face alignment by 3000 fps系列学习总结（二）

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;
end

shape_header = loadshape(shapepathlist{1}{1});

if isempty(shape_header)
    error('invalid shape file');
    mean_shape = [];
    return;
end

mean_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;
    end
    shape_min = min(shape_i, [], 1);
    shape_max = max(shape_i, [], 1);

    % translate to origin point
    shape_i = bsxfun(@minus, shape_i, shape_min);

    % resize shape
    shape_i = bsxfun(@rdivide, shape_i, shape_max - shape_min);

    mean_shape = mean_shape + shape_i;
    num_shapes = num_shapes + 1;
end

mean_shape = mean_shape ./ num_shapes;


img = 255 * ones(500, 500, 3);

drawshapes(img, 50 + 400 * mean_shape);

end

function shape = loadshape(path)
% function: load shape from pts file
file = fopen(path);
if file == -1
    shape = [];
    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:

{shapegt−[Region(1),Region(2)]}/[Region(3),Region(4)))]]⇒[0,1]×[0,1]

准备 ΔSt

我们知道3000FPS的核心思想是:

ΔSt=WtΦt(I,St−1)

其中 ΔSt=Sgt−St 为第t个阶段的残差；而 Φt(I,St−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 需进行相似性变换，而 Φt(I,St−1) 则不需要.
相似性变换的主要过程是:
先将 St ， S0 中心化变换，再求解如下变换矩阵:

S0=cRSt

,求解完cR后，对 ΔSt 施加同样的变换，即

St˜=cRΔSt

.我们将使用变化后的 St˜ 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landmark

dbsize = 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shape
    Data{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*4

    Data{i}.shapes_residual = zeros([size(Param.meanshape), augnumber]); %shapes_residual为shape 残差 维数:68*2*augnumber
    Data{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shape
    for sr = 1:params.augnumber
        if 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 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'; 
        else
            % randomly rotate the shape 
            % shape = resetshape(Data{i}.bbox_gt, Param.meanshape); % Data{indice_rotate(sr)}.shape_gt
            shape = 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 ~= 0
                shape = scaleshape(shape, scales(sr));
            end

            if params.augnumber_rotate ~= 0
                shape = rotateshape(shape);
            end

            if params.augnumber_shift ~= 0
                shape = translateshape(shape, Data{indice_shift(sr)}.shape_gt);
            end

            Data{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));
        end
    end
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，

S0¯¯¯¯S1¯¯¯¯=S0−mean(S0)=S1−mean(S1)}⇒S0¯¯¯¯=c1R1S1¯¯¯¯

因此根据

ΔS=Sg−S1

可推出

ΔS˜=c1R1ΔS

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 S0 和 S1 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:

S0_resize=S0∗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：

S0˜=S0_Resize−mean(S0_Resize)=S0∗Ratio−mean(S0)∗Ratio=(S0−mean(S0))∗Ratio=S0¯¯¯¯∗Ratio

经过计算得 S0˜=Ratio∗S0¯¯¯¯=c1˜R1˜S1¯¯¯¯ .（ ★ ）
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 c1˜R1˜ .
但我们想要的是 S0¯¯¯¯=c1R1S1¯¯¯¯ ,不用着急，( ★ )为我们指明了方向。
c1R1=c1˜R1˜/Ratio=c1˜R1˜/intermediate_bboxes .因此:

ΔS˜=c1˜R1˜/intermediate_bboxes∗ΔS

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';