准备初始数据

mean_shape

mean_shape就是训练图片所有ground_truth points的平均值.那么具体怎么做呢？是不是直接将特征点相加求平均值呢？
显然这样做是仓促和不准确的。因为图片之间人脸是各式各样的，收到光照、姿势等各方面的影响。因此我们求取平均值，应该在一个相对统一的框架下求取。如下先给出matlab代码:

function mean_shape = calc_meanshape(shapepathlistfile)fid = fopen(shapepathlistfile);
shapepathlist = textscan(fid, '%s', 'delimiter', '\n');if isempty(shapepathlist)error('no shape file found');mean_shape = [];return;
endshape_header = loadshape(shapepathlist{1}{1});if isempty(shape_header)error('invalid shape file');mean_shape = [];return;
endmean_shape = zeros(size(shape_header));num_shapes = 0;
for i = 1:length(shapepathlist{1})shape_i = double(loadshape(shapepathlist{1}{i}));if isempty(shape_i)continue;endshape_min = min(shape_i, [], 1);shape_max = max(shape_i, [], 1);% translate to origin pointshape_i = bsxfun(@minus, shape_i, shape_min);% resize shapeshape_i = bsxfun(@rdivide, shape_i, shape_max - shape_min);mean_shape = mean_shape + shape_i;num_shapes = num_shapes + 1;
endmean_shape = mean_shape ./ num_shapes;img = 255 * ones(500, 500, 3);drawshapes(img, 50 + 400 * mean_shape);endfunction shape = loadshape(path)
% function: load shape from pts file
file = fopen(path);
if file == -1shape = [];fclose(file);return;
end
shape = textscan(file, '%d16 %d16', 'HeaderLines', 3, 'CollectOutput', 2);
fclose(file);
shape = shape{1};
end

解析:

公式表示:

{s h a p e g t - [R e g i o n (1), R e g i o n (2)]} / [R e g i o n (3), R e g i o n (4)))]] \Rightarrow [0, 1] \times [0, 1]

$\{shape_{gt}-[Region(1),Region(2)]\}/[Region(3),Region(4)))]] \Rightarrow [0,1]\times[0,1]$

准备 $\Delta S^t$

我们知道3000FPS的核心思想是:

Δ S t = W t Φ t (I, S t - 1)

$\Delta S^t=W^t\Phi^t(I,S^{t-1})$
其中

ΔSt=Sgt−St $\Delta S^t=S^{gt}-S^{t}$ 为第t个阶段的残差；而

Φt(I,St−1) $\Phi^t(I,S^{t-1})$ 则为特征提取函数；W为线性回归矩阵。由《人脸配准坐标变换解析》我们可以看到所谓的

ΔSt $\Delta S^t$ 需进行相似性变换，而

Φt(I,St−1) $\Phi^t(I,S^{t-1})$ 则不需要.
相似性变换的主要过程是:
先将

St $S^t$ ，

S0 $S^0$ 中心化变换，再求解如下变换矩阵:

S 0 = c R S t

$S^0=cRS^t$ ,求解完cR后，对

ΔSt $\Delta S^t$ 施加同样的变换，即

S t ˜ = c R Δ S t

$\widetilde{S^t}=cR\Delta S^t$ .我们将使用变化后的

St˜ $\widetilde{S^t}$ 去求解线性回归矩阵W.
先贴代码: train_model.m 第103行起

Param.meanshape        = S0(Param.ind_usedpts, :); %选取特定的landmarkdbsize = length(Data);% load('Ts_bbox.mat');augnumber = Param.augnumber; %为每张人脸选取的init_shape的个数for i = 1:dbsize        % initializ the shape of current face image by randomly selecting multiple shapes from other face images       % indice = ceil(dbsize*rand(1, augnumber));  indice_rotate = ceil(dbsize*rand(1, augnumber));  indice_shift  = ceil(dbsize*rand(1, augnumber));  scales        = 1 + 0.2*(rand([1 augnumber]) - 0.5);Data{i}.intermediate_shapes = cell(1, Param.max_numstage); %中间shapeData{i}.intermediate_bboxes = cell(1, Param.max_numstage);Data{i}.intermediate_shapes{1} = zeros([size(Param.meanshape), augnumber]); %68*2*augnumber(augnumber为第i图片设置的初始shape的个数)Data{i}.intermediate_bboxes{1} = zeros([augnumber, size(Data{i}.bbox_gt, 2)]); %augnumber*4Data{i}.shapes_residual = zeros([size(Param.meanshape), augnumber]); %shapes_residual为shape 残差 维数:68*2*augnumberData{i}.tf2meanshape = cell(augnumber, 1);Data{i}.meanshape2tf = cell(augnumber, 1);% if Data{i}.isdet == 1%    Data{i}.bbox_facedet = Data{i}.bbox_facedet*ts_bbox;% end     % 如下一段的意思是如果augnumber=1，表明每个图片的Init_shape只有一个，因此这要设置成mean_shape即可,这时你会发现Data{i}.tf2meanshape{1}其实就是% 单位矩阵，因为他是从mean_shape转化到mean_shape。后面就不一样了.%；对于augnumber>1的其他init_shape将采用平移、旋转、% 缩放等方式产生更多的shape，也可以从其他图片的shape中挑选shapefor sr = 1:params.augnumberif sr == 1% estimate the similarity transformation from initial shape to mean shape% Data{i}.intermediate_shapes{1}(:,:, sr) = resetshape(Data{i}.bbox_gt, Param.meanshape);% Data{i}.intermediate_bboxes{1}(sr, :) = Data{i}.bbox_gt;Data{i}.intermediate_shapes{1}(:,:, sr) = resetshape(Data{i}.bbox_facedet, Param.meanshape);Data{i}.intermediate_bboxes{1}(sr, :) = Data{i}.bbox_facedet;%将mean shape reproject face detection bbox上meanshape_resize = resetshape(Data{i}.intermediate_bboxes{1}(sr, :), Param.meanshape); %meanshape_resize与 Data{i}.intermediate_shapes{1}(:,:, sr) 是相同的%计算当前的shape与mean shape之间的相似性变换         Data{i}.tf2meanshape{1} = fitgeotrans(bsxfun(@minus, Data{i}.intermediate_shapes{1}(1:end,:, 1), mean(Data{i}.intermediate_shapes{1}(1:end,:, 1))), ...(bsxfun(@minus, meanshape_resize(1:end, :), mean(meanshape_resize(1:end, :)))), 'NonreflectiveSimilarity');Data{i}.meanshape2tf{1} = fitgeotrans((bsxfun(@minus, meanshape_resize(1:end, :), mean(meanshape_resize(1:end, :)))), ...bsxfun(@minus, Data{i}.intermediate_shapes{1}(1:end,:, 1), mean(Data{i}.intermediate_shapes{1}(1:end,:, 1))), 'NonreflectiveSimilarity');% calculate the residual shape from initial shape to groundtruth shape under normalization scaleshape_residual = bsxfun(@rdivide, Data{i}.shape_gt - Data{i}.intermediate_shapes{1}(:,:, 1), [Data{i}.intermediate_bboxes{1}(1, 3) Data{i}.intermediate_bboxes{1}(1, 4)]);% transform the shape residual in the image coordinate to the mean shape coordinate[u, v] = transformPointsForward(Data{i}.tf2meanshape{1}, shape_residual(:, 1)', shape_residual(:, 2)'); Data{i}.shapes_residual(:, 1, 1) = u';Data{i}.shapes_residual(:, 2, 1) = v'; else% randomly rotate the shape            % shape = resetshape(Data{i}.bbox_gt, Param.meanshape);       % Data{indice_rotate(sr)}.shape_gtshape = resetshape(Data{i}.bbox_facedet, Param.meanshape);       % Data{indice_rotate(sr)}.shape_gt%根据随机选取的scale，rotation，translate计算新的初始shape然后投影到bbox上if params.augnumber_scale ~= 0shape = scaleshape(shape, scales(sr));endif params.augnumber_rotate ~= 0shape = rotateshape(shape);endif params.augnumber_shift ~= 0shape = translateshape(shape, Data{indice_shift(sr)}.shape_gt);endData{i}.intermediate_shapes{1}(:, :, sr) = shape;Data{i}.intermediate_bboxes{1}(sr, :) = getbbox(shape);meanshape_resize = resetshape(Data{i}.intermediate_bboxes{1}(sr, :), Param.meanshape); %将Data{i}.tf2meanshape{sr} = fitgeotrans(bsxfun(@minus, Data{i}.intermediate_shapes{1}(1:end,:, sr), mean(Data{i}.intermediate_shapes{1}(1:end,:, sr))), ...bsxfun(@minus, meanshape_resize(1:end, :), mean(meanshape_resize(1:end, :))), 'NonreflectiveSimilarity');Data{i}.meanshape2tf{sr} = fitgeotrans(bsxfun(@minus, meanshape_resize(1:end, :), mean(meanshape_resize(1:end, :))), ...bsxfun(@minus, Data{i}.intermediate_shapes{1}(1:end,:, sr), mean(Data{i}.intermediate_shapes{1}(1:end,:, sr))), 'NonreflectiveSimilarity');shape_residual = bsxfun(@rdivide, Data{i}.shape_gt - Data{i}.intermediate_shapes{1}(:,:, sr), [Data{i}.intermediate_bboxes{1}(sr, 3) Data{i}.intermediate_bboxes{1}(sr, 4)]);[u, v] = transformPointsForward(Data{i}.tf2meanshape{1}, shape_residual(:, 1)', shape_residual(:, 2)');Data{i}.shapes_residual(:, 1, sr) = u';Data{i}.shapes_residual(:, 2, sr) = v';% Data{i}.shapes_residual(:, :, sr) = tformfwd(Data{i}.tf2meanshape{sr}, shape_residual(:, 1), shape_residual(:, 2));endend
end

这段代码的理解需要结合上面给出的那篇文章《人脸配准坐标变换解析》。

按照《人脸配准坐标变换解析》文章所述，

S 0 ¯ ¯ ¯ ¯ S 1 ¯ ¯ ¯ ¯ = S 0 - m e a n (S 0) = S 1 - m e a n (S 1)} \Rightarrow S 0 ¯ ¯ ¯ ¯ = c 1 R 1 S 1 ¯ ¯ ¯ ¯

$\left.\begin{matrix}\overline{S_0}&=S_0-mean(S_0)\\ \overline{S_1}&=S_1-mean(S_1) \end{matrix}\right\}\Rightarrow \overline{S_0}=c_1R_1\overline{S_1}$
因此根据

Δ S = S g - S 1

$\Delta S=S_g-S_1$ 可推出

Δ S ˜ = c 1 R 1 Δ S

$\widetilde{\Delta S}=c_1R_1\Delta S$
但是现在问题比较特殊，需要多操作一下:
由：

 %将mean shape reproject face detection bbox上meanshape_resize = resetshape(Data{i}.intermediate_bboxes{1}(sr, :), Param.meanshape);

查看resetshape的定义知meanshape被映射到intermediate_bboxes中，使得 $S_0$ 和 $S_1$ 处于同样的尺度下和大致相似的位置上。用数学语言表达为:

S 0_r e s i z e = S 0 * R a t i o + [R e g i o n (1), R e g i o n (2)]

$S_0\_resize=S_0*Ratio+[Region(1),Region(2)]$ 这里Ratio实际上是intermediate_bboxes的大小。
于是同样按照上面的方法计算：

S 0 ˜ = S 0_R e s i z e - m e a n (S 0_R e s i z e) = S 0 * R a t i o - m e a n (S 0) * R a t i o = (S 0 - m e a n (S 0)) * R a t i o = S 0 ¯ ¯ ¯ ¯ * R a t i o

$\widetilde{S_0}=S_0\_Resize-mean(S_0\_Resize)=S_0*Ratio-mean(S_0)*Ratio=(S_0-mean(S_0))*Ratio= \overline{S_0}*Ratio$
经过计算得

S0˜=Ratio∗S0¯¯¯¯=c1˜R1˜S1¯¯¯¯ $\widetilde{S_0}=Ratio*\overline{S_0}=\widetilde{c_1}\widetilde{R_1} \overline{S_1}$ .（

★ $\bigstar$ ）
这也就是上面的代码：

 Data{i}.tf2meanshape{1} = fitgeotrans(bsxfun(@minus, Data{i}.intermediate_shapes{1}(1:end,:, 1), mean(Data{i}.intermediate_shapes{1}(1:end,:, 1))), ...(bsxfun(@minus, meanshape_resize(1:end, :), mean(meanshape_resize(1:end, :)))), 'NonreflectiveSimilarity');

Data{i}.tf2meanshape{1}即为这里算出的 $\widetilde{c_1}\widetilde{R_1}$ .
但我们想要的是 $\overline{S_0}=c_1R_1\overline{S_1}$ ,不用着急，( $\bigstar$ )为我们指明了方向。
$c_1R_1=\widetilde{c_1}\widetilde{R_1}/Ratio=\widetilde{c_1}\widetilde{R_1}/{intermediate\_{bboxes}}$ .因此:

Δ S ˜ = c 1 ˜ R 1 ˜ / i n t e r m e d i a t e_b b o x e s * Δ S

$\widetilde{\Delta S}=\widetilde{c_1}\widetilde{R_1}/{intermediate\_{bboxes}}*\Delta S$
也就是代码中提的:

 %计算当前的shape与mean shape之间的相似性变换         
Data{i}.tf2meanshape{1} = fitgeotrans(bsxfun(@minus, Data{i}.intermediate_shapes{1}(1:end,:, 1), mean(Data{i}.intermediate_shapes{1}(1:end,:, 1))),(bsxfun(@minus, meanshape_resize(1:end, :), mean(meanshape_resize(1:end, :)))), 'NonreflectiveSimilarity');Data{i}.meanshape2tf{1} = fitgeotrans((bsxfun(@minus, meanshape_resize(1:end, :), mean(meanshape_resize(1:end, :)))),bsxfun(@minus, Data{i}.intermediate_shapes{1}(1:end,:, 1), mean(Data{i}.intermediate_shapes{1}(1:end,:, 1))), 'NonreflectiveSimilarity');% calculate the residual shape from initial shape to groundtruth shape under normalization scale
shape_residual = bsxfun(@rdivide, Data{i}.shape_gt - Data{i}.intermediate_shapes{1}(:,:, 1), [Data{i}.intermediate_bboxes{1}(1, 3) Data{i}.intermediate_bboxes{1}(1, 4)]);% transform the shape residual in the image coordinate to the mean shape coordinate
[u, v] = transformPointsForward(Data{i}.tf2meanshape{1}, shape_residual(:, 1)', shape_residual(:, 2)'); Data{i}.shapes_residual(:, 1, 1) = u';Data{i}.shapes_residual(:, 2, 1) = v';