID, 2, k)));width = (Tr_Data{s}.width);height = (Tr_Data{s}.height);pixel_a_x = max(1, min(pixel_a_x, width)); %意思是 pixel_a_x应该介于1和width之间pixel_a_y = max(1, min(pixel_a_y, height));pixel_b_x = max(1, min(pixel_b_x, width));pixel_b_y = max(1, min(pixel_b_y, height));%取像素两种方法，一是img_gray(i,j)；二是img_gray(k)，k是按列数第k个元素pdfeats(:, i) = double(Tr_Data{s}.img_gray(pixel_a_y + (pixel_a_x-1)*height)) - double(Tr_Data{s}.img_gray(pixel_b_y + (pixel_b_x-1)*height));%./ double(Tr_Data{s}.img_gray(pixel_a_y + (pixel_a_x-1)*height)) + double(Tr_Data{s}.img_gray(pixel_b_y + (pixel_b_x-1)*height));% drawshapes(Tr_Data{s}.img_gray, [pixel_a_x pixel_a_y pixel_b_x pixel_b_y]);% hold off;shapes_residual(i, :) = Tr_Data{s}.shapes_residual(lmarkID, :, k);
endE_x_2 = mean(shapes_residual(:, 1).^2);
E_x = mean(shapes_residual(:, 1));E_y_2 = mean(shapes_residual(:, 2).^2);
E_y = mean(shapes_residual(:, 2));
% 整体方差，其中使用了方差的经典公式Dx=Ex^2-(Ex)^2
var_overall = length(ind_samples)*((E_x_2 - E_x^2) + (E_y_2 - E_y^2));% var_overall = length(ind_samples)*(var(shapes_residual(:, 1)) + var(shapes_residual(:, 2)));% max_step = min(length(ind_samples), params.max_numthreshs);
% step = floor(length(ind_samples)/max_step);
max_step = 1;var_reductions = zeros(params.max_numfeats(stage), max_step);
thresholds = zeros(params.max_numfeats(stage), max_step);[pdfeats_sorted] = sort(pdfeats, 2); %将数据打乱顺序，防止过拟合% shapes_residual = shapes_residual(ind, :);for i = 1:params.max_numfeats(stage) %暴力选举法，选出最合适的feature% for t = 1:max_stept = 1;ind = ceil(length(ind_samples)*(0.5 + 0.9*(rand(1) - 0.5)));threshold = pdfeats_sorted(i, ind);  % pdfeats_sorted(i, t*step); % thresholds(i, t) = threshold;ind_lc = (pdfeats(i, :) < threshold); %逻辑数组ind_rc = (pdfeats(i, :) >= threshold);% figure, hold on, plot(shapes_residual(ind_lc, 1), shapes_residual(ind_lc, 2), 'r.')% plot(shapes_residual(ind_rc, 1), shapes_residual(ind_rc, 2), 'g.')% close;% compute E_x_2_lc = mean(shapes_residual(ind_lc, 1).^2); %选出逻辑数组中为1的那些残差E_x_lc = mean(shapes_residual(ind_lc, 1));E_y_2_lc = mean(shapes_residual(ind_lc, 2).^2);E_y_lc = mean(shapes_residual(ind_lc, 2));var_lc = (E_x_2_lc + E_y_2_lc)- (E_x_lc^2 + E_y_lc^2);E_x_2_rc = (E_x_2*length(ind_samples) - E_x_2_lc*sum(ind_lc))/sum(ind_rc);E_x_rc = (E_x*length(ind_samples) - E_x_lc*sum(ind_lc))/sum(ind_rc);E_y_2_rc = (E_y_2*length(ind_samples) - E_y_2_lc*sum(ind_lc))/sum(ind_rc);E_y_rc = (E_y*length(ind_samples) - E_y_lc*sum(ind_lc))/sum(ind_rc);var_rc = (E_x_2_rc + E_y_2_rc)- (E_x_rc^2 + E_y_rc^2);var_reduce = var_overall - sum(ind_lc)*var_lc - sum(ind_rc)*var_rc;% var_reduce = var_overall - sum(ind_lc)*(var(shapes_residual(ind_lc, 1)) + var(shapes_residual(ind_lc, 2))) - sum(ind_rc)*(var(shapes_residual(ind_rc, 1)) + var(shapes_residual(ind_rc, 2)));var_reductions(i, t) = var_reduce;% end% plot(var_reductions(i, :));
end[~, ind_colmax] = max(var_reductions);%寻找最大差的序号
ind_max = 1;%{if var_max <= 0isvalid = 0;
elseisvalid = 1;
end
%}
isvalid = 1;thresh =  thresholds(ind_colmax(ind_max), ind_max); %当前阈值feat   = [anglepairs(ind_colmax(ind_max), :) radiuspairs(ind_colmax(ind_max), :)];lcind = ind_samples(find(pdfeats(ind_colmax(ind_max), :) < thresh));
rcind = ind_samples(find(pdfeats(ind_colmax(ind_max), :) >= thresh));end

问题：训练时默认一旦可以分割节点，则必然分割成两部分。那么会不会出现选取一个阈值将剩余的样本都归于一类呢？
说明:

如图所示外面有一个current 坐标系，里面有mean_shape的中心化归一化的坐标。最里面是以一个特征点为中心取的极坐标。这份代码取rr,θ\theta来标注在特征点附近取到的任意两个像素点的坐标.可以说有三个坐标系（按前面顺序，分别称为坐标系一、二、三）。里面两个坐标系的尺寸一样，但是坐标原点不一样。

假定在坐标系三下，取到一像素点坐标为（x,y），而特征点在坐标系二的坐标为（x0,y0x_0,y_0）,则像素点在坐标系二的坐标为(x˜,y˜\widetilde{x},\widetilde{y}),则有:

(x˜,y˜)=(x,y)+(x0,y0)

(\widetilde{x},\widetilde{y})=(x,y)+(x_0,y_0).
又由前面一篇文章《face alignment by 3000 fps系列学习总结（二）》中间进行的相似性变换，我们知道，将当前坐标由mean_shape的归一化中心化坐标转换为current_shape的中心化坐标，需要使用meanshape2tf变换。
即：

(x˜,y˜)/cR

(\widetilde{x},\widetilde{y})/cR
进一步的，取中心化后得

(x˜,y˜)/cR+mean(immediateshape)=(x,y)+(x0,y0)cR+mean(immediateshape)=(x,y)cR+(x0,y0)cR+mean(immediateshape)=(x,y)cR+immediate_shape_at(x0,y0)

\begin{align*} (\widetilde{x},\widetilde{y})/cR +mean(immediate_shape) &=\frac{(x,y)+(x_0,y_0)}{cR}+mean(immediate_shape)\\ &=\frac{(x,y)}{cR}+\frac{(x_0,y_0)}{cR}+mean(immediate_shape)\\ &=\frac{(x,y)}{cR}+immediate\_shape\_at(x_0,y_0) \end{align*}
我们又知道：

cR=c˜R˜/immediate_bbox

cR=\widetilde{c}\widetilde{R}/immediate\_bbox
所以上式= (x,y)∗immediate_bbox/{c˜R˜}+immediate_shape_at(x0,y0)(x,y)*immediate\_bbox/\{\widetilde{c}\widetilde{R}\}+immediate\_shape\_at(x_0,y_0)
最后一句就解析清了代码的步骤：

  % calculate the relative location under the coordinate of meanshape %x1=angles_cos(:, 1)).*radiuspairs(:, 1)pixel_a_x_imgcoord = (angles_cos(:, 1)).*radiuspairs(:, 1)*params.max_raio_radius(stage)*Tr_Data{s}.intermediate_bboxes{stage}(k, 3);pixel_a_y_imgcoord = (angles_sin(:, 1)).*radiuspairs(:, 1)*params.max_raio_radius(stage)*Tr_Data{s}.intermediate_bboxes{stage}(k, 4);pixel_b_x_imgcoord = (angles_cos(:, 2)).*radiuspairs(:, 2)*params.max_raio_radius(stage)*Tr_Data{s}.intermediate_bboxes{stage}(k, 3);pixel_b_y_imgcoord = (angles_sin(:, 2)).*radiuspairs(:, 2)*params.max_raio_radius(stage)*Tr_Data{s}.intermediate_bboxes{stage}(k, 4);% no transformation%{pixel_a_x_lmcoord = pixel_a_x_imgcoord;pixel_a_y_lmcoord = pixel_a_y_imgcoord;pixel_b_x_lmcoord = pixel_b_x_imgcoord;pixel_b_y_lmcoord = pixel_b_y_imgcoord;%}% transform the pixels from image coordinate (meanshape) to coordinate of current shape%以下计算出的都是中心化的坐标[pixel_a_x_lmcoord, pixel_a_y_lmcoord] = transformPointsForward(Tr_Data{s}.meanshape2tf{k}, pixel_a_x_imgcoord', pixel_a_y_imgcoord');    pixel_a_x_lmcoord = pixel_a_x_lmcoord';pixel_a_y_lmcoord = pixel_a_y_lmcoord';[pixel_b_x_lmcoord, pixel_b_y_lmcoord] = transformPointsForward(Tr_Data{s}.meanshape2tf{k}, pixel_b_x_imgcoord', pixel_b_y_imgcoord');pixel_b_x_lmcoord = pixel_b_x_lmcoord';pixel_b_y_lmcoord = pixel_b_y_lmcoord';     %转化为绝对坐标pixel_a_x = int32(bsxfun(@plus, pixel_a_x_lmcoord, Tr_Data{s}.intermediate_shapes{stage}(lmarkID, 1, k)));pixel_a_y = int32(bsxfun(@plus, pixel_a_y_lmcoord, Tr_Data{s}.intermediate_shapes{stage}(lmarkID, 2, k)));pixel_b_x = int32(bsxfun(@plus, pixel_b_x_lmcoord, Tr_Data{s}.intermediate_shapes{stage}(lmarkID, 1, k)));pixel_b_y = int32(bsxfun(@plus, pixel_b_y_lmcoord, Tr_Data{s}.intermediate_shapes{stage}(lmarkID, 2, k)));width = (Tr_Data{s}.width);height = (Tr_Data{s}.height);pixel_a_x = max(1, min(pixel_a_x, width)); %意思是 pixel_a_x应该介于1和width之间pixel_a_y = max(1, min(pixel_a_y, height));pixel_b_x = max(1, min(pixel_b_x, width));pixel_b_y = max(1, min(pixel_b_y, height));%取像素两种方法，一是img_gray(i,j)；二是img_gray(k)，k是按列数第k个元素pdfeats(:, i) = double(Tr_Data{s}.img_gray(pixel_a_y + (pixel_a_x-1)*height)) - double(Tr_Data{s}.img_gray(pixel_b_y + (pixel_b_x-1)*height));

如此我们训练全程就搞懂了。