Courra 吴恩达深度学习课程作业：卷积神经⽹络4.1（1）
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⽬录：
Convolutional Neural Networks: Step by Step
Welcome to Cour 4’s first assignment! In this assignment, you will implement convolutional (CONV) and pooling (POOL) layers in numpy, including both forward propagation and (optionally) backward propagation.
Notation:
- Superscript denotes an object of the layer.
- Example: is the layer activation. and are the layer parameters.
Superscript denotes an object from the example.
Example: is the training example input.
Lowerscript denotes the entry of a vector.
Example: denotes the entry of the activations in layer , assuming this is a fully connected (FC) layer.
, and denote respectively the height, width and number of channels of a given layer. If you want to reference a specific layer , you can also write , , ., and denote respectively the height, width and number of channels of the previous layer. If referencing a specific layer , this could also be denoted , , .We assume that you are already familiar with numpy and/or have completed the previous cours of the specialization. Let’s get started!1 - Packages
Let’s first import all the packages that you will need during this assignment.
- is the fundamental package for scientific computing with Python.
- is a library to plot graphs in Python.
- np.random.ed(1) is ud to keep all the random function calls consistent. It will help us grade your work.
2 - Outline of the Assignment
You will be implementing the building blocks of a convolutional neural network! Each function you will implement will have detailed instructions that will walk you through the steps needed:
Convolution functions, including:
Zero Padding
Convolve window
Convolution forward
Convolution backward (optional)
Pooling functions, including:
Pooling forward
Create mask
Distribute value
职位说明书Pooling backward (optional)
This notebook will ask you to implement the functions from scratch in numpy . In the next notebook, you will u the TensorFlow equivalents of the
functions to build the following model:
Note that for every forward function, there is its corresponding backward equivalent. Hence, at every step of your forward module you will store some import numpy as np
import h5py
import matplotlib.pyplot as plt
%matplotlib inline
%load_ext autoreload
%autoreload 2
lifehounp.random.ed(1)
parameters in a cache. The parameters are ud to compute gradients during backpropagation.
3 - Convolutional Neural Networks
Although programming frameworks make convolutions easy to u, they remain one of the hardest concepts to understand in Deep Learning. A convolution layer transforms an input volume into an output volume of different size, as shown below.
In this part, you will build every step of the convolution layer. You will first implement two helper functions: one for zero padding and the other for computing the convolution function itlf.
3.1 - Zero-Padding
Zero-padding adds zeros around the border of an image:
Figure 1 : Zero-Padding
Image (3 channels, RGB) with a padding of 2.
The main benefits of padding are the following:
It allows you to u a CONV layer without necessarily shrinking the height and width of the volumes. This is important for building deeper networks, since otherwi the height/width would shrink as you go to deeper layers. An important special ca is the “same” convolution, in which the height/width is聪的成语
exactly prerved after one layer.
It helps us keep more of the information at the border of an image. Without padding, very few values at the next layer would be affected by pixels as the edges of an image.
Exerci: Implement the following function, which pads all the images of a batch of examples X with zeros. . Note if you want to pad the array “a” of shape with pad = 1 for the 2nd dimension, pad = 3 for the 4th dimension and pad = 0 for the rest, you would do:
测试：
输出：
x.shape = (4, 3, 3, 2)
x_pad.shape = (4, 7, 7, 2)
x[1,1] = [[ 0.90085595 -0.68372786]
[-0.12289023 -0.93576943]
[-0.26788808 0.53035547]]
x_pad[1,1] = [[ 0. 0.]
[ 0. 0.]
[ 0. 0.]
[ 0. 0.]
[ 0. 0.]
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[ 0. 0.]
[ 0. 0.]]
Expected Output:
**x.shape**:(4, 3, 3, 2)
**x_pad.shape**:(4, 7, 7, 2)
**x[1,1]**:[[ 0.90085595 -0.68372786] [-0.12289023 -0.93576943] [-0.26788808 0.53035547]]
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**x_pad[1,1]**:[[ 0. 0.] [ 0. 0.] [ 0. 0.] [ 0. 0.] [ 0. 0.] [ 0. 0.] [ 0. 0.]]
3.2 - Single step of convolution
a = np.pad(a, ((0,0), (1,1), (0,0), (3,3), (0,0)), 'constant', constant_values = (..,..))
# GRADED FUNCTION: zero_pad
def zero_pad(X, pad):
"""
Pad with zeros all images of the datat 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) reprenting 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', constant_values=0)
### END CODE HERE ###
return X_pad
np.random.ed(1)
x = np.random.randn(4, 3, 3, 2)
x_pad = zero_pad(x, 2)
print ("x.shape =", x.shape)
print ("x_pad.shape =", x_pad.shape)
print ("x[1,1] =", x[1,1])
print ("x_pad[1,1] =", x_pad[1,1])
fig, axarr = plt.subplots(1, 2)
axarr[0].t_title('x')
axarr[0].imshow(x[0,:,:,0])
axarr[1].t_title('x_pad')
axarr[1].imshow(x_pad[0,:,:,0])
In this part, implement a single step of convolution, in which you apply the filter to a single position of the input. This will be ud to build a convolutional unit, which:
Takes an input volume
Applies a filter at every position of the input
Outputs another volume (usually of different size)
Figure 2 : Convolution operation
with a filter of 2x2 and a stride of 1 (stride = amount you move the window each time you slide)
In a computer vision application, each value in the matrix on the left corresponds to a single pixel value, and we convolve a 3x3 filter with the image by multiplying its values element-wi with the original matrix, then summing them up and adding a bias. In this first step of the exerci, you will implement a single step of convolution, corresponding to applying a filter to just one of the positions to get a single real-valued output.
Later in this notebook, you’ll apply this function to multiple positions of the input to implement the full convolutional operation.
Exerci: Implement conv_single_step(). .
测试：
输出：
Z = -6.99908945068
Expected Output:
**Z**-6.99908945068
3.3 - Convolutional Neural Networks - Forward pass
In the forward pass, you will take many filters and convolve them on the input. Each ‘convolution’ gives you a 2D matrix output. You will then stack the outputs to get a 3D volume:
height="498" width="510" src="urra-notebooks/ur/rfzkjgvqsrpkpyhrhhaeaq/notebooks/week1/images/conv_kiank.mp4"> # 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:
男士西服品牌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-wi product between a_slice and W. Do not add the bias yet.
s = a_slice_prev * W
# Sum over all entries of the volume s.
Z = np.sum(s)
# Add bias b to Z. Cast b to a float() so that Z results in a scalar value.
Z = Z + float(b)
# ## END CODE HERE ###
return Z
np.random.ed(1)
a_slice_prev = np.random.randn(4, 4, 3)
W = np.random.randn(4, 4, 3)
b = np.random.randn(1, 1, 1)
Z = conv_single_step(a_slice_prev, W, b)
print("Z =", Z)
Exerci: Implement the function below to convolve the filters W on an input activation A_prev. This function takes as input A_prev, the activations
output by the previous layer (for a batch of m inputs), F filters/weights denoted by W, and a bias vector denoted by b, where each filter has its own (single) bias. Finally you also have access to the hyperparameters dictionary which contains the stride and the padding.
Hint:
1. To lect a 2x2 slice at the upper left corner of a matrix “a_prev” (shape (5,5,3)), you would do:
a_slice_prev = a_prev[0:2,0:2,:]
This will be uful when you will define a_slice_prev below, using the start/end indexes you will define.
2. To define a_slice you will need to first define its corners vert_start, vert_end, horiz_start and horiz_end. This figure may be helpful for you to find how each of the corner can be defined using h, w, f and s in the code below.
Figure 3 : Definition of a slice using vertical and horizontal start/end (with a 2x2 filter)
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This figure shows only a single channel.
Reminder:
The formulas relating the output shape of the convolution to the input shape is:
For this exerci, we won’t worry about vectorization, and will just implement everything with for-loops.
