關於卷積操作是如何進行的就不必多說了，結合程式碼一步一步來看卷積層是怎麼實現的。

程式碼來源：https://github.com/eriklindernoren/ML-From-Scratch

先看一下其基本的元件函式，首先是determine_padding(filter_shape, output_shape="same")：

def determine_padding(filter_shape, output_shape="same"):

    # No padding
    if output_shape == "valid":
        return (0, 0), (0, 0)
    # Pad so that the output shape is the same as input shape (given that stride=1)
    elif output_shape == "same":
        filter_height, filter_width = filter_shape

        # Derived from:
        # output_height = (height + pad_h - filter_height) / stride + 1
        # In this case output_height = height and stride = 1. This gives the
        # expression for the padding below.
        pad_h1 = int(math.floor((filter_height - 1)/2))
        pad_h2 = int(math.ceil((filter_height - 1)/2))
        pad_w1 = int(math.floor((filter_width - 1)/2))
        pad_w2 = int(math.ceil((filter_width - 1)/2))

        return (pad_h1, pad_h2), (pad_w1, pad_w2)

說明：根據卷積核的形狀以及padding的方式來計算出padding的值，包括上、下、左、右，其中out_shape=valid表示不填充。

補充：

• math.floor(x)表示返回小於或等於x的最大整數。
• math.ceil(x)表示返回大於或等於x的最大整數。

帶入實際的引數來看下輸出：

pad_h,pad_w=determine_padding((3,3), output_shape="same")

輸出：(1,1),(1,1)

然後是image_to_column(images, filter_shape, stride, output_shape='same')函式

def image_to_column(images, filter_shape, stride, output_shape='same'):
    filter_height, filter_width = filter_shape
    pad_h, pad_w = determine_padding(filter_shape, output_shape)# Add padding to the image
    images_padded = np.pad(images, ((0, 0), (0, 0), pad_h, pad_w), mode='constant')# Calculate the indices where the dot products are to be applied between weights
    # and the image
    k, i, j = get_im2col_indices(images.shape, filter_shape, (pad_h, pad_w), stride)

    # Get content from image at those indices
    cols = images_padded[:, k, i, j]
    channels = images.shape[1]
    # Reshape content into column shape
    cols = cols.transpose(1, 2, 0).reshape(filter_height * filter_width * channels, -1)
    return cols

說明：輸入的images的形狀是[batchsize,channel,height,width]，類似於pytorch的影象格式的輸入。也就是說images_padded是在height和width上進行padding的。在其中呼叫了get_im2col_indices()函式，那我們接下來看看它是個什麼樣子的：

def get_im2col_indices(images_shape, filter_shape, padding, stride=1):
    # First figure out what the size of the output should be
    batch_size, channels, height, width = images_shape
    filter_height, filter_width = filter_shape
    pad_h, pad_w = padding
    out_height = int((height + np.sum(pad_h) - filter_height) / stride + 1)
    out_width = int((width + np.sum(pad_w) - filter_width) / stride + 1)

    i0 = np.repeat(np.arange(filter_height), filter_width)
    i0 = np.tile(i0, channels)
    i1 = stride * np.repeat(np.arange(out_height), out_width)
    j0 = np.tile(np.arange(filter_width), filter_height * channels)
    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(channels), filter_height * filter_width).reshape(-1, 1)return (k, i, j)

說明：單獨看很難理解，我們還是帶著帶著實際的引數一步步來看。

get_im2col_indices((1,3,32,32), (3,3), ((1,1),(1,1)), stride=1)

說明：看一下每一個變數的變化情況，out_width和out_height就不多說，是卷積之後的輸出的特徵圖的寬和高維度。

• i0：np.repeat(np.arange(3),3)：[0 ,0,0,1,1,1,2,2,2]
• i0：np.tile([0,0,0,1,1,1,2,2,2],3)：[0,0,0,1,1,1,2,2,2,0,0,0,1,1,1,2,2,2,0,0,0,1,1,1,2,2,2]，大小為：(27,)
• i1：1*np.repeat(np.arange(32),32)：[0,0,0......,31,31,31]，大小為：(1024,)
• j0：np.tile(np.arange(3),3*3)：[0,1,2,0,1,2,......]，大小為：(27,)
• j1：1*np.tile(np.arange(32),32)：[0,1,2,3,......,0,1,2,......,29,30,31]，大小為(1024,)
• i：i0.reshape(-1,1)+i1.reshape(1,-1)：大小(27,1024)
• j：j0.reshape(-1,1)+j1.reshape(1,-1)：大小(27,1024)
• k：np.repeat(np.arange(3),3*3).reshape(-1,1)：大小(27,1)

補充：

• numpy.pad(array, pad_width, mode, **kwargs)：array是要要被填充的資料，第二個引數指定填充的長度，mod用於指定填充的資料，預設是0，如果是constant，則需要指定填充的值。
• numpy.arange(start, stop, step, dtype = None)：舉例numpy.arange(3)，輸出[0,1,2]
• numpy.repeat(array,repeats,axis=None)：舉例numpy.repeat([0,1,2],3)，輸出：[0,0,0,1,1,1,2,2,2]
• numpy.tile(array,reps)：舉例numpy.tile([0,1,2],3)，輸出：[0,1,2,0,1,2,0,1,2]
• 具體的更復雜的用法還是得去查相關資料。這裡只列舉出與本程式碼相關的。

有了這些大小還是挺難理解的呀。那麼我們繼續，需要明確的是k是對通道進行操作，i是對特徵圖的高，j是對特徵圖的寬。使用3×3的卷積核在一個通道上進行卷積，每次執行3×3=9個畫素操作，共3個通道，所以共對9×3=27個畫素點進行操作。而影象大小是32×32，共1024個畫素。再回去看這三行程式碼：

    cols = images_padded[:, k, i, j]
    channels = images.shape[1]
    # Reshape content into column shape
    cols = cols.transpose(1, 2, 0).reshape(filter_height * filter_width * channels, -1)

images_padded的大小是(1,3,34,34)，則cols=images_padded的大小是(1,27,1024)

channels的大小是3

最終cols=cols.transpose(1,2,0).reshape(3*3*3,-1)的大小是(27,1024)。

當batchsize的大小不是1，假設是64時，那麼最終輸出的cols的大小就是：(27,1024×64)=(27,65536)。

最後就是卷積層的實現了：

首先有一個Layer通用基類，通過繼承該基類可以實現不同的層，例如卷積層、池化層、批量歸一化層等等：

class Layer(object):

    def set_input_shape(self, shape):
        """ Sets the shape that the layer expects of the input in the forward
        pass method """
        self.input_shape = shape

    def layer_name(self):
        """ The name of the layer. Used in model summary. """
        return self.__class__.__name__

    def parameters(self):
        """ The number of trainable parameters used by the layer """
        return 0

    def forward_pass(self, X, training):
        """ Propogates the signal forward in the network """
        raise NotImplementedError()

    def backward_pass(self, accum_grad):
        """ Propogates the accumulated gradient backwards in the network.
        If the has trainable weights then these weights are also tuned in this method.
        As input (accum_grad) it receives the gradient with respect to the output of the layer and
        returns the gradient with respect to the output of the previous layer. """
        raise NotImplementedError()

    def output_shape(self):
        """ The shape of the output produced by forward_pass """
        raise NotImplementedError()

對於子類繼承該基類必須要實現的方法，如果沒有實現使用raise NotImplementedError()丟擲異常。

接著就可以基於該基類實現Conv2D了：

class Conv2D(Layer):
    """A 2D Convolution Layer.
    Parameters:
    -----------
    n_filters: int
        The number of filters that will convolve over the input matrix. The number of channels
        of the output shape.
    filter_shape: tuple
        A tuple (filter_height, filter_width).
    input_shape: tuple
        The shape of the expected input of the layer. (batch_size, channels, height, width)
        Only needs to be specified for first layer in the network.
    padding: string
        Either 'same' or 'valid'. 'same' results in padding being added so that the output height and width
        matches the input height and width. For 'valid' no padding is added.
    stride: int
        The stride length of the filters during the convolution over the input.
    """
    def __init__(self, n_filters, filter_shape, input_shape=None, padding='same', stride=1):
        self.n_filters = n_filters
        self.filter_shape = filter_shape
        self.padding = padding
        self.stride = stride
        self.input_shape = input_shape
        self.trainable = True

    def initialize(self, optimizer):
        # Initialize the weights
        filter_height, filter_width = self.filter_shape
        channels = self.input_shape[0]
        limit = 1 / math.sqrt(np.prod(self.filter_shape))
        self.W  = np.random.uniform(-limit, limit, size=(self.n_filters, channels, filter_height, filter_width))
        self.w0 = np.zeros((self.n_filters, 1))
        # Weight optimizers
        self.W_opt  = copy.copy(optimizer)
        self.w0_opt = copy.copy(optimizer)

    def parameters(self):
        return np.prod(self.W.shape) + np.prod(self.w0.shape)

    def forward_pass(self, X, training=True):
        batch_size, channels, height, width = X.shape
        self.layer_input = X
        # Turn image shape into column shape
        # (enables dot product between input and weights)
        self.X_col = image_to_column(X, self.filter_shape, stride=self.stride, output_shape=self.padding)
        # Turn weights into column shape
        self.W_col = self.W.reshape((self.n_filters, -1))
        # Calculate output
        output = self.W_col.dot(self.X_col) + self.w0
        # Reshape into (n_filters, out_height, out_width, batch_size)
        output = output.reshape(self.output_shape() + (batch_size, ))
        # Redistribute axises so that batch size comes first
        return output.transpose(3,0,1,2)

    def backward_pass(self, accum_grad):
        # Reshape accumulated gradient into column shape
        accum_grad = accum_grad.transpose(1, 2, 3, 0).reshape(self.n_filters, -1)

        if self.trainable:
            # Take dot product between column shaped accum. gradient and column shape
            # layer input to determine the gradient at the layer with respect to layer weights
            grad_w = accum_grad.dot(self.X_col.T).reshape(self.W.shape)
            # The gradient with respect to bias terms is the sum similarly to in Dense layer
            grad_w0 = np.sum(accum_grad, axis=1, keepdims=True)

            # Update the layers weights
            self.W = self.W_opt.update(self.W, grad_w)
            self.w0 = self.w0_opt.update(self.w0, grad_w0)

        # Recalculate the gradient which will be propogated back to prev. layer
        accum_grad = self.W_col.T.dot(accum_grad)
        # Reshape from column shape to image shape
        accum_grad = column_to_image(accum_grad,
                                self.layer_input.shape,
                                self.filter_shape,
                                stride=self.stride,
                                output_shape=self.padding)

        return accum_grad

    def output_shape(self):
        channels, height, width = self.input_shape
        pad_h, pad_w = determine_padding(self.filter_shape, output_shape=self.padding)
        output_height = (height + np.sum(pad_h) - self.filter_shape[0]) / self.stride + 1
        output_width = (width + np.sum(pad_w) - self.filter_shape[1]) / self.stride + 1
        return self.n_filters, int(output_height), int(output_width)

假設輸入還是(1,3,32,32)的維度，使用16個3×3的卷積核進行卷積，那麼self.W的大小就是(16,3,3,3)，self.w0的大小就是(16,1)。

self.X_col的大小就是(27,1024)，self.W_col的大小是(16,27)，那麼output = self.W_col.dot(self.X_col) + self.w0的大小就是(16,1024)

最後是這麼使用的：

image = np.random.randint(0,255,size=(1,3,32,32)).astype(np.uint8)
input_shape=image.squeeze().shape
conv2d = Conv2D(16, (3,3), input_shape=input_shape, padding='same', stride=1)
conv2d.initialize(None)
output=conv2d.forward_pass(image,training=True)
print(output.shape)

輸出結果：(1,16,32,32)

計算下引數：

print(conv2d.parameters())

輸出結果：448

也就是448=3×3×3×16+16

再是一個padding=valid的：

image = np.random.randint(0,255,size=(1,3,32,32)).astype(np.uint8)
input_shape=image.squeeze().shape
conv2d = Conv2D(16, (3,3), input_shape=input_shape, padding='valid', stride=1)
conv2d.initialize(None)
output=conv2d.forward_pass(image,training=True)
print(output.shape)
print(conv2d.parameters())

需要注意的是cols的大小變化了，因為我們卷積之後的輸出是(1,16,30,30)

輸出：

cols的大小：(27,900)

(1,16,30,30)

448

最後是帶步長的：

image = np.random.randint(0,255,size=(1,3,32,32)).astype(np.uint8)
input_shape=image.squeeze().shape
conv2d = Conv2D(16, (3,3), input_shape=input_shape, padding='valid', stride=2)
conv2d.initialize(None)
output=conv2d.forward_pass(image,training=True)
print(output.shape)
print(conv2d.parameters())

cols的大小：(27,225)

(1,16,15,15)

448

最後補充下：

卷積層引數計算公式 ：params=卷積核高×卷積核寬×通道數目×卷積核數目+偏置項（卷積核數目）

卷積之後影象大小計算公式：

輸出影象的高=(輸入影象的高+padding（高）×2-卷積核高)/步長+1

輸出影象的寬=(輸入影象的寬+padding（寬）×2-卷積核寬)/步長+1

get_im2col_indices()函式中的變換操作是清楚了，至於為什麼這麼變換的原因還需要好好去琢磨。至於反向傳播和優化optimizer等研究好了之後再更新了。

&n