1、填充零的padding函式實現

# GRADED FUNCTION: zero_pad

def zero_pad(X, pad):
    """
    Pad with zeros all images of the dataset X. The padding is applied to the height and width of an image, 
    as illustrated in Figure 1.

    Argument:
    X -- python numpy array of shape (m, n_H, n_W, n_C) representing a batch of m images
    pad -- integer, amount of padding around each image on vertical and horizontal dimensions

    Returns:
    X_pad -- padded image of shape (m, n_H + 2*pad, n_W + 2*pad, n_C)
    """
 


    ### START CODE HERE ### (≈ 1 line)
    X_pad = np.pad(X,((0,0),(pad,pad),(pad,pad),(0,0)),'constant')
    ### END CODE HERE ###

    return X_pad

2、一步卷積網路實現：
2.1 單個輸入片卷積

# GRADED FUNCTION: conv_single_step

def conv_single_step(a_slice_prev, W, b):
    """
    Apply one filter defined by parameters W on a single slice (a_slice_prev) of the output activation 
    of the previous layer.

    Arguments:
    a_slice_prev -- slice of input data of shape (f, f, n_C_prev)
    W -- Weight parameters contained in a window - matrix of shape (f, f, n_C_prev)
    b -- Bias parameters contained in a window - matrix of shape (1, 1, 1)

    Returns:
    Z -- a scalar value, result of convolving the sliding window (W, b) on a slice x of the input data
    """
 


    ### START CODE HERE ### (≈ 2 lines of code)
    # Element-wise product between a_slice and W. Add bias.
    s = np.multiply(a_slice_prev,W) + b#a_slice_prev*W + b
    # Sum over all entries of the volume s
    Z = np.sum(s)
    ### END CODE HERE ###

    return Z

2.2 前向傳播卷積實現：

# GRADED FUNCTION: conv_forward
 


def conv_forward(A_prev, W, b, hparameters):
    """
    Implements the forward propagation for a convolution function

    Arguments:
    A_prev -- output activations of the previous layer, numpy array of shape (m, n_H_prev, n_W_prev, n_C_prev)
    W -- Weights, numpy array of shape (f, f, n_C_prev, n_C)
    b -- Biases, numpy array of shape (1, 1, 1, n_C)
    hparameters -- python dictionary containing "stride" and "pad"

    Returns:
    Z -- conv output, numpy array of shape (m, n_H, n_W, n_C)
    cache -- cache of values needed for the conv_backward() function
    """

    ### START CODE HERE ###
    # Retrieve dimensions from A_prev's shape (≈1 line)  
    (m, n_H_prev, n_W_prev, n_C_prev) = A_prev.shape #  None

    # Retrieve dimensions from W's shape (≈1 line)
    (f, f, n_C_prev, n_C) = W.shape  #None

    # Retrieve information from "hparameters" (≈2 lines)
    stride = hparameters['stride']   #None
    pad = hparameters['pad']

    # Compute the dimensions of the CONV output volume using the formula given above. Hint: use int() to floor. (≈2 lines)
    n_H = np.int((n_H_prev + 2.0*pad -f)/stride) + 1
    n_W = np.int((n_W_prev +2.0*pad-f)/stride) + 1

    # Initialize the output volume Z with zeros. (≈1 line)
    Z = np.zeros([m,n_H,n_W,n_C])

    # Create A_prev_pad by padding A_prev
    A_prev_pad = zero_pad(A_prev,pad)

    for i in range(m):                               # loop over the batch of training examples
        a_prev_pad = A_prev_pad[i]                               # Select ith training example's padded activation
        for h in range(n_H):                           # loop over vertical axis of the output volume
            for w in range(n_W):                       # loop over horizontal axis of the output volume
                for c in range(n_C):                   # loop over channels (= #filters) of the output volume

                    # Find the corners of the current "slice" (≈4 lines)
                    vert_start = h*stride
                    vert_end = vert_start + f
                    horiz_start = w*stride
                    horiz_end = horiz_start + f

                    # Use the corners to define the (3D) slice of a_prev_pad (See Hint above the cell). (≈1 line)
                    a_slice_prev = a_prev_pad[vert_start:vert_end,horiz_start:horiz_end,:]

                    # Convolve the (3D) slice with the correct filter W and bias b, to get back one output neuron. (≈1 line)
                    Z[i, h, w, c] = conv_single_step(a_slice_prev,W[:,:,:,c],b[:,:,:,c])

    ### END CODE HERE ###

    # Making sure your output shape is correct
    assert(Z.shape == (m, n_H, n_W, n_C))

    # Save information in "cache" for the backprop
    cache = (A_prev, W, b, hparameters)

    return Z, cache

3、池化函式實現：

# GRADED FUNCTION: pool_forward

def pool_forward(A_prev, hparameters, mode = "max"):
    """
    Implements the forward pass of the pooling layer

    Arguments:
    A_prev -- Input data, numpy array of shape (m, n_H_prev, n_W_prev, n_C_prev)
    hparameters -- python dictionary containing "f" and "stride"
    mode -- the pooling mode you would like to use, defined as a string ("max" or "average")

    Returns:
    A -- output of the pool layer, a numpy array of shape (m, n_H, n_W, n_C)
    cache -- cache used in the backward pass of the pooling layer, contains the input and hparameters 
    """

    # Retrieve dimensions from the input shape
    (m, n_H_prev, n_W_prev, n_C_prev) = A_prev.shape

    # Retrieve hyperparameters from "hparameters"
    f = hparameters["f"]
    stride = hparameters["stride"]

    # Define the dimensions of the output
    n_H = int(1 + (n_H_prev - f) / stride)
    n_W = int(1 + (n_W_prev - f) / stride)
    n_C = n_C_prev

    # Initialize output matrix A
    A = np.zeros((m, n_H, n_W, n_C))              

    ### START CODE HERE ###
    for i in range(m):                         # loop over the training examples
        for h in range(n_H):                     # loop on the vertical axis of the output volume
            for w in range(n_W):                 # loop on the horizontal axis of the output volume
                for c in range (n_C):            # loop over the channels of the output volume

                    # Find the corners of the current "slice" (≈4 lines)
                    vert_start = h*stride
                    vert_end = vert_start + f
                    horiz_start = w*stride
                    horiz_end = horiz_start + f

                    # Use the corners to define the current slice on the ith training example of A_prev, channel c. (≈1 line)
                    a_prev_slice = A_prev[i,vert_start:vert_end,horiz_start:horiz_end,c]

                    # Compute the pooling operation on the slice. Use an if statment to differentiate the modes. Use np.max/np.mean.
                    if mode == "max":
                        A[i, h, w, c] = a_prev_slice.max()
                    elif mode == "average":
                        A[i, h, w, c] = a_prev_slice.mean()

    ### END CODE HERE ###

    # Store the input and hparameters in "cache" for pool_backward()
    cache = (A_prev, hparameters)

    # Making sure your output shape is correct
    assert(A.shape == (m, n_H, n_W, n_C))

    return A, cache

4、向後卷積實現：

def conv_backward(dZ, cache):
    """
    Implement the backward propagation for a convolution function

    Arguments:
    dZ -- gradient of the cost with respect to the output of the conv layer (Z), numpy array of shape (m, n_H, n_W, n_C)
    cache -- cache of values needed for the conv_backward(), output of conv_forward()

    Returns:
    dA_prev -- gradient of the cost with respect to the input of the conv layer (A_prev),
               numpy array of shape (m, n_H_prev, n_W_prev, n_C_prev)
    dW -- gradient of the cost with respect to the weights of the conv layer (W)
          numpy array of shape (f, f, n_C_prev, n_C)
    db -- gradient of the cost with respect to the biases of the conv layer (b)
          numpy array of shape (1, 1, 1, n_C)
    """

    ### START CODE HERE ###
    # Retrieve information from "cache"
    (A_prev, W, b, hparameters) = cache   #None

    # Retrieve dimensions from A_prev's shape
    (m, n_H_prev, n_W_prev, n_C_prev) = A_prev.shape

    # Retrieve dimensions from W's shape
    (f, f, n_C_prev, n_C) = W.shape

    # Retrieve information from "hparameters"
    stride = hparameters['stride']
    pad = hparameters['pad']

    # Retrieve dimensions from dZ's shape
    (m, n_H, n_W, n_C) = dZ.shape

    # Initialize dA_prev, dW, db with the correct shapes
    dA_prev = np.zeros(A_prev.shape)                           
    dW = np.zeros(W.shape)
    db = np.zeros((1,1,1,n_C))

    # Pad A_prev and dA_prev
    A_prev_pad = zero_pad(A_prev,pad)
    dA_prev_pad = zero_pad(dA_prev,pad)

    for i in range(m):                       # loop over the training examples

        # select ith training example from A_prev_pad and dA_prev_pad
        a_prev_pad = A_prev_pad[i]
        da_prev_pad = dA_prev_pad[i]

        for h in range(n_H):                   # loop over vertical axis of the output volume
            for w in range(n_W):               # loop over horizontal axis of the output volume
                for c in range(n_C):           # loop over the channels of the output volume

                    # Find the corners of the current "slice"
                    vert_start = h*stride
                    vert_end = vert_start+f
                    horiz_start = w*stride
                    horiz_end = horiz_start + f

                    # Use the corners to define the slice from a_prev_pad
                    a_slice = a_prev_pad[vert_start:vert_end,horiz_start:horiz_end,:]

                    # Update gradients for the window and the filter's parameters using the code formulas given above
                    da_prev_pad[vert_start:vert_end, horiz_start:horiz_end, :] += W[:,:,:,c]*dZ[i,h,w,c]
                    dW[:,:,:,c] += a_slice*dZ[i,h,w,c]
                    db[:,:,:,c] += dZ[i,h,w,c]

        # Set the ith training example's dA_prev to the unpaded da_prev_pad (Hint: use X[pad:-pad, pad:-pad, :])
        dA_prev[i, :, :, :] = dA_prev_pad[i,pad: -pad,pad: -pad,:]
    ### END CODE HERE ###

    # Making sure your output shape is correct
    assert(dA_prev.shape == (m, n_H_prev, n_W_prev, n_C_prev))

    return dA_prev, dW, db

5 後向傳播池化實現
5.1 max pooling 後向傳播實現：

def create_mask_from_window(x):
    """
    Creates a mask from an input matrix x, to identify the max entry of x.

    Arguments:
    x -- Array of shape (f, f)

    Returns:
    mask -- Array of the same shape as window, contains a True at the position corresponding to the max entry of x.
    """

    ### START CODE HERE ### (≈1 line)
    mask = x==x.max()
    ### END CODE HERE ###

    return mask

5.2、 average_pooling 池化後向實現：

def distribute_value(dz, shape):
    """
    Distributes the input value in the matrix of dimension shape

    Arguments:
    dz -- input scalar
    shape -- the shape (n_H, n_W) of the output matrix for which we want to distribute the value of dz

    Returns:
    a -- Array of size (n_H, n_W) for which we distributed the value of dz
    """

    ### START CODE HERE ###
    # Retrieve dimensions from shape (≈1 line)
    (n_H, n_W) = shape

    # Compute the value to distribute on the matrix (≈1 line)
    average = dz/n_H/n_W

    # Create a matrix where every entry is the "average" value (≈1 line)
    a = np.ones((n_H,n_W))*average
    ### END CODE HERE ###

    return a

5.3 池化函式組合實現：

def pool_backward(dA, cache, mode = "max"):
    """
    Implements the backward pass of the pooling layer

    Arguments:
    dA -- gradient of cost with respect to the output of the pooling layer, same shape as A
    cache -- cache output from the forward pass of the pooling layer, contains the layer's input and hparameters 
    mode -- the pooling mode you would like to use, defined as a string ("max" or "average")

    Returns:
    dA_prev -- gradient of cost with respect to the input of the pooling layer, same shape as A_prev
    """

    ### START CODE HERE ###

    # Retrieve information from cache (≈1 line)
    (A_prev, hparameters) = cache

    # Retrieve hyperparameters from "hparameters" (≈2 lines)
    stride = hparameters['stride']
    f = hparameters['f']

    # Retrieve dimensions from A_prev's shape and dA's shape (≈2 lines)
    m, n_H_prev, n_W_prev, n_C_prev =A_prev.shape
    m, n_H, n_W, n_C = A.shape

    # Initialize dA_prev with zeros (≈1 line)
    dA_prev = np.zeros(A_prev.shape)

    for i in range(m):                       # loop over the training examples

        # select training example from A_prev (≈1 line)
        a_prev = A_prev[i]

        for h in range(n_H):                   # loop on the vertical axis
            for w in range(n_W):               # loop on the horizontal axis
                for c in range(n_C):           # loop over the channels (depth)

                    # Find the corners of the current "slice" (≈4 lines)
                    vert_start = h*stride
                    vert_end = vert_start + f
                    horiz_start = w*stride
                    horiz_end = horiz_start + f

                    # Compute the backward propagation in both modes.
                    if mode == "max":

                        # Use the corners and "c" to define the current slice from a_prev (≈1 line)
                        a_prev_slice =a_prev[vert_start:vert_end,horiz_start:horiz_end,c]
                        # Create the mask from a_prev_slice (≈1 line)
                        mask = create_mask_from_window(a_prev_slice)
                        # Set dA_prev to be dA_prev + (the mask multiplied by the correct entry of dA) (≈1 line)
                        dA_prev[i, vert_start: vert_end, horiz_start: horiz_end, c] += mask*dA[i,vert_start:vert_end,horiz_start:horiz_end,c]

                    elif mode == "average":

                        # Get the value a from dA (≈1 line)
                        da = dA[i,h,w,c]
                        # Define the shape of the filter as fxf (≈1 line)
                        shape = distribute_value(da,(f,f))
                        # Distribute it to get the correct slice of dA_prev. i.e. Add the distributed value of da. (≈1 line)
                        dA_prev[i, vert_start: vert_end, horiz_start: horiz_end, c] += dA[i,vert_start:vert_end,horiz_start:horiz_end,c]*shape

    ### END CODE ###

    # Making sure your output shape is correct
    assert(dA_prev.shape == A_prev.shape)

    return dA_prev