吴恩达 深度学习 编程作业（4-4）- Face Recognition for the Happy House & Art: Neural Style Transfer

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吴恩达 Coursera 课程 DeepLearning.ai 编程作业系列，本文为《卷积神经网络》部分的第四周“特殊应用：人脸识别和神经风格迁移”的课程作业。

另外，本节课程笔记在此：《 吴恩达Coursera深度学习课程 DeepLearning.ai 提炼笔记（4-4）– 特殊应用：人脸识别和神经风格迁移》，如有任何建议和问题，欢迎留言。

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Part 1：Face Recognition for the Happy House

Welcome to the first assignment of week 4! Here you will build a face recognition system. Many of the ideas presented here are from FaceNet. In lecture, we also talked about DeepFace.

Face recognition problems commonly fall into two categories:

• Face Verification - “is this the claimed person?”. For example, at some airports, you can pass through customs by letting a system scan your passport and then verifying that you (the person carrying the passport) are the correct person. A mobile phone that unlocks using your face is also using face verification. This is a 1:1 matching problem.
• Face Recognition - “who is this person?”. For example, the video lecture showed a face recognition video (https://www.youtube.com/watch?v=wr4rx0Spihs) of Baidu employees entering the office without needing to otherwise identify themselves. This is a 1:K matching problem.

FaceNet learns a neural network that encodes a face image into a vector of 128 numbers. By comparing two such vectors, you can then determine if two pictures are of the same person.

In this assignment, you will:
- Implement the triplet loss function
- Use a pretrained model to map face images into 128-dimensional encodings
- Use these encodings to perform face verification and face recognition

In this exercise, we will be using a pre-trained model which represents ConvNet activations using a “channels first” convention, as opposed to the “channels last” convention used in lecture and previous programming assignments. In other words, a batch of images will be of shape (m,nC,nH,nW) ( m , n C , n H , n W ) instead of (m,nH,nW,nC) ( m , n H , n W , n C ) . Both of these conventions have a reasonable amount of traction among open-source implementations; there isn’t a uniform standard yet within the deep learning community.

Let’s load the required packages.

from keras.models import Sequential
from keras.layers import Conv2D, ZeroPadding2D, Activation, Input, concatenate
from keras.models import Model
from keras.layers.normalization import BatchNormalization
from keras.layers.pooling import MaxPooling2D, AveragePooling2D
from keras.layers.merge import Concatenate
from keras.layers.core import Lambda, Flatten, Dense
from keras.initializers import glorot_uniform
from keras.engine.topology import Layer
from keras import backend as K
K.set_image_data_format('channels_first')
import cv2
import os
import numpy as np
from numpy import genfromtxt
import pandas as pd
import tensorflow as tf
from fr_utils import *
from inception_blocks_v2 import *

%matplotlib inline
%load_ext autoreload
%autoreload 2

np.set_printoptions(threshold=np.nan)

You can get the data sets and other functions from here：https://pan.baidu.com/s/1boZGtcj 。

0 - Naive Face Verification

In Face Verification, you’re given two images and you have to tell if they are of the same person. The simplest way to do this is to compare the two images pixel-by-pixel. If the distance between the raw images are less than a chosen threshold, it may be the same person!

Figure 1

Of course, this algorithm performs really poorly, since the pixel values change dramatically due to variations in lighting, orientation of the person’s face, even minor changes in head position, and so on.

You’ll see that rather than using the raw image, you can learn an encoding f(img) f ( i m g ) so that element-wise comparisons of this encoding gives more accurate judgements as to whether two pictures are of the same person.

1 - Encoding face images into a 128-dimensional vector

1.1 - Using an ConvNet to compute encodings

The FaceNet model takes a lot of data and a long time to train. So following common practice in applied deep learning settings, let’s just load weights that someone else has already trained. The network architecture follows the Inception model from Szegedy et al.. We have provided an inception network implementation. You can look in the file inception_blocks.py to see how it is implemented (do so by going to “File->Open…” at the top of the Jupyter notebook).

The key things you need to know are:

• This network uses 96x96 dimensional RGB images as its input. Specifically, inputs a face image (or batch of m m face images) as a tensor of shape (m,nC,nH,nW)=(m,3,96,96) ( m , n C , n H , n W ) = ( m , 3 , 96 , 96 )
• It outputs a matrix of shape (m,128) ( m , 128 ) that encodes each input face image into a 128-dimensional vector

Run the cell below to create the model for face images.

FRmodel = faceRecoModel(input_shape=(3, 96, 96))

print("Total Params:", FRmodel.count_params())

Total Params: 3743280

By using a 128-neuron fully connected layer as its last layer, the model ensures that the output is an encoding vector of size 128. You then use the encodings the compare two face images as follows:

Figure 2:
By computing a distance between two encodings and thresholding, you can determine if the two pictures represent the same person

So, an encoding is a good one if:
- The encodings of two images of the same person are quite similar to each other
- The encodings of two images of different persons are very different

The triplet loss function formalizes this, and tries to “push” the encodings of two images of the same person (Anchor and Positive) closer together, while “pulling” the encodings of two images of different persons (Anchor, Negative) further apart.

Figure 3:
In the next part, we will call the pictures from left to right: Anchor (A), Positive (P), Negative (N)

1.2 - The Triplet Loss

For an image x x , we denote its encoding f(x) f ( x ) , where f f is the function computed by the neural network.

Training will use triplets of images (A,P,N) ( A , P , N ) :

• A is an “Anchor” image–a picture of a person.
• P is a “Positive” image–a picture of the same person as the Anchor image.
• N is a “Negative” image–a picture of a different person than the Anchor image.

These triplets are picked from our training dataset. We will write (A(i),P(i),N(i)) ( A ( i ) , P ( i ) , N ( i ) ) to denote the i i -th training example.

You’d like to make sure that an image A(i) A ( i ) of an individual is closer to the Positive P(i) P ( i ) than to the Negative image N(i) N ( i ) ) by at least a margin α α :

∣∣f(A(i))−f(P(i))∣∣22+α<∣∣f(A(i))−f(N(i))∣∣22 ∣ ∣ f ( A ( i ) ) − f ( P ( i ) ) ∣ ∣ 2 2 + α <∣∣ f ( A ( i ) ) − f ( N ( i ) ) ∣ ∣ 2 2

You would thus like to minimize the following “triplet cost”:

J=∑i=1m[∣∣f(A(i))−f(P(i))∣∣22(1)−∣∣f(A(i))−f(N(i))∣∣22(2)+α]+(3) (3) J = ∑ i = 1 m [ ∣ ∣ f ( A ( i ) ) − f ( P ( i ) ) ∣ ∣ 2 2 ⏟ (1) − ∣ ∣ f ( A ( i ) ) − f ( N ( i ) ) ∣ ∣ 2 2 ⏟ (2) + α ] +

Here, we are using the notation “ [z]+ [ z ] + ” to denote max(z,0) m a x ( z , 0 ) .

Notes:
- The term (1) is the squared distance between the anchor “A” and the positive “P” for a given triplet; you want this to be small.
- The term (2) is the squared distance between the anchor “A” and the negative “N” for a given triplet, you want this to be relatively large, so it thus makes sense to have a minus sign preceding it.
- α α is called the margin. It is a hyperparameter that you should pick manually. We will use α=0.2 α = 0.2 .

Most implementations also normalize the encoding vectors to have norm equal one (i.e., ∣∣f(img)∣∣2 ∣ ∣ f ( i m g ) ∣ ∣ 2 =1); you won’t have to worry about that here.

Exercise: Implement the triplet loss as defined by formula (3). Here are the 4 steps:
1. Compute the distance between the encodings of “anchor” and “positive”: ∣∣f(A(i))−f(P(i))∣∣22 ∣ ∣ f ( A ( i ) ) − f ( P ( i ) ) ∣ ∣ 2 2
2. Compute the distance between the encodings of “anchor” and “negative”: ∣∣f(A(i))−f(N(i))∣∣22 ∣ ∣ f ( A ( i ) ) − f ( N ( i ) ) ∣ ∣ 2 2
3. Compute the formula per training example: ∣∣f(A(i))−f(P(i))∣−∣∣f(A(i))−f(N(i))∣∣22+α ∣ ∣ f ( A ( i ) ) − f ( P ( i ) ) ∣ − ∣∣ f ( A ( i ) ) − f ( N ( i ) ) ∣ ∣ 2 2 + α
3. Compute the full formula by taking the max with zero and summing over the training examples:

J=∑i=1m[∣∣f(A(i))−f(P(i))∣∣22−∣∣f(A(i))−f(N(i))∣∣22+α]+(3) (3) J = ∑ i = 1 m [ ∣ ∣ f ( A ( i ) ) − f ( P ( i ) ) ∣ ∣ 2 2 − ∣∣ f ( A ( i ) ) − f ( N ( i ) ) ∣ ∣ 2 2 + α ] +

Useful functions: tf.reduce_sum(), tf.square(), tf.subtract(), tf.add(), tf.maximum().
For steps 1 and 2, you will need to sum over the entries of ∣∣f(A(i))−f(P(i))∣∣22 ∣ ∣ f ( A ( i ) ) − f ( P ( i ) ) ∣ ∣ 2 2 and ∣∣f(A(i))−f(N(i))∣∣22 ∣ ∣ f ( A ( i ) ) − f ( N ( i ) ) ∣ ∣ 2 2 while for step 4 you will need to sum over the training examples.

# GRADED FUNCTION: triplet_loss

def triplet_loss(y_true, y_pred, alpha = 0.2):
    """
    Implementation of the triplet loss as defined by formula (3)

    Arguments:
    y_true -- true labels, required when you define a loss in Keras, you don't need it in this function.
    y_pred -- python list containing three objects:
            anchor -- the encodings for the anchor images, of shape (None, 128)
            positive -- the encodings for the positive images, of shape (None, 128)
            negative -- the encodings for the negative images, of shape (None, 128)

    Returns:
    loss -- real number, value of the loss
    """

    anchor, positive, negative = y_pred[0], y_pred[1], y_pred[2]

    ### START CODE HERE ### (≈ 4 lines)
    # Step 1: Compute the (encoding) distance between the anchor and the positive, you will need to sum over axis=-1
    pos_dist = tf.reduce_sum(tf.square(tf.subtract(anchor, positive)))
    # Step 2: Compute the (encoding) distance between the anchor and the negative, you will need to sum over axis=-1
    neg_dist = tf.reduce_sum(tf.square(tf.subtract(anchor, negative)))
    # Step 3: subtract the two previous distances and add alpha.
    basic_loss = tf.add(tf.subtract(pos_dist,neg_dist), alpha)
    # Step 4: Take the maximum of basic_loss and 0.0. Sum over the training examples.
    loss = tf.reduce_sum(tf.maximum(basic_loss, 0.))
    ### END CODE HERE ###

    return loss

with tf.Session() as test:
    tf.set_random_seed(1)
    y_true = (None, None, None)
    y_pred = (tf.random_normal([3, 128], mean=6, stddev=0.1, seed = 1),
              tf.random_normal([3, 128], mean=1, stddev=1, seed = 1),
              tf.random_normal([3, 128], mean=3, stddev=4, seed = 1))
    loss = triplet_loss(y_true, y_pred)

    print("loss = " + str(loss.eval()))

loss = 350.026

2 - Loading the trained model

FaceNet is trained by minimizing the triplet loss. But since training requires a lot of data and a lot of computation, we won’t train it from scratch here. Instead, we load a previously trained model. Load a model using the following cell; this might take a couple of minutes to run.

FRmodel.compile(optimizer = 'adam', loss = triplet_loss, metrics = ['accuracy'])
load_weights_from_FaceNet(FRmodel)

Here’re some examples of distances between the encodings between three individuals:

Figure 4:
Example of distance outputs between three individuals’ encodings

Let’s now use this model to perform face verification and face recognition!

3 - Applying the model

Back to the Happy House! Residents are living blissfully since you implemented happiness recognition for the house in an earlier assignment.

However, several issues keep coming up: The Happy House became so happy that every happy person in the neighborhood is coming to hang out in your living room. It is getting really crowded, which is having a negative impact on the residents of the house. All these random happy people are also eating all your food.

So, you decide to change the door entry policy, and not just let random happy people enter anymore, even if they are happy! Instead, you’d like to build a Face verification system so as to only let people from a specified list come in. To get admitted, each person has to swipe an ID card (identification card) to identify themselves at the door. The face recognition system then checks that they are who they claim to be.

3.1 - Face Verification

Let’s build a database containing one encoding vector for each person allowed to enter the happy house. To generate the encoding we use img_to_encoding(image_path, model) which basically runs the forward propagation of the model on the specified image.

Run the following code to build the database (represented as a python dictionary). This database maps each person’s name to a 128-dimensional encoding of their face.

database = {}
database["danielle"] = img_to_encoding("images/danielle.png", FRmodel)
database["younes"] = img_to_encoding("images/younes.jpg", FRmodel)
database["tian"] = img_to_encoding("images/tian.jpg", FRmodel)
database["andrew"] = img_to_encoding("images/andrew.jpg", FRmodel)
database["kian"] = img_to_encoding("images/kian.jpg", FRmodel)
database["dan"] = img_to_encoding("images/dan.jpg", FRmodel)
database["sebastiano"] = img_to_encoding("images/sebastiano.jpg", FRmodel)
database["bertrand"] = img_to_encoding("images/bertrand.jpg", FRmodel)
database["kevin"] = img_to_encoding("images/kevin.jpg", FRmodel)
database["felix"] = img_to_encoding("images/felix.jpg", FRmodel)
database["benoit"] = img_to_encoding("images/benoit.jpg", FRmodel)
database["arnaud"] = img_to_encoding("images/arnaud.jpg", FRmodel)

Now, when someone shows up at your front door and swipes their ID card (thus giving you their name), you can look up their encoding in the database, and use it to check if the person standing at the front door matches the name on the ID.

Exercise: Implement the verify() function which checks if the front-door camera picture (image_path) is actually the person called “identity”. You will have to go through the following steps:
1. Compute the encoding of the image from image_path
2. Compute the distance about this encoding and the encoding of the identity image stored in the database
3. Open the door if the distance is less than 0.7, else do not open.

As presented above, you should use the L2 distance (np.linalg.norm). (Note: In this implementation, compare the L2 distance, not the square of the L2 distance, to the threshold 0.7.)

# GRADED FUNCTION: verify

def verify(image_path, identity, database, model):
    """
    Function that verifies if the person on the "image_path" image is "identity".

    Arguments:
    image_path -- path to an image
    identity -- string, name of the person you'd like to verify the identity. Has to be a resident of the Happy house.
    database -- python dictionary mapping names of allowed people's names (strings) to their encodings (vectors).
    model -- your Inception model instance in Keras

    Returns:
    dist -- distance between the image_path and the image of "identity" in the database.
    door_open -- True, if the door should open. False otherwise.
    """

    ### START CODE HERE ###

    # Step 1: Compute the encoding for the image. Use img_to_encoding() see example above. (≈ 1 line)
    encoding = img_to_encoding(image_path, model)

    # Step 2: Compute distance with identity's image (≈ 1 line)
    dist = np.linalg.norm(encoding - database[identity])

    # Step 3: Open the door if dist < 0.7, else don't open (≈ 3 lines)
    if dist < 0.7:
        print("It's " + str(identity) + ", welcome home!")
        door_open = True
    else:
        print("It's not " + str(identity) + ", please go away")
        door_open = False

    ### END CODE HERE ###

    return dist, door_open

Younes is trying to enter the Happy House and the camera takes a picture of him (“images/camera_0.jpg”). Let’s run your verification algorithm on this picture:

verify("images/camera_0.jpg", "younes", database, FRmodel)

It's younes, welcome home!

(0.65939283, True)

Benoit, who broke the aquarium last weekend, has been banned from the house and removed from the database. He stole Kian’s ID card and came back to the house to try to present himself as Kian. The front-door camera took a picture of Benoit (“images/camera_2.jpg). Let’s run the verification algorithm to check if benoit can enter.

verify("images/camera_2.jpg", "kian", database, FRmodel)

It's not kian, please go away

(0.86224014, False)

3.2 - Face Recognition

Your face verification system is mostly working well. But since Kian got his ID card stolen, when he came back to the house that evening he couldn’t get in!

To reduce such shenanigans, you’d like to change your face verification system to a face recognition system. This way, no one has to carry an ID card anymore. An authorized person can just walk up to the house, and the front door will unlock for them!

You’ll implement a face recognition system that takes as input an image, and figures out if it is one of the authorized persons (and if so, who). Unlike the previous face verification system, we will no longer get a person’s name as another input.

Exercise: Implement who_is_it(). You will have to go through the following steps:
1. Compute the target encoding of the image from image_path
2. Find the encoding from the database that has smallest distance with the target encoding.
- Initialize the min_dist variable to a large enough number (100). It will help you keep track of what is the closest encoding to the input’s encoding.
- Loop over the database dictionary’s names and encodings. To loop use for (name, db_enc) in database.items().
- Compute L2 distance between the target “encoding” and the current “encoding” from the database.
- If this distance is less than the min_dist, then set min_dist to dist, and identity to name.

# GRADED FUNCTION: who_is_it

def who_is_it(image_path, database, model):
    """
    Implements face recognition for the happy house by finding who is the person on the image_path image.

    Arguments:
    image_path -- path to an image
    database -- database containing image encodings along with the name of the person on the image
    model -- your Inception model instance in Keras

    Returns:
    min_dist -- the minimum distance between image_path encoding and the encodings from the database
    identity -- string, the name prediction for the person on image_path
    """

    ### START CODE HERE ### 

    ## Step 1: Compute the target "encoding" for the image. Use img_to_encoding() see example above. ## (≈ 1 line)
    encoding = img_to_encoding(image_path, model)

    ## Step 2: Find the closest encoding ##

    # Initialize "min_dist" to a large value, say 100 (≈1 line)
    min_dist = 100

    # Loop over the database dictionary's names and encodings.
    for (name, db_enc) in database.items():

        # Compute L2 distance between the target "encoding" and the current "emb" from the database. (≈ 1 line)
        dist = np.linalg.norm(encoding - db_enc)

        # If this distance is less than the min_dist, then set min_dist to dist, and identity to name. (≈ 3 lines)
        if dist < min_dist:
            min_dist = dist
            identity = name

    ### END CODE HERE ###

    if min_dist > 0.7:
        print("Not in the database.")
    else:
        print ("it's " + str(identity) + ", the distance is " + str(min_dist))

    return min_dist, identity

Younes is at the front-door and the camera takes a picture of him (“images/camera_0.jpg”). Let’s see if your who_it_is() algorithm identifies Younes.

who_is_it("images/camera_0.jpg", database, FRmodel)

it's younes, the distance is 0.659393

(0.65939283, 'younes')

You can change “camera_0.jpg” (picture of younes) to “camera_1.jpg” (picture of bertrand) and see the result.

Your Happy House is running well. It only lets in authorized persons, and people don’t need to carry an ID card around anymore!

You’ve now seen how a state-of-the-art face recognition system works.

Although we won’t implement it here, here’re some ways to further improve the algorithm:
- Put more images of each person (under different lighting conditions, taken on different days, etc.) into the database. Then given a new image, compare the new face to multiple pictures of the person. This would increae accuracy.
- Crop the images to just contain the face, and less of the “border” region around the face. This preprocessing removes some of the irrelevant pixels around the face, and also makes the algorithm more robust.

What you should remember:
- Face verification solves an easier 1:1 matching problem; face recognition addresses a harder 1:K matching problem.
- The triplet loss is an effective loss function for training a neural network to learn an encoding of a face image.
- The same encoding can be used for verification and recognition. Measuring distances between two images’ encodings allows you to determine whether they are pictures of the same person.

Congrats on finishing this assignment!

References:

• Florian Schroff, Dmitry Kalenichenko, James Philbin (2015). FaceNet: A Unified Embedding for Face Recognition and Clustering
• Yaniv Taigman, Ming Yang, Marc’Aurelio Ranzato, Lior Wolf (2014). DeepFace: Closing the gap to human-level performance in face verification
• The pretrained model we use is inspired by Victor Sy Wang’s implementation and was loaded using his code: https://github.com/iwantooxxoox/Keras-OpenFace.
• Our implementation also took a lot of inspiration from the official FaceNet github repository: https://github.com/davidsandberg/facenet

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Part 2：Deep Learning & Art: Neural Style Transfer

Welcome to the second assignment of this week. In this assignment, you will learn about Neural Style Transfer. This algorithm was created by Gatys et al. (2015) (https://arxiv.org/abs/1508.06576).

In this assignment, you will:
- Implement the neural style transfer algorithm
- Generate novel artistic images using your algorithm

Most of the algorithms you’ve studied optimize a cost function to get a set of parameter values. In Neural Style Transfer, you’ll optimize a cost function to get pixel values!

import os
import sys
import scipy.io
import scipy.misc
import matplotlib.pyplot as plt
from matplotlib.pyplot import imshow
from PIL import Image
from nst_utils import *
import numpy as np
import tensorflow as tf

%matplotlib inline

You can get the data sets and other functions from here.

1 - Problem Statement

Neural Style Transfer (NST) is one of the most fun techniques in deep learning. As seen below, it merges two images, namely, a “content” image (C) and a “style” image (S), to create a “generated” image (G). The generated image G combines the “content” of the image C with the “style” of image S.

In this example, you are going to generate an image of the Louvre museum in Paris (content image C), mixed with a painting by Claude Monet, a leader of the impressionist movement (style image S).

Let’s see how you can do this.

2 - Transfer Learning

Neural Style Transfer (NST) uses a previously trained convolutional network, and builds on top of that. The idea of using a network trained on a different task and applying it to a new task is called transfer learning.

Following the original NST paper (https://arxiv.org/abs/1508.06576), we will use the VGG network. Specifically, we’ll use VGG-19, a 19-layer version of the VGG network. This model has already been trained on the very large ImageNet database, and thus has learned to recognize a variety of low level features (at the earlier layers) and high level features (at the deeper layers).

Run the following code to load parameters from the VGG model. This may take a few seconds.

model = load_vgg_model("pretrained-model/imagenet-vgg-verydeep-19.mat")
print(model)

{'input': , 'conv1_1': , 'conv1_2': , 'avgpool1': , 'conv2_1': , 'conv2_2': , 'avgpool2': , 'conv3_1': , 'conv3_2': , 'conv3_3': , 'conv3_4': , 'avgpool3': , 'conv4_1': , 'conv4_2': , 'conv4_3': , 'conv4_4': , 'avgpool4': , 'conv5_1': , 'conv5_2': , 'conv5_3': , 'conv5_4': , 'avgpool5': }

The model is stored in a python dictionary where each variable name is the key and the corresponding value is a tensor containing that variable’s value. To run an image through this network, you just have to feed the image to the model. In TensorFlow, you can do so using the tf.assign function. In particular, you will use the assign function like this:

model["input"].assign(image)

This assigns the image as an input to the model. After this, if you want to access the activations of a particular layer, say layer 4_2 when the network is run on this image, you would run a TensorFlow session on the correct tensor conv4_2, as follows:

sess.run(model["conv4_2"])

3 - Neural Style Transfer

We will build the NST algorithm in three steps:

• Build the content cost function Jcontent(C,G) J c o n t e n t ( C , G )
• Build the style cost function Jstyle(S,G) J s t y l e ( S , G )
• Put it together to get J(G)=αJcontent(C,G)+βJstyle(S,G) J ( G ) = α J c o n t e n t ( C , G ) + β J s t y l e ( S , G ) .

3.1 - Computing the content cost

In our running example, the content image C will be the picture of the Louvre Museum in Paris. Run the code below to see a picture of the Louvre.

content_image = scipy.misc.imread("images/louvre.jpg")
imshow(content_image)

The content image (C) shows the Louvre museum’s pyramid surrounded by old Paris buildings, against a sunny sky with a few clouds.

* 3.1.1 - How do you ensure the generated image G matches the content of the image C?*

As we saw in lecture, the earlier (shallower) layers of a ConvNet tend to detect lower-level features such as edges and simple textures, and the later (deeper) layers tend to detect higher-level features such as more complex textures as well as object classes.

We would like the “generated” image G to have similar content as the input image C. Suppose you have chosen some layer’s activations to represent the content of an image. In practice, you’ll get the most visually pleasing results if you choose a layer in the middle of the network–neither too shallow nor too deep. (After you have finished this exercise, feel free to come back and experiment with using different layers, to see how the results vary.)

So, suppose you have picked one particular hidden layer to use. Now, set the image C as the input to the pretrained VGG network, and run forward propagation. Let a(C) a ( C ) be the hidden layer activations in the layer you had chosen. (In lecture, we had written this as a[l](C) a [ l ] ( C ) , but here we’ll drop the superscript [l] [ l ] to simplify the notation.) This will be a nH×nW×nC n H × n W × n C tensor. Repeat this process with the image G: Set G as the input, and run forward progation. Let

a(G) a ( G )

be the corresponding hidden layer activation. We will define as the content cost function as:

Jcontent(C,G)=14×nH×nW×nC∑all entries(a(C)−a(G))2(1) (1) J c o n t e n t ( C , G ) = 1 4 × n H × n W × n C ∑ all entries ( a ( C ) − a ( G ) ) 2

Here, nH,nW n H , n W and nC n C are the height, width and number of channels of the hidden layer you have chosen, and appear in a normalization term in the cost. For clarity, note that a(C) a ( C ) and a(G) a ( G ) are the volumes corresponding to a hidden layer’s activations. In order to compute the cost Jcontent(C,G) J c o n t e n t ( C , G ) , it might also be convenient to unroll these 3D volumes into a 2D matrix, as shown below. (Technically this unrolling step isn’t needed to compute Jcontent J c o n t e n t , but it will be good practice for when you do need to carry out a similar operation later for computing the style const Jstyle J s t y l e .)

Exercise: Compute the “content cost” using TensorFlow.

Instructions: The 3 steps to implement this function are:
1. Retrieve dimensions from a_G:
- To retrieve dimensions from a tensor X, use: X.get_shape().as_list()
2. Unroll a_C and a_G as explained in the picture above
- If you are stuck, take a look at Hint1 and Hint2.
3. Compute the content cost:
- If you are stuck, take a look at Hint3, Hint4 and Hint5.

# GRADED FUNCTION: compute_content_cost

def compute_content_cost(a_C, a_G):
    """
    Computes the content cost

    Arguments:
    a_C -- tensor of dimension (1, n_H, n_W, n_C), hidden layer activations representing content of the image C 
    a_G -- tensor of dimension (1, n_H, n_W, n_C), hidden layer activations representing content of the image G

    Returns: 
    J_content -- scalar that you compute using equation 1 above.
    """

    ### START CODE HERE ###
    # Retrieve dimensions from a_G (≈1 line)
    m, n_H, n_W, n_C = a_G.get_shape().as_list()

    # Reshape a_C and a_G (≈2 lines)
    a_C_unrolled = tf.reshape(a_C, [n_H*n_W, n_C])
    a_G_unrolled = tf.reshape(a_G, [n_H*n_W, n_C])

    # compute the cost with tensorflow (≈1 line)
    J_content = 1./(4 * n_H * n_W * n_C)*tf.reduce_sum(tf.square(tf.subtract(a_C_unrolled, a_G_unrolled)))
    ### END CODE HERE ###

    return J_content

tf.reset_default_graph()

with tf.Session() as test:
    tf.set_random_seed(1)
    a_C = tf.random_normal([1, 4, 4, 3], mean=1, stddev=4)
    a_G = tf.random_normal([1, 4, 4, 3], mean=1, stddev=4)
    J_content = compute_content_cost(a_C, a_G)
    print("J_content = " + str(J_content.eval()))

J_content = 6.76559

What you should remember:
- The content cost takes a hidden layer activation of the neural network, and measures how different a(C) a ( C ) and a(G) a ( G ) are.
- When we minimize the content cost later, this will help make sure G G has similar content as C C .

3.2 - Computing the style cost

For our running example, we will use the following style image:

style_image = scipy.misc.imread("images/monet_800600.jpg")
imshow(style_image)

This painting was painted in the style of impressionism.

Lets see how you can now define a “style” const function Jstyle(S,G) J s t y l e ( S , G ) .

3.2.1 - Style matrix

The style matrix is also called a “Gram matrix.” In linear algebra, the Gram matrix G of a set of vectors (v1,…,vn) ( v 1 , … , v n ) is the matrix of dot products, whose entries are Gij=vTivj=np.dot(vi,vj) G i j = v i T v j = n p . d o t ( v i , v j ) . In other words, Gij G i j compares how similar vi v i is to vj v j : If they are highly similar, you would expect them to have a large dot product, and thus for Gij G i j to be large.

Note that there is an unfortunate collision in the variable names used here. We are following common terminology used in the literature, but G G is used to denote the Style matrix (or Gram matrix) as well as to denote the generated image G G . We will try to make sure which G G we are referring to is always clear from the context.

In NST, you can compute the Style matrix by multiplying the “unrolled” filter matrix with their transpose:

The result is a matrix of dimension (nC,nC) ( n C , n C ) where nC n C is the number of filters. The value Gij G i j measures how similar the activations of filter i i are to the activations of filter j j .

One important part of the gram matrix is that the diagonal elements such as Gii G i i also measures how active filter i i is. For example, suppose filter i i is detecting vertical textures in the image. Then Gii G i i measures how common vertical textures are in the image as a whole: If Gii G i i is large, this means that the image has a lot of vertical texture.

By capturing the prevalence of different types of features ( Gii G i i ), as well as how much different features occur together ( Gij G i j ), the Style matrix G G measures the style of an image.

Exercise:
Using TensorFlow, implement a function that computes the Gram matrix of a matrix A. The formula is: The gram matrix of A is GA=AAT G A = A A T . If you are stuck, take a look at Hint 1 and Hint 2.

# GRADED FUNCTION: gram_matrix

def gram_matrix(A):
    """
    Argument:
    A -- matrix of shape (n_C, n_H*n_W)

    Returns:
    GA -- Gram matrix of A, of shape (n_C, n_C)
    """

    ### START CODE HERE ### (≈1 line)
    GA = tf.matmul(A, tf.transpose(A))
    ### END CODE HERE ###

    return GA

tf.reset_default_graph()

with tf.Session() as test:
    tf.set_random_seed(1)
    A = tf.random_normal([3, 2*1], mean=1, stddev=4)
    GA = gram_matrix(A)

    print("GA = " + str(GA.eval()))

GA = [[  6.42230511  -4.42912197  -2.09668207]
 [ -4.42912197  19.46583748  19.56387138]
 [ -2.09668207  19.56387138  20.6864624 ]]

3.2.2 - Style cost

After generating the Style matrix (Gram matrix), your goal will be to minimize the distance between the Gram matrix of the “style” image S and that of the “generated” image G. For now, we are using only a single hidden layer a[l] a [ l ] , and the corresponding style cost for this layer is defined as:

J[l]style(S,G)=14×nC2×(nH×nW)2∑i=1nC∑j=1nC(G(S)ij−G(G)ij)2(2) (2) J s t y l e [ l ] ( S , G ) = 1 4 × n C 2 × ( n H × n W ) 2 ∑ i = 1 n C ∑ j = 1 n C ( G i j ( S ) − G i j ( G ) ) 2

where G(S) G ( S ) and G(G) G ( G ) are respectively the Gram matrices of the “style” image and the “generated” image, computed using the hidden layer activations for a particular hidden layer in the network.

Exercise: Compute the style cost for a single layer.

Instructions: The 3 steps to implement this function are:
1. Retrieve dimensions from the hidden layer activations a_G:
- To retrieve dimensions from a tensor X, use: X.get_shape().as_list()
2. Unroll the hidden layer activations a_S and a_G into 2D matrices, as explained in the picture above.
- You may find Hint1 and Hint2 useful.
3. Compute the Style matrix of the images S and G. (Use the function you had previously written.)
4. Compute the Style cost:
- You may find Hint3, Hint4 and Hint5 useful.

# GRADED FUNCTION: compute_layer_style_cost

def compute_layer_style_cost(a_S, a_G):
    """
    Arguments:
    a_S -- tensor of dimension (1, n_H, n_W, n_C), hidden layer activations representing style of the image S 
    a_G -- tensor of dimension (1, n_H, n_W, n_C), hidden layer activations representing style of the image G

    Returns: 
    J_style_layer -- tensor representing a scalar value, style cost defined above by equation (2)
    """

    ### START CODE HERE ###
    # Retrieve dimensions from a_G (≈1 line)
    m, n_H, n_W, n_C = a_G.get_shape().as_list()

    # Reshape the images to have them of shape (n_C, n_H*n_W) (≈2 lines)
    a_S = tf.transpose(tf.reshape(a_S, [n_H*n_W, n_C]))
    a_G = tf.transpose(tf.reshape(a_G, [n_H*n_W, n_C]))

    # Computing gram_matrices for both images S and G (≈2 lines)
    GS = gram_matrix(a_S)
    GG = gram_matrix(a_G)

    # Computing the loss (≈1 line)
    J_style_layer = 1./(4 * n_C * n_C * n_H * n_W * n_H * n_W) * tf.reduce_sum(tf.square(tf.subtract(GS, GG)))

    ### END CODE HERE ###

    return J_style_layer

tf.reset_default_graph()

with tf.Session() as test:
    tf.set_random_seed(1)
    a_S = tf.random_normal([1, 4, 4, 3], mean=1, stddev=4)
    a_G = tf.random_normal([1, 4, 4, 3], mean=1, stddev=4)
    J_style_layer = compute_layer_style_cost(a_S, a_G)

    print("J_style_layer = " + str(J_style_layer.eval()))

J_style_layer = 9.19028

3.2.3 Style Weights

So far you have captured the style from only one layer. We’ll get better results if we “merge” style costs from several different layers. After completing this exercise, feel free to come back and experiment with different weights to see how it changes the generated image G G . But for now, this is a pretty reasonable default:

STYLE_LAYERS = [
    ('conv1_1', 0.2),
    ('conv2_1', 0.2),
    ('conv3_1', 0.2),
    ('conv4_1', 0.2),
    ('conv5_1', 0.2)]

You can combine the style costs for different layers as follows:

Jstyle(S,G)=∑lλ[l]J[l]style(S,G) J s t y l e ( S , G ) = ∑ l λ [ l ] J s t y l e [ l ] ( S , G )

where the values for λ[l] λ [ l ] are given in STYLE_LAYERS.

We’ve implemented a compute_style_cost(…) function. It simply calls your compute_layer_style_cost(...) several times, and weights their results using the values in STYLE_LAYERS. Read over it to make sure you understand what it’s doing.

def compute_style_cost(model, STYLE_LAYERS):
    """
    Computes the overall style cost from several chosen layers

    Arguments:
    model -- our tensorflow model
    STYLE_LAYERS -- A python list containing:
                        - the names of the layers we would like to extract style from
                        - a coefficient for each of them

    Returns: 
    J_style -- tensor representing a scalar value, style cost defined above by equation (2)
    """

    # initialize the overall style cost
    J_style = 0

    for layer_name, coeff in STYLE_LAYERS:

        # Select the output tensor of the currently selected layer
        out = model[layer_name]

        # Set a_S to be the hidden layer activation from the layer we have selected, by running the session on out
        a_S = sess.run(out)

        # Set a_G to be the hidden layer activation from same layer. Here, a_G references model[layer_name] 
        # and isn't evaluated yet. Later in the code, we'll assign the image G as the model input, so that
        # when we run the session, this will be the activations drawn from the appropriate layer, with G as input.
        a_G = out

        # Compute style_cost for the current layer
        J_style_layer = compute_layer_style_cost(a_S, a_G)

        # Add coeff * J_style_layer of this layer to overall style cost
        J_style += coeff * J_style_layer

    return J_style

Note: In the inner-loop of the for-loop above, a_G is a tensor and hasn’t been evaluated yet. It will be evaluated and updated at each iteration when we run the TensorFlow graph in model_nn() below.

What you should remember:
- The style of an image can be represented using the Gram matrix of a hidden layer’s activations. However, we get even better results combining this representation from multiple different layers. This is in contrast to the content representation, where usually using just a single hidden layer is sufficient.
- Minimizing the style cost will cause the image G G to follow the style of the image S S .

3.3 - Defining the total cost to optimize

Finally, let’s create a cost function that minimizes both the style and the content cost. The formula is:

J(G)=αJcontent(C,G)+βJstyle(S,G) J ( G ) = α J c o n t e n t ( C , G ) + β J s t y l e ( S , G )

Exercise: Implement the total cost function which includes both the content cost and the style cost.

# GRADED FUNCTION: total_cost

def total_cost(J_content, J_style, alpha = 10, beta = 40):
    """
    Computes the total cost function

    Arguments:
    J_content -- content cost coded above
    J_style -- style cost coded above
    alpha -- hyperparameter weighting the importance of the content cost
    beta -- hyperparameter weighting the importance of the style cost

    Returns:
    J -- total cost as defined by the formula above.
    """

    ### START CODE HERE ### (≈1 line)
    J = alpha * J_content + beta * J_style
    ### END CODE HERE ###

    return J

tf.reset_default_graph()

with tf.Session() as test:
    np.random.seed(3)
    J_content = np.random.randn()    
    J_style = np.random.randn()
    J = total_cost(J_content, J_style)
    print("J = " + str(J))

J = 35.34667875478276

What you should remember:
- The total cost is a linear combination of the content cost Jcontent(C,G) J c o n t e n t ( C , G ) and the style cost Jstyle(S,G) J s t y l e ( S , G )
- α α and β β are hyperparameters that control the relative weighting between content and style

4 - Solving the optimization problem

Finally, let’s put everything together to implement Neural Style Transfer!

Here’s what the program will have to do:

1. Create an Interactive Session
2. Load the content image
3. Load the style image
4. Randomly initialize the image to be generated
5. Load the VGG16 model
6. Build the TensorFlow graph:
   
   • Run the content image through the VGG16 model and compute the content cost
   • Run the style image through the VGG16 model and compute the style cost
   • Compute the total cost
   • Define the optimizer and the learning rate
7. Initialize the TensorFlow graph and run it for a large number of iterations, updating the generated image at every step.

Lets go through the individual steps in detail.

You’ve previously implemented the overall cost J(G) J ( G ) . We’ll now set up TensorFlow to optimize this with respect to G G . To do so, your program has to reset the graph and use an “Interactive Session“. Unlike a regular session, the “Interactive Session” installs itself as the default session to build a graph. This allows you to run variables without constantly needing to refer to the session object, which simplifies the code.

Lets start the interactive session.

# Reset the graph
tf.reset_default_graph()

# Start interactive session
sess = tf.InteractiveSession()

Let’s load, reshape, and normalize our “content” image (the Louvre museum picture):

content_image = scipy.misc.imread("images/louvre_small.jpg")
content_image = reshape_and_normalize_image(content_image)

Let’s load, reshape and normalize our “style” image (Claude Monet’s painting):

style_image = scipy.misc.imread("images/monet.jpg")
style_image = reshape_and_normalize_image(style_image)

Now, we initialize the “generated” image as a noisy image created from the content_image. By initializing the pixels of the generated image to be mostly noise but still slightly correlated with the content image, this will help the content of the “generated” image more rapidly match the content of the “content” image. (Feel free to look in nst_utils.py to see the details of generate_noise_image(...); to do so, click “File–>Open…” at the upper-left corner of this Jupyter notebook.)

generated_image = generate_noise_image(content_image)
imshow(generated_image[0])

Next, as explained in part (2), let’s load the VGG16 model.

model = load_vgg_model("pretrained-model/imagenet-vgg-verydeep-19.mat")

To get the program to compute the content cost, we will now assign a_C and a_G to be the appropriate hidden layer activations. We will use layer conv4_2 to compute the content cost. The code below does the following:

1. Assign the content image to be the input to the VGG model.
2. Set a_C to be the tensor giving the hidden layer activation for layer “conv4_2”.
3. Set a_G to be the tensor giving the hidden layer activation for the same layer.
4. Compute the content cost using a_C and a_G.

# Assign the content image to be the input of the VGG model.  
sess.run(model['input'].assign(content_image))

# Select the output tensor of layer conv4_2
out = model['conv4_2']

# Set a_C to be the hidden layer activation from the layer we have selected
a_C = sess.run(out)

# Set a_G to be the hidden layer activation from same layer. Here, a_G references model['conv4_2'] 
# and isn't evaluated yet. Later in the code, we'll assign the image G as the model input, so that
# when we run the session, this will be the activations drawn from the appropriate layer, with G as input.
a_G = out

# Compute the content cost
J_content = compute_content_cost(a_C, a_G)

Note: At this point, a_G is a tensor and hasn’t been evaluated. It will be evaluated and updated at each iteration when we run the Tensorflow graph in model_nn() below.

# Assign the input of the model to be the "style" image 
sess.run(model['input'].assign(style_image))

# Compute the style cost
J_style = compute_style_cost(model, STYLE_LAYERS)

Exercise: Now that you have J_content and J_style, compute the total cost J by calling total_cost(). Use alpha = 10 and beta = 40.

### START CODE HERE ### (1 line)
J = total_cost(J_content, J_style, 10, 40)
### END CODE HERE ###

You’d previously learned how to set up the Adam optimizer in TensorFlow. Lets do that here, using a learning rate of 2.0. See reference

# define optimizer (1 line)
optimizer = tf.train.AdamOptimizer(2.0)

# define train_step (1 line)
train_step = optimizer.minimize(J)

Exercise: Implement the model_nn() function which initializes the variables of the tensorflow graph, assigns the input image (initial generated image) as the input of the VGG16 model and runs the train_step for a large number of steps.

def model_nn(sess, input_image, num_iterations = 200):

    # Initialize global variables (you need to run the session on the initializer)
    ### START CODE HERE ### (1 line)
    sess.run(tf.global_variables_initializer())
    ### END CODE HERE ###

    # Run the noisy input image (initial generated image) through the model. Use assign().
    ### START CODE HERE ### (1 line)
    sess.run(model["input"].assign(input_image))
    ### END CODE HERE ###

    for i in range(num_iterations):

        # Run the session on the train_step to minimize the total cost
        ### START CODE HERE ### (1 line)
        sess.run(train_step)
        ### END CODE HERE ###

        # Compute the generated image by running the session on the current model['input']
        ### START CODE HERE ### (1 line)
        generated_image =sess.run(model["input"])
        ### END CODE HERE ###

        # Print every 20 iteration.
        if i%20 == 0:
            Jt, Jc, Js = sess.run([J, J_content, J_style])
            print("Iteration " + str(i) + " :")
            print("total cost = " + str(Jt))
            print("content cost = " + str(Jc))
            print("style cost = " + str(Js))

            # save current generated image in the "/output" directory
            save_image("output/" + str(i) + ".png", generated_image)

    # save last generated image
    save_image('output/generated_image.jpg', generated_image)

    return generated_image

Run the following cell to generate an artistic image. It should take about 3min on CPU for every 20 iterations but you start observing attractive results after ≈140 iterations. Neural Style Transfer is generally trained using GPUs.

model_nn(sess, generated_image)

Iteration 0 :
total cost = 5.05035e+09
content cost = 7877.67
style cost = 1.26257e+08
Iteration 20 :
total cost = 9.43272e+08
content cost = 15187.1
style cost = 2.3578e+07
Iteration 40 :
total cost = 4.84905e+08
content cost = 16785.1
style cost = 1.21184e+07

You’re done! After running this, in the upper bar of the notebook click on “File” and then “Open”. Go to the “/output” directory to see all the saved images. Open “generated_image” to see the generated image! :)

You should see something the image presented below on the right:

We didn’t want you to wait too long to see an initial result, and so had set the hyperparameters accordingly. To get the best looking results, running the optimization algorithm longer (and perhaps with a smaller learning rate) might work better. After completing and submitting this assignment, we encourage you to come back and play more with this notebook, and see if you can generate even better looking images.

Here are few other examples:

• The beautiful ruins of the ancient city of Persepolis (Iran) with the style of Van Gogh (The Starry Night)
  

  

• The tomb of Cyrus the great in Pasargadae with the style of a Ceramic Kashi from Ispahan.
  

  

• A scientific study of a turbulent fluid with the style of a abstract blue fluid painting.
  

  

5 - Test with your own image (Optional/Ungraded)

Finally, you can also rerun the algorithm on your own images!

To do so, go back to part 4 and change the content image and style image with your own pictures. In detail, here’s what you should do:

1. Click on “File -> Open” in the upper tab of the notebook
2. Go to “/images” and upload your images (requirement: (WIDTH = 300, HEIGHT = 225)), rename them “my_content.png” and “my_style.png” for example.
3. Change the code in part (3.4) from :

content_image = scipy.misc.imread("images/louvre.jpg")
style_image = scipy.misc.imread("images/claude-monet.jpg")

to:

content_image = scipy.misc.imread("images/my_content.jpg")
style_image = scipy.misc.imread("images/my_style.jpg")

1. Rerun the cells (you may need to restart the Kernel in the upper tab of the notebook).

You can also tune your hyperparameters:
- Which layers are responsible for representing the style? STYLE_LAYERS
- How many iterations do you want to run the algorithm? num_iterations
- What is the relative weighting between content and style? alpha/beta

6 - Conclusion

Great job on completing this assignment! You are now able to use Neural Style Transfer to generate artistic images. This is also your first time building a model in which the optimization algorithm updates the pixel values rather than the neural network’s parameters. Deep learning has many different types of models and this is only one of them!

What you should remember:
- Neural Style Transfer is an algorithm that given a content image C and a style image S can generate an artistic image
- It uses representations (hidden layer activations) based on a pretrained ConvNet.
- The content cost function is computed using one hidden layer’s activations.
- The style cost function for one layer is computed using the Gram matrix of that layer’s activations. The overall style cost function is obtained using several hidden layers.
- Optimizing the total cost function results in synthesizing new images.

This was the final programming exercise of this course. Congratulations–you’ve finished all the programming exercises of this course on Convolutional Networks! We hope to also see you in Course 5, on Sequence models!

References:

The Neural Style Transfer algorithm was due to Gatys et al. (2015). Harish Narayanan and Github user “log0” also have highly readable write-ups from which we drew inspiration. The pre-trained network used in this implementation is a VGG network, which is due to Simonyan and Zisserman (2015). Pre-trained weights were from the work of the MathConvNet team.

• Leon A. Gatys, Alexander S. Ecker, Matthias Bethge, (2015). A Neural Algorithm of Artistic Style (https://arxiv.org/abs/1508.06576)
• Harish Narayanan, Convolutional neural networks for artistic style transfer. https://harishnarayanan.org/writing/artistic-style-transfer/
• Log0, TensorFlow Implementation of “A Neural Algorithm of Artistic Style”. http://www.chioka.in/tensorflow-implementation-neural-algorithm-of-artistic-style
• Karen Simonyan and Andrew Zisserman (2015). Very deep convolutional networks for large-scale image recognition (https://arxiv.org/pdf/1409.1556.pdf)
• MatConvNet. http://www.vlfeat.org/matconvnet/pretrained/

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