【CNN从零开始系列之二】网络的搭建

初始化网络结构

前面一篇文章讲解了数据集的准备，其实它也是网络的第一层输入层，主要是将数据格式转化为下一层网络可处理的形式。我们先看看整体网络结构，定义如下：

conv1 = Convolution(data, kernel_size = 5, num_output = 20)#卷积层
pool1 = Max_pooling(conv1, kernel_size = 2, stride = 2)#池化层
conv2 = Convolution(pool1, kernel_size = 5, num_output = 50)
pool2 = Max_pooling(conv2, kernel_size = 2, stride = 2)
flat1 = Flatten(pool2)#将矩阵扁平化
fc1 = Full_connection(flat1, 500)#全连接层
relu1 = Relu(fc1)#激活函数
fc2 = Full_connection(relu1, 10)
softmax = Softmax(fc2)

我们会讲解每一层如何定义，以及它们的向前传播和反向传播的实现。

卷积层

在Convolution这个类中，定义了当前神经节点数量，前一个网络层实体，，当前层输入矩阵大小和卷积以后输出矩阵大小，卷积核，padding等信息，，权重的采用Xavier初始化梯度下降采用动量更新法，考虑前一次梯度的影响，同时考虑梯度衰减，具体卷积层的实现如下：

def conv(X, W, b, stride = 1, padding = 0):#卷积的计算过程，向前传播的具体计算
    n_filters, d_filter, kernel_size, _ = W.shape
    n_x, d_x, h_x, w_x = X.shape
    h_out = (h_x - kernel_size + 2 * padding) / stride + 1
    w_out = (w_x - kernel_size + 2 * padding) / stride + 1
    h_out, w_out = int(h_out), int(w_out)
    X_col = im2col(X, kernel_size, padding=padding, stride=stride)
    W_col = W.reshape(n_filters, -1)
    out = (np.dot(W_col, X_col).T + b).T
    out = out.reshape(n_filters, h_out, w_out, n_x)
    out = out.transpose(3, 0, 1, 2)
    return out

def im2col(x, kernel_size, padding=0, stride=1):#图片转矩阵
    p = padding
    x_padded = np.pad(x, ((0, 0), (0, 0), (p, p), (p, p)), mode='constant')
    k, i, j = get_im2col_indices(x.shape, kernel_size, padding, stride)
    cols = x_padded[:, k, i, j]
    C = x.shape[1]
    cols = cols.transpose(1, 2, 0).reshape(kernel_size ** 2 * C, -1)
    return cols

def get_im2col_indices(x_shape, kernel_size, padding=0, stride=1):
    N, C, H, W = x_shape
    assert (H + 2 * padding - kernel_size) % stride == 0
    assert (W + 2 * padding - kernel_size) % stride == 0
    out_height = int((H + 2 * padding - kernel_size) / stride + 1)
    out_width = int((W + 2 * padding - kernel_size) / stride + 1)
    i0 = np.repeat(np.arange(kernel_size), kernel_size)
    i0 = np.tile(i0, C)
    i1 = stride * np.repeat(np.arange(out_height), out_width)
    j0 = np.tile(np.arange(kernel_size), kernel_size * C)
    j1 = stride * np.tile(np.arange(out_width), out_height)
    i = i0.reshape(-1, 1) + i1.reshape(1, -1)
    j = j0.reshape(-1, 1) + j1.reshape(1, -1)
    k = np.repeat(np.arange(C), kernel_size * kernel_size).reshape(-1, 1)
    return (k.astype(int), i.astype(int), j.astype(int))

def col2im(x, img_shape, kernel_size, padding = 0, stride = 1):
    x_row_num, x_col_num = x.shape
    channels, img_height, img_width = img_shape
    x_width = img_width - kernel_size + padding + 1
    x_height = img_height - kernel_size + padding + 1
    assert channels * kernel_size ** 2 == x_row_num
    assert x_width * x_height == x_col_num
    x_reshape = x.T.reshape(x_height, x_width, channels, kernel_size, kernel_size)
    output_padded = np.zeros((channels, img_height + 2 * padding, img_width + 2 * padding))
    for i in range(x_height):
        for j in range(x_width):
            output_padded[:, i * stride : i * stride + kernel_size, j * stride : j * stride + kernel_size] = \
                    output_padded[:, i * stride : i * stride + kernel_size, j * stride : j * stride + kernel_size] + \
                    x_reshape[i, j, ...]
    return output_padded[:, padding : img_height + padding, padding : img_width + padding]
class Convolution:

    def __init__(self, layer, kernel_size = 1, num_output = 1, padding = 0):
        self.upper_layer = layer#前一层
        self.num = layer.num
        self.num_input = layer.num_output#网络的输入为前一层网络的输入
        self.input_h = layer.output_h
        self.input_w = layer.output_w
        self.output_h = self.input_h + 2 * padding - kernel_size + 1
        self.output_w = self.input_w + 2 * padding - kernel_size + 1#卷积具体过程
        self.num_output = num_output
        self.kernel_size = kernel_size
        self.padding = padding
        scale = get_XavierScale(kernel_size)
        self.weight = np.random.rand(num_output, self.num_input, kernel_size, kernel_size)
        self.weight = (self.weight - 0.5) * 2 * scale#采用Xavier初始化
        self.weight_diff_his = np.zeros(self.weight.shape)#历史梯度w，用来更新当前梯度动量更新法
        self.bias = np.zeros((num_output))
        self.bias_diff_his = np.zeros(self.bias.shape)

    def forward(self):
        self.input_data = self.upper_layer.forward()
        self.num = self.upper_layer.num
        self.output_data = conv(self.input_data, self.weight, self.bias, padding = self.padding)
        return self.output_data

    def backward(self, diff):
        weight_diff,bias_diff=get_diff()#采用动量梯度下降
        self.weight_diff_his = 0.9 * self.weight_diff_his + weight_diff.reshape(self.weight.shape)
        self.weight = self.weight * 0.9995 - self.weight_diff_his#梯度衰减
        self.bias_diff_his = 0.9 * self.bias_diff_his + 2 * bias_diff
        self.bias = self.bias * 0.9995 - self.bias_diff_his
        self.upper_layer.backward(self.diff)

    def get_output(self):
        return self.output_data
    def get_XavierScale(self,kernel_size):
        return math.sqrt(3. / (self.num_input * kernel_size**2)) 
    def get_diff(self):
        self.diff = np.zeros(self.input_data.shape)
        weight_diff = np.zeros(self.weight.shape)
        weight_diff = weight_diff.reshape(weight_diff.shape[0], -1)
        bias_diff = np.zeros((self.num_output))
        weight_reshape_T = self.weight.reshape(self.weight.shape[0], -1).T
        for i in range(self.num):
            input_data_col = im2col(self.input_data[[i]], self.kernel_size, self.padding)
            weight_diff = weight_diff + diff[i].reshape(diff[i].shape[0], -1).dot(input_data_col.T)
            bias_diff = bias_diff + np.sum(diff[i].reshape(diff[i].shape[0], -1), 1)
            tmp_diff = weight_reshape_T.dot(diff[i].reshape(diff[i].shape[0], -1))
            self.diff[i, ...] = col2im(tmp_diff, self.input_data.shape[1:], self.kernel_size, padding = self.padding)
        return weight_diff,bias_diff

池化层

我们采用的是max池化,代码里面很详细，我就不废话了.具体如下：

def im2col(X, kernel_size = 1, stride = 1):
    num, channels, height, width = X.shape
    surplus_height = (height - kernel_size) % stride
    surplus_width = (width - kernel_size) % stride
    pad_h = (kernel_size - surplus_height) % kernel_size
    pad_w = (kernel_size - surplus_width) % kernel_size
    X = np.pad(X, ((0,0),(0,0),(0,pad_h),(0,pad_w)), mode='constant')
    k,i,j = get_im2col_indices(X.shape, kernel_size, stride = stride)
    X_col = X[:,k,i,j].reshape(num * channels, kernel_size**2, -1)
    X_col = X_col.transpose(0,2,1)
    return X_col.reshape(-1, kernel_size**2)

def get_im2col_indices(x_shape, kernel_size, padding=0, stride=1):
    N, C, H, W = x_shape
    assert (H + 2 * padding - kernel_size) % stride == 0
    assert (W + 2 * padding - kernel_size) % stride == 0
    out_height = int((H + 2 * padding - kernel_size) / stride + 1)
    out_width = int((W + 2 * padding - kernel_size) / stride + 1)
    i0 = np.repeat(np.arange(kernel_size), kernel_size)
    i0 = np.tile(i0, C)
    i1 = stride * np.repeat(np.arange(out_height), out_width)
    j0 = np.tile(np.arange(kernel_size), kernel_size * C)
    j1 = stride * np.tile(np.arange(out_width), out_height)
    i = i0.reshape(-1, 1) + i1.reshape(1, -1)
    j = j0.reshape(-1, 1) + j1.reshape(1, -1)
    k = np.repeat(np.arange(C), kernel_size * kernel_size).reshape(-1, 1)
    return (k.astype(int), i.astype(int), j.astype(int))

def col2ims(x, img_shape, kernel_size, stride):
    x_row_num, x_col_num = x.shape
    img_n, img_c, img_h, img_w = img_shape
    o_h = int(math.ceil((img_h - kernel_size + 0.) / stride)) + 1
    o_w = int(math.ceil((img_w - kernel_size + 0.) / stride)) + 1
    assert img_n * img_c * o_h * o_w == x_row_num
    assert kernel_size**2 == x_col_num
    surplus_h = (img_h - kernel_size) % stride
    surplus_w = (img_w - kernel_size) % stride
    pad_h = (kernel_size - surplus_h) % stride
    pad_w = (kernel_size - surplus_w) % stride
    output_padded = np.zeros((img_n, img_c, img_h + pad_h, img_w + pad_w))
    x_reshape = x.reshape(img_n, img_c, o_h, o_w, kernel_size, kernel_size)
    for n in range(img_n):
        for i in range(o_h):
            for j in range(o_w):
                output_padded[n, :, i * stride : i * stride + kernel_size, j * stride : j * stride + kernel_size] = \
                        output_padded[n, :, i * stride : i * stride + kernel_size, j * stride : j * stride + kernel_size] + \
                        x_reshape[n, :, i, j, ...]
    return output_padded[:, :, 0 : img_h + pad_h, 0 : img_w + pad_w]

class Max_pooling:

    def __init__(self, layer, kernel_size = 1, stride = 1):
        self.num = layer.num
        self.num_output = layer.num_output
        self.num_input = layer.num_output
        self.input_h = layer.output_h
        self.input_w = layer.output_w
        self.output_h = int(math.ceil((self.input_h - kernel_size + 0.) / stride)) + 1
        self.output_w = int(math.ceil((self.input_w - kernel_size + 0.) / stride)) + 1
        self.upper_layer = layer
        self.kernel_size = kernel_size
        self.stride = stride
    
    def forward(self):
        self.input_data = self.upper_layer.forward()
        self.num = self.upper_layer.num
        input_col = im2col(self.input_data, self.kernel_size, self.stride)
        tmp_index = np.tile(np.arange(input_col.shape[1]),input_col.shape[0]).reshape(input_col.shape)
        c=input_col.argmax(1).reshape(-1,1)
        d=input_col.argmax(1)
        self.max_index = tmp_index == input_col.argmax(1).reshape(-1,1)
        self.output_data = input_col[self.max_index].reshape(self.num, self.num_input, self.output_h, self.output_w)
        return self.output_data

    def backward(self, diff):
        diff_col = np.zeros((self.num * self.num_input * self.output_h * self.output_w, self.kernel_size**2))
        c=diff.reshape(-1)
        diff_col[self.max_index] = diff.reshape(-1)
        self.diff = col2ims(diff_col, self.input_data.shape, self.kernel_size, self.stride)
        self.upper_layer.backward(self.diff)

    def get_output(self):
        return self.output_data

扁平化

将矩阵扁平化，使得更好的被全连接层处理运算。具体实现如下：

class Flatten:

    def __init__(self, layer):
        self.upper_layer = layer
        self.num_input = layer.num_output
        self.input_w = layer.output_w
        self.input_h = layer.output_h
        self.output_w = 1
        self.output_h = 1
        self.num_output = self.num_input * self.input_h * self.input_w
        self.num = layer.num
    
    def forward(self):
        self.input_data = self.upper_layer.forward()
        self.num = self.upper_layer.num
        self.output_data = self.input_data.reshape(self.input_data.shape[0], -1)
        return self.output_data

    def backward(self, diff):
        self.diff = diff.reshape(self.input_data.shape)
        self.upper_layer.backward(self.diff)

    def get_output(self):
        return self.output_data

激活函数

激活函数采用relu和softmax，具体实现如下：

class Relu:
   
    def __init__(self, layer):
        self.upper_layer = layer
        self.num_output = layer.num_output
        self.num = layer.num
        self.output_w = layer.output_w
        self.output_h = layer.output_h

    def forward(self):
        self.input_data = self.upper_layer.forward()
        self.num = self.upper_layer.num
        self.output_data = self.input_data.copy()
        self.output_data[self.output_data < 0] = 0
        return self.output_data

    def backward(self, diff):
        self.diff = diff.copy()
        self.diff[self.input_data < 0] == 0
        self.upper_layer.backward(self.diff)

    def get_output(self):
        return self.output_data

class Softmax:
    
    def __init__(self, layer):
        self.upper_layer = layer

    def forward(self):
        self.input_data = self.upper_layer.forward()
        self.num = self.upper_layer.num
        _, self.dim = self.input_data.shape
        t = np.exp(self.input_data - self.input_data.max(1).reshape(-1,1))
        self.softmax_data = t / t.sum(1).reshape(-1,1)
        self.softmax_data[self.softmax_data < 1e-30] = 1e-30
        return self.softmax_data

    def calc_loss(self, label):
        s = np.tile(np.arange(self.dim), self.num).reshape(self.input_data.shape)
        gt_index = s == label.reshape(-1, 1) 
        loss = 0 - np.average(np.log(self.softmax_data[gt_index]))
        self.diff = self.softmax_data.copy()
        self.diff[gt_index] = self.diff[gt_index] - 1.
        self.diff = self.diff / self.num
        return loss

    def backward(self, lr):
        self.upper_layer.backward(self.diff * lr)

网络的结构已经实现定义好了，最后我们来讲如何训练和测试网络模型。