1. batch norm
输入batch norm层的数据为[N, C, H, W], 该层计算得到均值为C个，方差为C个，输出数据为[N, C, H, W].
<1> 形象点说，均值的计算过程为：
(1)
即对batch中相同索引的通道数取平均值，所以最终计算得到的均值为C个，方差的计算过程与此相同。
<2> batch norm层的作用：
a. 均值：
(2)
b. 方差：
(3)
c. 归一化：
(4)
2. caffe中batch_norm_layer.cpp中的LayerSetUp函数：
1 template <typename Dtype>
2 void BatchNormLayer<Dtype>::LayerSetUp(const vector<Blob<Dtype>*>& bottom,
3 const vector<Blob<Dtype>*>& top) {
4 BatchNormParameter param = this->layer_param_.batch_norm_param(); //读取deploy中moving_average_fraction参数值
5 moving_average_fraction_ = param.moving_average_fraction(); //改变量在batch_norm_layer.hpp中的定义为bool use_global_stats_
6 use_global_stats_ = this->phase_ == TEST; //channel在batch_norm_layer.hpp中的定义为int channels_
7 if (param.has_use_global_stats())
8 use_global_stats_ = param.use_global_stats();
9 if (bottom[0]->num_axes() == 1)
10 channels_ = 1;
11 else
12 channels_ = bottom[0]->shape(1);
13 eps_ = param.eps();
14 if (this->blobs_.size() > 0) {
15 LOG(INFO) << "Skipping parameter initialization";
16 } else { //blobs的个数为三个，其中: //blobs_[0]的尺寸为channels_,保存输入batch中各通道的均值； //blobs_[1]的尺寸为channels_,保存输入batch中各通道的方差； //blobs_[2]的尺寸为1， 保存moving_average_fraction参数； //对上面三个blobs_初始化为0.
17 this->blobs_.resize(3);
18 vector<int> sz;
19 sz.push_back(channels_);
20 this->blobs_[0].reset(new Blob<Dtype>(sz));
21 this->blobs_[1].reset(new Blob<Dtype>(sz));
22 sz[0] = 1;
23 this->blobs_[2].reset(new Blob<Dtype>(sz));
24 for (int i = 0; i < 3; ++i) {
25 caffe_set(this->blobs_[i]->count(), Dtype(0),
26 this->blobs_[i]->mutable_cpu_data());
27 }
28 }
29 // Mask statistics from optimization by setting local learning rates
30 // for mean, variance, and the bias correction to zero.
31 for (int i = 0; i < this->blobs_.size(); ++i) {
32 if (this->layer_param_.param_size() == i) {
33 ParamSpec* fixed_param_spec = this->layer_param_.add_param();
34 fixed_param_spec->set_lr_mult(0.f);
35 } else {
36 CHECK_EQ(this->layer_param_.param(i).lr_mult(), 0.f)
37 << "Cannot configure batch normalization statistics as layer "
38 << "parameters.";
39 }
40 }
41 }
3. caffe中batch_norm_layer.cpp中的Reshape函数：
1 void BatchNormLayer<Dtype>::Reshape(const vector<Blob<Dtype>*>& bottom,
2 const vector<Blob<Dtype>*>& top) {
3 if (bottom[0]->num_axes() >= 1)
4 CHECK_EQ(bottom[0]->shape(1), channels_);
5 top[0]->ReshapeLike(*bottom[0]); //batch_norm_layer.hpp对如下变量进行了定义： //Blob<Dtype> mean_, variance_, temp_, x_norm_; //blob<Dtype> batch_sum_multiplier_; //blob<Dtype> sum_by_chans_; //blob<Dtype> spatial_sum_multiplier_;
6 vector<int> sz;
7 sz.push_back(channels_); //mean blob和variance blob的尺寸为channel
8 mean_.Reshape(sz);
9 variance_.Reshape(sz); //temp_ blob和x_norm_ blob的尺寸、数据和输入blob相同
10 temp_.ReshapeLike(*bottom[0]);
11 x_norm_.ReshapeLike(*bottom[0]); //sz[0]的值为N，batch_sum_multiplier_ blob的尺寸为N
12 sz[0] = bottom[0]->shape(0);
13 batch_sum_multiplier_.Reshape(sz); //spatial_dim = N*C*H*W / C*N = H*W
14 int spatial_dim = bottom[0]->count()/(channels_*bottom[0]->shape(0));
15 if (spatial_sum_multiplier_.num_axes() == 0 ||
16 spatial_sum_multiplier_.shape(0) != spatial_dim) {
17 sz[0] = spatial_dim; //spatial_sum_multiplier_的尺寸为H*W, 并且初始化为1
18 spatial_sum_multiplier_.Reshape(sz);
19 Dtype* multiplier_data = spatial_sum_multiplier_.mutable_cpu_data();
20 caffe_set(spatial_sum_multiplier_.count(), Dtype(1), multiplier_data);
21 } //numbychans = C*N
22 int numbychans = channels_*bottom[0]->shape(0);
23 if (num_by_chans_.num_axes() == 0 ||
24 num_by_chans_.shape(0) != numbychans) {
25 sz[0] = numbychans; //num_by_chans_的尺寸为C*N,并且初始化为1
26 num_by_chans_.Reshape(sz);
27 caffe_set(batch_sum_multiplier_.count(), Dtype(1),
28 batch_sum_multiplier_.mutable_cpu_data());
29 }
30 }
形象点说上面各blob变量的尺寸：
mean_和variance_：元素个数为channel的向量
temp_和x_norm_: 和输入blob的尺寸相同，为N*C*H*W
batch_sum_multiplier_: 元素个数为N的向量
spatial_sum_multiplier_: 元素个数为H*W的矩阵，并且每个元素的值为1
num_by_chans_:元素个数为C*N的矩阵，并且每个元素的值为1
4. caffe中batch_norm_layer.cpp中的Forward_cpu函数：
1 void BatchNormLayer<Dtype>::Forward_cpu(const vector<Blob<Dtype>*>& bottom,
2 const vector<Blob<Dtype>*>& top) {
3 const Dtype* bottom_data = bottom[0]->cpu_data();
4 Dtype* top_data = top[0]->mutable_cpu_data(); //num = N
5 int num = bottom[0]->shape(0); //spatial_dim = N*C*H*W/N*C = H*W
6 int spatial_dim = bottom[0]->count()/(bottom[0]->shape(0)*channels_);
7
8 if (bottom[0] != top[0]) {
9 caffe_copy(bottom[0]->count(), bottom_data, top_data);
10 }
11
12 if (use_global_stats_) {
13 // use the stored mean/variance estimates. //在测试模式下，scale_factor=1/this->blobs_[2]->cpu_data()[0]
14 const Dtype scale_factor = this->blobs_[2]->cpu_data()[0] == 0 ?
15 0 : 1 / this->blobs_[2]->cpu_data()[0]; //mean_ blob = scale_factor * this->blobs_[0]->cpu_data() //variance_ blob = scale_factor * this_blobs_[1]->cpu_data() //因为blobs_变量定义在类中，所以每次调用某一batch norm层时，blobs_[0], blobs_[1], blobs_[2]都会更新
16 caffe_cpu_scale(variance_.count(), scale_factor,
17 this->blobs_[0]->cpu_data(), mean_.mutable_cpu_data());
18 caffe_cpu_scale(variance_.count(), scale_factor,
19 this->blobs_[1]->cpu_data(), variance_.mutable_cpu_data());
20 } else {
21 // compute mean //在训练模式下计算一个batch的均值
22 caffe_cpu_gemv<Dtype>(CblasNoTrans, channels_ * num, spatial_dim,
23 1. / (num * spatial_dim), bottom_data,
24 spatial_sum_multiplier_.cpu_data(), 0.,
25 num_by_chans_.mutable_cpu_data());
26 caffe_cpu_gemv<Dtype>(CblasTrans, num, channels_, 1.,
27 num_by_chans_.cpu_data(), batch_sum_multiplier_.cpu_data(), 0.,
28 mean_.mutable_cpu_data());
29 }
30 //由上面两步可以得到：无论是训练，还是测试模式下输入batch的均值 //对batch中的每个数据减去对应通道的均值
31 // subtract mean
32 caffe_cpu_gemm<Dtype>(CblasNoTrans, CblasNoTrans, num, channels_, 1, 1,
33 batch_sum_multiplier_.cpu_data(), mean_.cpu_data(), 0.,
34 num_by_chans_.mutable_cpu_data());
35 caffe_cpu_gemm<Dtype>(CblasNoTrans, CblasNoTrans, channels_ * num,
36 spatial_dim, 1, -1, num_by_chans_.cpu_data(),
37 spatial_sum_multiplier_.cpu_data(), 1., top_data);
38
39 if (!use_global_stats_) { //计算训练模式下的方差
40 // compute variance using var(X) = E((X-EX)^2)
41 caffe_sqr<Dtype>(top[0]->count(), top_data,
42 temp_.mutable_cpu_data()); // (X-EX)^2
43 caffe_cpu_gemv<Dtype>(CblasNoTrans, channels_ * num, spatial_dim,
44 1. / (num * spatial_dim), temp_.cpu_data(),
45 spatial_sum_multiplier_.cpu_data(), 0.,
46 num_by_chans_.mutable_cpu_data());
47 caffe_cpu_gemv<Dtype>(CblasTrans, num, channels_, 1.,
48 num_by_chans_.cpu_data(), batch_sum_multiplier_.cpu_data(), 0.,
49 variance_.mutable_cpu_data()); // E((X_EX)^2)
50
51 // compute and save moving average //在训练阶段，由以上计算步骤可以得到：batch中每个channel的均值和方差 //blobs_[2] = 1 + blobs_[2]*moving_average_fraction_ //第一个batch时，blobs_[2]=0, 计算后的blobs_[2] = 1 //第二个batch时，blobs_[2]=1, 计算后的blobs_[2] = 1 + 1*moving_average_fraction_ = 1.9
52 this->blobs_[2]->mutable_cpu_data()[0] *= moving_average_fraction_;
53 this->blobs_[2]->mutable_cpu_data()[0] += 1; //blobs_[0] = 1 * mean_ + moving_average_fraction_ * blobs_[0] //其中mean_是本次batch的均值，blobs_[0]是上次batch的均值
54 caffe_cpu_axpby(mean_.count(), Dtype(1), mean_.cpu_data(),
55 moving_average_fraction_, this->blobs_[0]->mutable_cpu_data()); //m = N*C*H*W/C = N*H*W
56 int m = bottom[0]->count()/channels_; //bias_correction_factor = m/m-1
57 Dtype bias_correction_factor = m > 1 ? Dtype(m)/(m-1) : 1; //blobs_[1] = bias_correction_factor * variance_ + moving_average_fraction_ * blobs_[1]
58 caffe_cpu_axpby(variance_.count(), bias_correction_factor,
59 variance_.cpu_data(), moving_average_fraction_,
60 this->blobs_[1]->mutable_cpu_data());
61 }
62 //给上一步计算得到的方差加上一个常数eps_,防止方差作为分母在归一化的时候值出现为0的情况，同时开方63 // normalize variance
64 caffe_add_scalar(variance_.count(), eps_, variance_.mutable_cpu_data());
65 caffe_sqrt(variance_.count(), variance_.cpu_data(),
66 variance_.mutable_cpu_data());
67
68 // replicate variance to input size //top_data目前保存的是输入blobs - mean的值，下面几行代码的意思是给每个元素除以对应方差
69 caffe_cpu_gemm<Dtype>(CblasNoTrans, CblasNoTrans, num, channels_, 1, 1,
70 batch_sum_multiplier_.cpu_data(), variance_.cpu_data(), 0.,
71 num_by_chans_.mutable_cpu_data());
72 caffe_cpu_gemm<Dtype>(CblasNoTrans, CblasNoTrans, channels_ * num,
73 spatial_dim, 1, 1., num_by_chans_.cpu_data(),
74 spatial_sum_multiplier_.cpu_data(), 0., temp_.mutable_cpu_data());
75 caffe_div(temp_.count(), top_data, temp_.cpu_data(), top_data);
76 // TODO(cdoersch): The caching is only needed because later in-place layers
77 // might clobber the data. Can we skip this if they won't?
78 caffe_copy(x_norm_.count(), top_data,
79 x_norm_.mutable_cpu_data());
80 }
caffe_cpu_gemv的原型为：
1 caffe_cpu_gemv<float>(const CBLAS_TRANSPOSE TransA, const int M, const int N, const float alpha, const float *A, const float *x, const float beta, float *y)
实现的功能是矩阵和向量相乘：Y = alpha * A * x + beta * Y
其中，A矩阵的维度为M*N, x向量的维度为N*1, Y向量的维度为M*1.
在训练阶段，forward cpu函数执行如下步骤：
(1) 均值计算，均值计算的过程如下，分为两步：
<1> 计算batch中每个元素的每个channel通道的和；
1 caffe_cpu_gemv<Dtype>(CblasNoTrans, channels_ * num, spatial_dim, 1. / (num * spatial_dim), bottom_data, spatial_sum_multiplier_.cpu_data(), 0., num_by_chans_.mutable_cpu_data());
其中：xN-1,C-1,H-1,W-1表示的含义为：N-1表示batch中的第N-1个样本，C-1表示该样本对应的第C-1个通道，H-1表示该通道中第H-1行，W-1表示该通道中第W-1列；
sumN-1,C-1表示的含义为：batch中第N-1个样本的第C-1个通道中所有元素之和。
<2> 计算batch中每个通道的均值：
1 caffe_cpu_gemv<Dtype>(CblasTrans, num, channels_, 1., num_by_chans_.cpu_data(), batch_sum_multiplier_.cpu_data(), 0., mean_.mutable_cpu_data());
(2) 对batch中的每个数据减去其对应通道的均值；
<1> 得到均值矩阵
1 caffe_cpu_gemm<Dtype>(CblasNoTrans, CblasNoTrans, num, channels_, 1, 1, batch_sum_multiplier_.cpu_data(), mean_.cpu_data(), 0., num_by_chans_.mutable_cpu_data());
<2> 每个元素减去对应均值
1 caffe_cpu_gemm<Dtype>(CblasNoTrans, CblasNoTrans, channels_ * num, spatial_dim, 1, -1, num_by_chans_.cpu_data(), spatial_sum_multiplier_.cpu_data(), 1., top_data);
(3) 每个通道的方差计算，计算方式和均值的计算方式相同；
(4) 输入blob除以对应方差，得到归一化后的值。