Learning Distributed Word  Representations with Neural Network: an implementation in Octave

In this article, the problem of learning word representations with neural network from scratch is going to be described. This problem appeared as an assignment in the Coursera course Neural Networks for Machine Learning, taught by  Prof.  Geoffrey Hinton from the University of Toronto in 2012.  This problem also appeared as an assignment in this course from the same university.  The problem description is taken from the assignment pdf.

 

Problem Statement

In this article we will design a neural net language model. The model will learn to
predict the next word given the previous three words. The network looks like the following:

f1.png

  • The dataset provided consists of 4-grams (A 4-gram is a sequence of 4 adjacent words in a sentence). These 4-grams were extracted from a large collection of text.
  • The 4-grams are chosen so that all the words involved come from a small
    vocabulary of 250 words. Note that for the purposes of this assignment special characters such as commas, full-stops, parentheses etc. are also considered words.
  • Few of the 250 words in the vocabulary are shown as the output from the matlab / octave code below.

load data.mat
data.vocab
ans =
{
[1,1] = all
[1,2] = set
[1,3] = just
[1,4] = show
[1,5] = being
[1,6] = money
[1,7] = over
[1,8] = both
[1,9] = years
[1,10] = four
[1,11] = through
[1,12] = during
[1,13] = go
[1,14] = still
[1,15] = children
[1,16] = before
[1,17] = police
[1,18] = office
[1,19] = million
[1,20] = also
.
.
[1,246] = so
[1,247] = time
[1,248] = five
[1,249] = the
[1,250] = left
}

  • The training set consists of 372,550 4-grams. The validation and test sets have 46,568 4-grams each.
  • Let’s first look at the raw sentences file, first few lines of the file is shown below. It contains the raw sentences from which these 4-grams were extracted. It can be seen that the kind of sentences we are dealing with here are fairly simple ones.

The raw sentences file: first few lines

No , he says now .
And what did he do ?
The money ‘s there .
That was less than a year ago .
But he made only the first .
There ‘s still time for them to do it .
But he should nt have .
They have to come down to the people .
I do nt know where that is .
No , I would nt .
Who Will It Be ?
And no , I was not the one .
You could do a Where are they now ?
There ‘s no place like it that I know of .
Be here now , and so on .
It ‘s not you or him , it ‘s both of you .
So it ‘s not going to get in my way .
When it ‘s time to go , it ‘s time to go .
No one ‘s going to do any of it for us .
Well , I want more .
Will they make it ?
Who to take into school or not take into school ?
But it ‘s about to get one just the same .
We all have it .

  • The training data extracted from this raw text is a matrix of 372550 X 4. This means there are 372550 training cases and 4 words (corresponding to each 4-gram) per training case.
  • Each entry is an integer that is the index of a word in the vocabulary. So each row represents a sequence of 4 words. The following octave / matlab code shows how the training dataset looks like.

 


load data.mat
[train_x, train_t, valid_x, valid_t, test_x, test_t, vocab] = load_data(100);

% 3-gram features for a training data-tuple
train_x(:,13,14)
%ans =
%46
%58
%32
data.vocab{train_x(:,13,14)}
%ans = now
%ans = where
%ans = do

% target for the same data tuple from training dataset
train_t(:,13,14)
%ans = 91
data.vocab{train_t(:,13,14)}
%ans = we

  • The validation and test data are also similar. They contain 46,568 4-grams each.
  • Before starting the training, all three need to be separated into inputs and targets and the training set needs to be split into mini-batches.
  • The data needs to get loaded and then separated into inputs and target. After that,  mini-batches of size 100 for the training set are created.
  • First we need to train the model for one epoch (one pass through the training set using forward propagation). Once implemented the cross-entropy loss will start decreasing.
  • At this point, we can try changing the hyper-parameters (number of epochs, number of hidden units, learning rates, momentum, etc) to see what effect that has on the training and validation cross entropy.
  • The training method will output a ‘model’ (weight matrices, biases for each layer in the network).

 

Description of the Network

f1

  • As shown above, the network consists of an input layer, embedding layer, hidden layer and output layer.
  • The input layer consists of three word indices. The same ‘word_embedding_weights’ are used to map each index to a distributed feature representation. These mapped features constitute the embedding layer. More details can be found here.
  • This layer is connected to the hidden layer, which in turn is connected to the output layer.
  • The output layer is a softmax over the 250 words.
  • The training consists of two steps:  (1) forward propagation: computes (predicts) the output probabilities of the words in the vocabulary as the next word given a 3-gram as input. (2) back-propagation: propagates the error in prediction from the output layer to the input layer through the hidden layers.

 


Forward Propagation


  • The forward propagation is pretty straight-forward and can be implemented as shown in the following code:
    
    function [embedding_layer_state, hidden_layer_state, output_layer_state] = ...
     fprop(input_batch, word_embedding_weights, embed_to_hid_weights,...
     hid_to_output_weights, hid_bias, output_bias)
    % This method forward propagates through a neural network.
    % Inputs:
    % input_batch: The input data as a matrix of size numwords X batchsize where,
    % numwords is the number of words, batchsize is the number of data points.
    % So, if input_batch(i, j) = k then the ith word in data point j is word
    % index k of the vocabulary.
    %
    % word_embedding_weights: Word embedding as a matrix of size
    % vocab_size X numhid1, where vocab_size is the size of the vocabulary
    % numhid1 is the dimensionality of the embedding space.
    %
    % embed_to_hid_weights: Weights between the word embedding layer and hidden
    % layer as a matrix of soze numhid1*numwords X numhid2, numhid2 is the
    % number of hidden units.
    %
    % hid_to_output_weights: Weights between the hidden layer and output softmax
    % unit as a matrix of size numhid2 X vocab_size
    %
    % hid_bias: Bias of the hidden layer as a matrix of size numhid2 X 1.
    %
    % output_bias: Bias of the output layer as a matrix of size vocab_size X 1.
    %
    % Outputs:
    % embedding_layer_state: State of units in the embedding layer as a matrix of
    % size numhid1*numwords X batchsize
    %
    % hidden_layer_state: State of units in the hidden layer as a matrix of size
    % numhid2 X batchsize
    %
    % output_layer_state: State of units in the output layer as a matrix of size
    % vocab_size X batchsize
    %
    
    [numwords, batchsize] = size(input_batch);
    [vocab_size, numhid1] = size(word_embedding_weights);
    numhid2 = size(embed_to_hid_weights, 2);
    
    %% COMPUTE STATE OF WORD EMBEDDING LAYER.
    % Look up the inputs word indices in the word_embedding_weights matrix.
    embedding_layer_state = reshape(...
     word_embedding_weights(reshape(input_batch, 1, []),:)',...
     numhid1 * numwords, []);
    
    %% COMPUTE STATE OF HIDDEN LAYER.
    % Compute inputs to hidden units.
    inputs_to_hidden_units = embed_to_hid_weights' * embedding_layer_state + ...
     repmat(hid_bias, 1, batchsize);
    
    % Apply logistic activation function.
    hidden_layer_state = 1 ./ (1 + exp(-inputs_to_hidden_units)); %zeros(numhid2, batchsize);
    
    %% COMPUTE STATE OF OUTPUT LAYER.
    % Compute inputs to softmax.
    inputs_to_softmax = hid_to_output_weights' * hidden_layer_state + repmat(output_bias, 1, batchsize); %zeros(vocab_size, batchsize);
    
    % Subtract maximum.
    % Remember that adding or subtracting the same constant from each input to a
    % softmax unit does not affect the outputs. Here we are subtracting maximum to
    % make all inputs <= 0. This prevents overflows when computing their
    % exponents.
    inputs_to_softmax = inputs_to_softmax...
     - repmat(max(inputs_to_softmax), vocab_size, 1);
    
    % Compute exp.
    output_layer_state = exp(inputs_to_softmax);
    
    % Normalize to get probability distribution.
    output_layer_state = output_layer_state ./ repmat(...
     sum(output_layer_state, 1), vocab_size, 1);
    
    

     


Back-Propagation


 

  •  The back-propagation is much more involved. The math for the back-propagation is shown below for a simple 2-layer network, taken from this lecture note.

 

backprop_softmax.png

  • As the model trains it prints out some numbers that tell how well the training is going.
  • The model shows the average per-case cross entropy (CE) obtained on the training set. The average CE is computed every 100 mini-batches. The average CE over the entire training set is reported at the end of every epoch.
  • After every 1000 mini-batches of training, the model is run on the validation set. Recall, that the validation set consists of data that is not used for training. It is used to see how well the model does on unseen data. The cross entropy on validation set is reported.
  • The validation error is expected to decrease with increasing epochs till the model starts getting over-fitted with the training data. Hence, the training is stopped immediately when the validation error starts increasing to prevent over-fitting.
  • At the end of training, the model is run both on the validation set and on the test set and the cross entropy on both is reported.

 


Some Applications



1. Predict next word


  • Once the model has been trained, it can be used to produce some predictions for the next word given a set of 3 previous words.
  • The next example shows when the model is given a 3-gram ‘life’, ‘in’, ‘new’ as input and asked to predict the next word, it predicts the word ‘york’ to be most likely word with the highest (~0.94) probability and the words such as ‘year’, ‘life’ and ‘world’ with low probabilities.
  • It also shows how the forward propagation is used to compute the prediction: the distribution for the next word given the 3-gram. First the words are projected into the embedding space, flattened and then the weight-matrices are multiplied sequentially followed by application of the softmax function to compute the likelihood of each word being a next word following the 3-gram.

 

fp

 


2. Generate stylized pseudo-random text


Here are the steps to generate a piece of pseudo-random  text:

  1. Given 3 words to start from, initialize the text with those 3 words.
  2. Next, the model is asked to predict k most probable words as a candidate word following the last 3 words.
  3. Choose one of the most probable words predicted randomly and insert it at the end of the text.
  4. Repeat steps 2-3 to generate more words otherwise stop.

Here is the code that by default generates top 3 predictions for each 3-gram sliding window and chooses one of predicted words tandomly:


function gen_rand_text(words, model, k=3)

probs = [];
i = 4;
while (i < 20 || word != '.')
[word, prob] = predict_next_word(words{i-3}, words{i-2}, words{i-1}, model, k);                   words = {words{:}, word};
probs = [probs; prob];
i = i + 1;
end
fprintf(1, "%s ", words{:}) ;
fprintf(1, '\n');
fprintf(1, "%.2f ", round(probs.*100)./100) ;
fprintf(1, '\n');

end

Starting with the words  'i was going‘, here are some texts that were generated using the model:

f3.png

Starting with the words  ‘life in new‘, here is a piece of text that was generated using the model:

f4.png


3. Find nearest words


  •  The word embedding weight matrix can be used to represent a word in the embedding space and then the distances from every other word in the vocabulary are computed in this word representation space. Then the closest words are returned.
  • As can be seen from the following animation examples, the semantically closer words are chosen mostly as the nearest words given a word. Also, higher the number of epochs, better the ordering of the words in terms of semantic similarity.
  • For example, the closest semantically similar word (i.e. with least distance) for the word ‘between’ is the word ‘among‘, whereas the nearest words for ‘day’ are ‘year’ and ‘week’. Also, the word ‘and’ is nearer to the word ‘but’ than the word ‘or’.

 

betweencangamehislifelittlemanmoneynewofficeschooltwo

day.gif



4. Visualization in 2-dimension with t-SNE


  •  In all the above examples, the dimension of the word embedding space was 50. Using t-SNE plot (t-distributed stochastic nearest neighbor embedding by Laurens van der Maaten) the words can be projected into a 2 dimensional space and visualized, by keeping the (semantically) nearer words in the distributed representation space nearer in the projected space.
  • As can be seen from the following figures, the semantically close words (highlighted with ellipses) are placed near to each other in the visualization, since in the distributed representation space they were close to each other.
  • Also, the next animation visualizes how the neighborhood of each word changes with training epochs (the model is trained up to 10 epochs).

 

img_5_3700img_99_3700tsne



5. Solving Word-Analogy Problem


  •  with the distributed representation: In this type of problems 2 words (w1, w2) from the vocabulary are given where the first is relate to the second one with some semantic relation.  Now, a third word (w3, from the vocabulary) is given and a fourth word that has similar semantic relation with the third word is to be found from the vocabulary.
  • The following figure shows the word analogy problem and a possible solution using an exhaustive search in the embedding space for a word that has the distance (with the third word) that is closest to the distance in between the first and second word in the representation space.

f2.png

  • The next code shows results of a few word-analogy example problems and the solutions found using the distributed representation space. As can be seen, despite the fact that the dataset was quite small and there were only 250 words in the vocabulary, the algorithm worked quite well to find the answers for the examples shown.
    
    analogy('year', 'years', 'day', model); % singular-plural relation
    %year:years::day:days
    %dist_E('year','years')=1.119368, dist_E('day', 'days')= 1.169186
    
    analogy('on', 'off', 'new', model) % antonyms relation
    %on:off::new:old
    %dist_E('on','off')=2.013958, dist_E('new','old')=2.265665
    
    analogy('use', 'used', 'do', model) % present-past relation
    %use:used::do:did
    %dist_E('use','used')=2.556175, dist_E('do','did')=2.456098
    
    analogy('he', 'his', 'they', model) % pronoun-relations
    %he:his::they:their
    %dist_E('he','his')=3.824808, dist_E('they','their')=3.825453
    
    analogy('today', 'yesterday', 'now', model)
    %today:yesterday::now:then
    %dist_E('today','yesterday')=1.045192, dist_E('now','then')=1.220935
    

     


Model Selection


  • Now the model is trained 4 times by changing the values of the hyper-parameters d (dimension of the representation space) and h (the number of nodes in the hidden layer), by trying all possible combinations d=8, d=32 and h=64, h=256.
  • The following figures show the cross-entropy errors on the training and validation sets for the models.As can be seen from the following figures,  the models with hidden layer size 64 are trained till 3 epochs, whereas the models with hidden layer size 256 are trained for 4 epochs (since higher numbers of parameters to train).
  • The least validation error (also least training error) is obtained for the model with d=32 and h=256, so this is the best model.

 

training_CEvalidation_CE

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Autonomous Driving – Car detection with YOLO Model with Keras in Python

In this article, object detection using the very powerful YOLO model will be described, particularly in the context of car detection for autonomous driving. This problem appeared as an assignment in the coursera course Convolution Networks which is a part of the Deep Learning Specialization (taught by Prof. Andrew Ng.,  from Stanford and deeplearning.ai, the lecture videos corresponding to the YOLO algorithm can be found here).  The problem description is taken straightaway from the assignment.

Given a set of images (a car detection dataset), the goal is to detect objects (cars) in those images using a pre-trained YOLO (You Only Look Once) model, with bounding boxes. Many of the ideas are from the two original YOLO papers: Redmon et al., 2016  and Redmon and Farhadi, 2016 .

Some Theory

Let’s first clear the concepts regarding classification, localization, detection and how the object detection problem can be transformed to supervised machine learning problem and subsequently can be solved using a deep convolution neural network. As can be seen from the next figure,

  • Image classification with localization aims to find the location of an object in an image by not only classifying the image (e.g., a binary classification problem: whether there is a car in an image or not), but also finding a bounding box around the object, if one found.
  • Detection goes a level further by aiming to identify multiple instances of same/ different types of objects, by marking their locations (the localization problem usually tries to find a single object location).
  • The localization problem can be converted to a supervised machine learning multi-class classification problem in the following way: in addition to the class label of the object to be identified, the output vector corresponding to an input training image must also contain the location (bounding box coordinates relative to image size) of the object.
  • A typical output data vector will contain 8 entries for a 4-class classification, as shown in the next figure, the first entry will correspond to whether or not an object of any from the 3 classes of objects. In case one is present in an image, the next 4 entries will define the bounding box containing the object, followed by 3 binary values for the 3 class labels indicating the class of the object. In case none of the objects are present, the first entry will be 0 and the others will be ignored.

 

f1.png

  • Now moving from localization to detection, one can proceed in two steps as shown below in the next figure: first use small tightly cropped images to train a convolution neural net for image classification and then use sliding windows of different window sizes (smaller to larger) to classify a test image within that window using the convnet learnt and run the windows sequentially through the entire image, but it’s infeasibly slow computationally.
  • However, as shown in the next figure, the convolutional implementation of the sliding windows by replacing the fully-connected layers by 1×1 filters makes it possible to simultaneously classify the image-subset inside all possible sliding windows parallelly, making it much more efficient computationally.

 

f2.png

  • The convolutional sliding windows, although computationally much more efficient, still has the problem of detecting the accurate bounding boxes, since the boxes don’t align with the sliding windows and the object shapes also tend to be different.
  • YOLO algorithm overcomes this limitation by dividing a training image into grids and assigning an object to a grid if and only if the center of the object falls inside the grid, that way each object in a training image can get assigned to exactly one grid and then the corresponding bounding box is represented by the coordinates relative to the grid. The next figure described the details of the algorithm.
  • In the test images, multiple adjacent grids may think that an object actually belongs to them, in order to resolve the iou (intersection of union) measure is used to find the maximum overlap and the non-maximum-suppression algorithm is used to discard all the other bounding boxes with low-confidence of containing an object, keeping the one with the highest confidence among the competing ones and discard the others.
  • Still there is a problem of multiple objects falling in the same grid. Multiple anchor boxes (of different shapes) are used to resolve the problem, each anchor box of a particular shape being likely to eventually detect  an object of a particular shape.

 

f3.png

The following figure shows the slides taken from the presentation You Only Look Once: Unified, Real-Time Object Detection in the CVPR 2016 summarizing the algorithm:

yolo_cvpr.png

Problem Statement

Let’s assume that we are working on a self-driving car. As a critical component of this project, we’d like to first build a car detection system. To collect data, we’ve mounted a camera to the hood (meaning the front) of the car, which takes pictures of the road ahead every few seconds while we drive around.

The above pictures are taken from a car-mounted camera while driving around Silicon Valley.  We would like to especially thank drive.ai for providing this dataset! Drive.ai is a company building the brains of self-driving vehicles.

driveai.png

We’ve gathered all these images into a folder and have labelled them by drawing bounding boxes around every car we found. Here’s an example of what our bounding boxes look like.

Definition of a box
box_label.png

 

If we have 80 classes that we want YOLO to recognize, we can represent the class label c either as an integer from 1 to 80, or as an 80-dimensional vector (with 80 numbers) one component of which is 1 and the rest of which are 0. Here we will use both representations, depending on which is more convenient for a particular step.

In this exercise, we shall learn how YOLO works, then apply it to car detection. Because the YOLO model is very computationally expensive to train, we will load pre-trained weights for our use.  The instructions for how to do it can be obtained from here and here.

 

YOLO

YOLO (“you only look once“) is a popular algorithm because it achieves high accuracy while also being able to run in real-time. This algorithm “only looks once” at the image in the sense that it requires only one forward propagation pass through the network to make predictions. After non-max suppression, it then outputs recognized objects together with the bounding boxes.

Model details

First things to know:

  • The input is a batch of images of shape (m, 608, 608, 3).
  • The output is a list of bounding boxes along with the recognized classes. Each bounding box is represented by 6 numbers (pc,bx,by,bh,bw,c) as explained above. If we expand c into an 80-dimensional vector, each bounding box is then represented by 85 numbers.

We will use 5 anchor boxes. So we can think of the YOLO architecture as the following: IMAGE (m, 608, 608, 3) -> DEEP CNN -> ENCODING (m, 19, 19, 5, 85).

Let’s look in greater detail at what this encoding represents.

Encoding architecture for YOLO

architecture.png

If the center/midpoint of an object falls into a grid cell, that grid cell is responsible for detecting that object.

Since we are using 5 anchor boxes, each of the 19 x19 cells thus encodes information about 5 boxes. Anchor boxes are defined only by their width and height.

For simplicity, we will flatten the last two last dimensions of the shape (19, 19, 5, 85) encoding. So the output of the Deep CNN is (19, 19, 425).

Flattening the last two last dimensions

flatten.png

 

Now, for each box (of each cell) we will compute the following element-wise product and extract a probability that the box contains a certain class.

Find the class detected by each box

probability_extraction.png

Here’s one way to visualize what YOLO is predicting on an image:

  • For each of the 19×19 grid cells, find the maximum of the probability scores (taking a max across both the 5 anchor boxes and across different classes).
  • Color that grid cell according to what object that grid cell considers the most likely.

Doing this results in this picture:

proba_map.png

Each of the 19×19 grid cells colored according to which class has the largest predicted probability in that cell.

Note that this visualization isn’t a core part of the YOLO algorithm itself for making predictions; it’s just a nice way of visualizing an intermediate result of the algorithm.

Another way to visualize YOLO’s output is to plot the bounding boxes that it outputs. Doing that results in a visualization like this:

anchor_map.png

Each cell gives us 5 boxes. In total, the model predicts: 19x19x5 = 1805 boxes just by looking once at the image (one forward pass through the network)! Different colors denote different classes.

In the figure above, we plotted only boxes that the model had assigned a high probability to, but this is still too many boxes. You’d like to filter the algorithm’s output down to a much smaller number of detected objects. To do so, we’ll use non-max suppression. Specifically, we’ll carry out these steps:

  • Get rid of boxes with a low score (meaning, the box is not very confident about detecting a class).
  • Select only one box when several boxes overlap with each other and detect the same object.

 

Filtering with a threshold on class scores

We are going to apply a first filter by thresholding. We would like to get rid of any box for which the class “score” is less than a chosen threshold.

The model gives us a total of 19x19x5x85 numbers, with each box described by 85 numbers. It’ll be convenient to rearrange the (19,19,5,85) (or (19,19,425)) dimensional tensor into the following variables:

  • box_confidence: tensor of shape (19×19,5,1) containing pc (confidence probability that there’s some object) for each of the 5 boxes predicted in each of the 19×19 cells.
  • boxes: tensor of shape (19×19,5,4) containing (bx,by,bh,bw) for each of the 5 boxes per cell.
  • box_class_probs: tensor of shape (19×19,5,80) containing the detection probabilities (c1,c2,…c80) for each of the 80 classes for each of the 5 boxes per cell.

Exercise: Implement yolo_filter_boxes().

  • Compute box scores by doing the element-wise product as described in the above figure.
  • For each box, find:
    • the index of the class with the maximum box score.
    • the corresponding box score.
  • Create a mask by using a threshold.  The mask should be True for the boxes you want to keep.
  • Use TensorFlow to apply the mask to box_class_scores, boxes and box_classes to filter out the boxes we don’t want.
    We should be left with just the subset of boxes we want to keep.

Let’s first load the packages and dependencies that are going to be useful.

import argparse
import os
import matplotlib.pyplot as plt
from matplotlib.pyplot import imshow
import scipy.io
import scipy.misc
import numpy as np
import pandas as pd
import PIL
import tensorflow as tf
from keras import backend as K
from keras.layers import Input, Lambda, Conv2D
from keras.models import load_model, Model
from yolo_utils import read_classes, read_anchors, generate_colors, preprocess_image, draw_boxes, scale_boxes
from yad2k.models.keras_yolo import yolo_head, yolo_boxes_to_corners, preprocess_true_boxes, yolo_loss, yolo_body

 


def yolo_filter_boxes(box_confidence, boxes, box_class_probs, threshold = .6):
 """Filters YOLO boxes by thresholding on object and class confidence.

 Arguments:
 box_confidence -- tensor of shape (19, 19, 5, 1)
 boxes -- tensor of shape (19, 19, 5, 4)
 box_class_probs -- tensor of shape (19, 19, 5, 80)
 threshold -- real value, if [ highest class probability score = threshold)

 # Step 4: Apply the mask to scores, boxes and classes

return scores, boxes, classes

 

Non-max suppression

Even after filtering by thresholding over the classes scores, we still end up a lot of overlapping boxes. A second filter for selecting the right boxes is called non-maximum suppression (NMS).

non-max-suppression.png

n this example, the model has predicted 3 cars, but it’s actually 3 predictions of the same car. Running non-max suppression (NMS) will select only the most accurate (highest probability) one of the 3 boxes.

Non-max suppression uses the very important function called “Intersection over Union”, or IoU.

Definition of “Intersection over Union”

iou.png

 

Exercise: Implement iou(). Some hints:

  • In this exercise only, we define a box using its two corners (upper left and lower right): (x1, y1, x2, y2) rather than the midpoint and height/width.
  • To calculate the area of a rectangle we need to multiply its height (y2 – y1) by its width (x2 – x1)
  • We’ll also need to find the coordinates (xi1, yi1, xi2, yi2) of the intersection of two boxes. Remember that:
    xi1 = maximum of the x1 coordinates of the two boxes
    yi1 = maximum of the y1 coordinates of the two boxes
    xi2 = minimum of the x2 coordinates of the two boxes
    yi2 = minimum of the y2 coordinates of the two boxes

In this code, we use the convention that (0,0) is the top-left corner of an image, (1,0) is the upper-right corner, and (1,1) the lower-right corner.


def iou(box1, box2):
 """Implement the intersection over union (IoU) between box1 and box2

 Arguments:
 box1 -- first box, list object with coordinates (x1, y1, x2, y2)
 box2 -- second box, list object with coordinates (x1, y1, x2, y2)
 """

# Calculate the (y1, x1, y2, x2) coordinates of the intersection of box1 and box2. Calculate its Area.

# Calculate the Union area by using Formula: Union(A,B) = A + B - Inter(A,B)

# compute the IoU

return iou

 

We are now ready to implement non-max suppression. The key steps are:

  • Select the box that has the highest score.
  • Compute its overlap with all other boxes, and remove boxes that overlap it more than iou_threshold.
  • Go back to step 1 and iterate until there’s no more boxes with a lower score than the current selected box.

This will remove all boxes that have a large overlap with the selected boxes. Only the “best” boxes remain.

Exercise: Implement yolo_non_max_suppression() using TensorFlow. TensorFlow has two built-in functions that are used to implement non-max suppression (so we don’t actually need to use your iou() implementation):

def yolo_non_max_suppression(scores, boxes, classes, max_boxes = 10, iou_threshold = 0.5):
 """
 Applies Non-max suppression (NMS) to set of boxes

 Arguments:
 scores -- tensor of shape (None,), output of yolo_filter_boxes()
 boxes -- tensor of shape (None, 4), output of yolo_filter_boxes() that have been scaled to the image size (see later)
 classes -- tensor of shape (None,), output of yolo_filter_boxes()
 max_boxes -- integer, maximum number of predicted boxes you'd like
 iou_threshold -- real value, "intersection over union" threshold used for NMS filtering

 Returns:
 scores -- tensor of shape (, None), predicted score for each box
 boxes -- tensor of shape (4, None), predicted box coordinates
 classes -- tensor of shape (, None), predicted class for each box

 Note: The "None" dimension of the output tensors has obviously to be less than max_boxes. Note also that this
 function will transpose the shapes of scores, boxes, classes. This is made for convenience.
 """

 max_boxes_tensor = K.variable(max_boxes, dtype='int32') # tensor to be used in tf.image.non_max_suppression()
 K.get_session().run(tf.variables_initializer([max_boxes_tensor])) # initialize variable max_boxes_tensor

 # Use tf.image.non_max_suppression() to get the list of indices corresponding to boxes you keep

 # Use K.gather() to select only nms_indices from scores, boxes and classes

 return scores, boxes, classes

 

Wrapping up the filtering

It’s time to implement a function taking the output of the deep CNN (the 19x19x5x85 dimensional encoding) and filtering through all the boxes using the functions we’ve just implemented.

Exercise: Implement yolo_eval() which takes the output of the YOLO encoding and filters the boxes using score threshold and NMS. There’s just one last implementational detail we have to know. There’re a few ways of representing boxes, such as via their corners or via their midpoint and height/width. YOLO converts between a few such formats at different times, using the following functions (which are provided):

boxes = yolo_boxes_to_corners(box_xy, box_wh)

which converts the yolo box coordinates (x,y,w,h) to box corners’ coordinates (x1, y1, x2, y2) to fit the input of yolo_filter_boxes

boxes = scale_boxes(boxes, image_shape)

YOLO’s network was trained to run on 608×608 images. If we are testing this data on a different size image – for example, the car detection dataset had 720×1280 images – his step rescales the boxes so that they can be plotted on top of the original 720×1280 image.


def yolo_eval(yolo_outputs, image_shape = (720., 1280.), max_boxes=10, score_threshold=.6, iou_threshold=.5):
 """
 Converts the output of YOLO encoding (a lot of boxes) to your predicted boxes along with their scores, box coordinates and classes.

 Arguments:
 yolo_outputs -- output of the encoding model (for image_shape of (608, 608, 3)), contains 4 tensors:
 box_confidence: tensor of shape (None, 19, 19, 5, 1)
 box_xy: tensor of shape (None, 19, 19, 5, 2)
 box_wh: tensor of shape (None, 19, 19, 5, 2)
 box_class_probs: tensor of shape (None, 19, 19, 5, 80)
 image_shape -- tensor of shape (2,) containing the input shape, in this notebook we use (608., 608.) (has to be float32 dtype)
 max_boxes -- integer, maximum number of predicted boxes you'd like
 score_threshold -- real value, if [ highest class probability score < threshold], then get rid of the corresponding box
 iou_threshold -- real value, "intersection over union" threshold used for NMS filtering

 Returns:
 scores -- tensor of shape (None, ), predicted score for each box
 boxes -- tensor of shape (None, 4), predicted box coordinates
 classes -- tensor of shape (None,), predicted class for each box
 """

 # Retrieve outputs of the YOLO model

 # Convert boxes to be ready for filtering functions 

 # Use one of the functions you've implemented to perform Score-filtering with a threshold of score_threshold

 # Scale boxes back to original image shape.

 # Use one of the functions you've implemented to perform Non-max suppression with a threshold of iou_threshold 

 return scores, boxes, classes

 

Summary for YOLO:

  • Input image (608, 608, 3)
  • The input image goes through a CNN, resulting in a (19,19,5,85) dimensional output.
  • After flattening the last two dimensions, the output is a volume of shape (19, 19, 425):
    • Each cell in a 19×19 grid over the input image gives 425 numbers.
    • 425 = 5 x 85 because each cell contains predictions for 5 boxes, corresponding to 5 anchor boxes, as seen in lecture.
    • 85 = 5 + 80 where 5 is because (pc,bx,by,bh,bw) has 5 numbers, and and 80 is the number of classes we’d like to detect.
  • We then select only few boxes based on:
    • Score-thresholding: throw away boxes that have detected a class with a score less than the threshold.
    • Non-max suppression: Compute the Intersection over Union and avoid selecting overlapping boxes.
  • This gives us YOLO’s final output.

 

Test YOLO pretrained model on images

In this part, we are going to use a pre-trained model and test it on the car detection dataset. As usual, we start by creating a session to start your graph. Run the following cell.

sess = K.get_session()

Defining classes, anchors and image shape.

Recall that we are trying to detect 80 classes, and are using 5 anchor boxes. We have gathered the information about the 80 classes and 5 boxes in two files “coco_classes.txt” and “yolo_anchors.txt”. Let’s load these quantities into the model by running the next cell.

The car detection dataset has 720×1280 images, which we’ve pre-processed into 608×608 images.

class_names = read_classes(“coco_classes.txt”)
anchors = read_anchors(“yolo_anchors.txt”)
image_shape = (720., 1280.)

 

Loading a pretrained model

Training a YOLO model takes a very long time and requires a fairly large dataset of labelled bounding boxes for a large range of target classes. We are going to load an existing pretrained Keras YOLO model stored in “yolo.h5”. (These weights come from the official YOLO website, and were converted using a function written by Allan Zelener.  Technically, these are the parameters from the “YOLOv2” model, but we will more simply refer to it as “YOLO” in this notebook.)

yolo_model = load_model(“yolo.h5”)

This loads the weights of a trained YOLO model. Here’s a summary of the layers our model contains.

yolo_model.summary()

____________________________________________________________________________________________
Layer (type) Output Shape Param # Connected to
===========================================================================
input_1 (InputLayer) (None, 608, 608, 3) 0
____________________________________________________________________________________________
conv2d_1 (Conv2D) (None, 608, 608, 32) 864 input_1[0][0]
____________________________________________________________________________________________
batch_normalization_1 (BatchNor (None, 608, 608, 32) 128 conv2d_1[0][0]
____________________________________________________________________________________________
leaky_re_lu_1 (LeakyReLU) (None, 608, 608, 32) 0 batch_normalization_1[0][0]
____________________________________________________________________________________________
max_pooling2d_1 (MaxPooling2D) (None, 304, 304, 32) 0 leaky_re_lu_1[0][0]
____________________________________________________________________________________________
conv2d_2 (Conv2D) (None, 304, 304, 64) 18432 max_pooling2d_1[0][0]
____________________________________________________________________________________________
batch_normalization_2 (BatchNor (None, 304, 304, 64) 256 conv2d_2[0][0]
____________________________________________________________________________________________
leaky_re_lu_2 (LeakyReLU) (None, 304, 304, 64) 0 batch_normalization_2[0][0]
____________________________________________________________________________________________
max_pooling2d_2 (MaxPooling2D) (None, 152, 152, 64) 0 leaky_re_lu_2[0][0]
____________________________________________________________________________________________
conv2d_3 (Conv2D) (None, 152, 152, 128 73728 max_pooling2d_2[0][0]
____________________________________________________________________________________________
batch_normalization_3 (BatchNor (None, 152, 152, 128 512 conv2d_3[0][0]
____________________________________________________________________________________________
leaky_re_lu_3 (LeakyReLU) (None, 152, 152, 128 0 batch_normalization_3[0][0]
____________________________________________________________________________________________
conv2d_4 (Conv2D) (None, 152, 152, 64) 8192 leaky_re_lu_3[0][0]
____________________________________________________________________________________________
batch_normalization_4 (BatchNor (None, 152, 152, 64) 256 conv2d_4[0][0]
____________________________________________________________________________________________
leaky_re_lu_4 (LeakyReLU) (None, 152, 152, 64) 0 batch_normalization_4[0][0]
____________________________________________________________________________________________
conv2d_5 (Conv2D) (None, 152, 152, 128 73728 leaky_re_lu_4[0][0]
____________________________________________________________________________________________
batch_normalization_5 (BatchNor (None, 152, 152, 128 512 conv2d_5[0][0]
____________________________________________________________________________________________
leaky_re_lu_5 (LeakyReLU) (None, 152, 152, 128 0 batch_normalization_5[0][0]
____________________________________________________________________________________________
max_pooling2d_3 (MaxPooling2D) (None, 76, 76, 128) 0 leaky_re_lu_5[0][0]
____________________________________________________________________________________________
conv2d_6 (Conv2D) (None, 76, 76, 256) 294912 max_pooling2d_3[0][0]
____________________________________________________________________________________________
batch_normalization_6 (BatchNor (None, 76, 76, 256) 1024 conv2d_6[0][0]
____________________________________________________________________________________________
leaky_re_lu_6 (LeakyReLU) (None, 76, 76, 256) 0 batch_normalization_6[0][0]
____________________________________________________________________________________________
conv2d_7 (Conv2D) (None, 76, 76, 128) 32768 leaky_re_lu_6[0][0]
____________________________________________________________________________________________
batch_normalization_7 (BatchNor (None, 76, 76, 128) 512 conv2d_7[0][0]
____________________________________________________________________________________________
leaky_re_lu_7 (LeakyReLU) (None, 76, 76, 128) 0 batch_normalization_7[0][0]
____________________________________________________________________________________________
conv2d_8 (Conv2D) (None, 76, 76, 256) 294912 leaky_re_lu_7[0][0]
____________________________________________________________________________________________
batch_normalization_8 (BatchNor (None, 76, 76, 256) 1024 conv2d_8[0][0]
____________________________________________________________________________________________
leaky_re_lu_8 (LeakyReLU) (None, 76, 76, 256) 0 batch_normalization_8[0][0]
____________________________________________________________________________________________
max_pooling2d_4 (MaxPooling2D) (None, 38, 38, 256) 0 leaky_re_lu_8[0][0]
____________________________________________________________________________________________
conv2d_9 (Conv2D) (None, 38, 38, 512) 1179648 max_pooling2d_4[0][0]
____________________________________________________________________________________________
batch_normalization_9 (BatchNor (None, 38, 38, 512) 2048 conv2d_9[0][0]
____________________________________________________________________________________________
leaky_re_lu_9 (LeakyReLU) (None, 38, 38, 512) 0 batch_normalization_9[0][0]
____________________________________________________________________________________________
conv2d_10 (Conv2D) (None, 38, 38, 256) 131072 leaky_re_lu_9[0][0]
____________________________________________________________________________________________
batch_normalization_10 (BatchNo (None, 38, 38, 256) 1024 conv2d_10[0][0]
____________________________________________________________________________________________
leaky_re_lu_10 (LeakyReLU) (None, 38, 38, 256) 0 batch_normalization_10[0][0]
____________________________________________________________________________________________
conv2d_11 (Conv2D) (None, 38, 38, 512) 1179648 leaky_re_lu_10[0][0]
____________________________________________________________________________________________
batch_normalization_11 (BatchNo (None, 38, 38, 512) 2048 conv2d_11[0][0]
____________________________________________________________________________________________
leaky_re_lu_11 (LeakyReLU) (None, 38, 38, 512) 0 batch_normalization_11[0][0]
____________________________________________________________________________________________
conv2d_12 (Conv2D) (None, 38, 38, 256) 131072 leaky_re_lu_11[0][0]
____________________________________________________________________________________________
batch_normalization_12 (BatchNo (None, 38, 38, 256) 1024 conv2d_12[0][0]
____________________________________________________________________________________________
leaky_re_lu_12 (LeakyReLU) (None, 38, 38, 256) 0 batch_normalization_12[0][0]
____________________________________________________________________________________________
conv2d_13 (Conv2D) (None, 38, 38, 512) 1179648 leaky_re_lu_12[0][0]
____________________________________________________________________________________________
batch_normalization_13 (BatchNo (None, 38, 38, 512) 2048 conv2d_13[0][0]
____________________________________________________________________________________________
leaky_re_lu_13 (LeakyReLU) (None, 38, 38, 512) 0 batch_normalization_13[0][0]
____________________________________________________________________________________________
max_pooling2d_5 (MaxPooling2D) (None, 19, 19, 512) 0 leaky_re_lu_13[0][0]
____________________________________________________________________________________________
conv2d_14 (Conv2D) (None, 19, 19, 1024) 4718592 max_pooling2d_5[0][0]
____________________________________________________________________________________________
batch_normalization_14 (BatchNo (None, 19, 19, 1024) 4096 conv2d_14[0][0]
____________________________________________________________________________________________
leaky_re_lu_14 (LeakyReLU) (None, 19, 19, 1024) 0 batch_normalization_14[0][0]
____________________________________________________________________________________________
conv2d_15 (Conv2D) (None, 19, 19, 512) 524288 leaky_re_lu_14[0][0]
____________________________________________________________________________________________
batch_normalization_15 (BatchNo (None, 19, 19, 512) 2048 conv2d_15[0][0]
____________________________________________________________________________________________
leaky_re_lu_15 (LeakyReLU) (None, 19, 19, 512) 0 batch_normalization_15[0][0]
____________________________________________________________________________________________
conv2d_16 (Conv2D) (None, 19, 19, 1024) 4718592 leaky_re_lu_15[0][0]
____________________________________________________________________________________________
batch_normalization_16 (BatchNo (None, 19, 19, 1024) 4096 conv2d_16[0][0]
____________________________________________________________________________________________
leaky_re_lu_16 (LeakyReLU) (None, 19, 19, 1024) 0 batch_normalization_16[0][0]
____________________________________________________________________________________________
conv2d_17 (Conv2D) (None, 19, 19, 512) 524288 leaky_re_lu_16[0][0]
____________________________________________________________________________________________
batch_normalization_17 (BatchNo (None, 19, 19, 512) 2048 conv2d_17[0][0]
____________________________________________________________________________________________
leaky_re_lu_17 (LeakyReLU) (None, 19, 19, 512) 0 batch_normalization_17[0][0]
____________________________________________________________________________________________
conv2d_18 (Conv2D) (None, 19, 19, 1024) 4718592 leaky_re_lu_17[0][0]
____________________________________________________________________________________________
batch_normalization_18 (BatchNo (None, 19, 19, 1024) 4096 conv2d_18[0][0]
____________________________________________________________________________________________
leaky_re_lu_18 (LeakyReLU) (None, 19, 19, 1024) 0 batch_normalization_18[0][0]
____________________________________________________________________________________________
conv2d_19 (Conv2D) (None, 19, 19, 1024) 9437184 leaky_re_lu_18[0][0]
____________________________________________________________________________________________
batch_normalization_19 (BatchNo (None, 19, 19, 1024) 4096 conv2d_19[0][0]
____________________________________________________________________________________________
conv2d_21 (Conv2D) (None, 38, 38, 64) 32768 leaky_re_lu_13[0][0]
____________________________________________________________________________________________
leaky_re_lu_19 (LeakyReLU) (None, 19, 19, 1024) 0 batch_normalization_19[0][0]
____________________________________________________________________________________________
batch_normalization_21 (BatchNo (None, 38, 38, 64) 256 conv2d_21[0][0]
____________________________________________________________________________________________
conv2d_20 (Conv2D) (None, 19, 19, 1024) 9437184 leaky_re_lu_19[0][0]
____________________________________________________________________________________________
leaky_re_lu_21 (LeakyReLU) (None, 38, 38, 64) 0 batch_normalization_21[0][0]
____________________________________________________________________________________________
batch_normalization_20 (BatchNo (None, 19, 19, 1024) 4096 conv2d_20[0][0]
____________________________________________________________________________________________
space_to_depth_x2 (Lambda) (None, 19, 19, 256) 0 leaky_re_lu_21[0][0]
____________________________________________________________________________________________
leaky_re_lu_20 (LeakyReLU) (None, 19, 19, 1024) 0 batch_normalization_20[0][0]
____________________________________________________________________________________________
concatenate_1 (Concatenate) (None, 19, 19, 1280) 0 space_to_depth_x2[0][0]
leaky_re_lu_20[0][0]
____________________________________________________________________________________________
conv2d_22 (Conv2D) (None, 19, 19, 1024) 11796480 concatenate_1[0][0]
____________________________________________________________________________________________
batch_normalization_22 (BatchNo (None, 19, 19, 1024) 4096 conv2d_22[0][0]
____________________________________________________________________________________________
leaky_re_lu_22 (LeakyReLU) (None, 19, 19, 1024) 0 batch_normalization_22[0][0]
____________________________________________________________________________________________
conv2d_23 (Conv2D) (None, 19, 19, 425) 435625 leaky_re_lu_22[0][0]
===========================================================================
Total params: 50,983,561
Trainable params: 50,962,889
Non-trainable params: 20,672
____________________________________________________________________________________________

model.png

Reminder: this model converts a pre-processed batch of input images (shape: (m, 608, 608, 3)) into a tensor of shape (m, 19, 19, 5, 85) as explained in the above Figure.

Convert output of the model to usable bounding box tensors

The output of yolo_model is a (m, 19, 19, 5, 85) tensor that needs to pass through non-trivial processing and conversion. The following code does this.

yolo_outputs = yolo_head(yolo_model.output, anchors, len(class_names))

We added yolo_outputs to your graph. This set of 4 tensors is ready to be used as input by our yolo_eval function.

Filtering boxes

yolo_outputs gave us all the predicted boxes of yolo_model in the correct format. We’re now ready to perform filtering and select only the best boxes. Lets now call yolo_eval, which you had previously implemented, to do this.

scores, boxes, classes = yolo_eval(yolo_outputs, image_shape)

Run the graph on an image

Let the fun begin. We have created a (sess) graph that can be summarized as follows:

  1. yolo_model.input is given to yolo_model. The model is used to compute the output yolo_model.output
  2. yolo_model.output is processed by yolo_head. It gives us yolo_outputs
  3. yolo_outputs goes through a filtering function, yolo_eval. It outputs your predictions: scores, boxes, classes

Exercise: Implement predict() which runs the graph to test YOLO on an image. We shall need to run a TensorFlow session, to have it compute scores, boxes, classes.

The code below also uses the following function:

image, image_data = preprocess_image(“images/” + image_file, model_image_size = (608, 608))

which outputs:

  • image: a python (PIL) representation of your image used for drawing boxes. You won’t need to use it.
  • image_data: a numpy-array representing the image. This will be the input to the CNN.

Important note: when a model uses BatchNorm (as is the case in YOLO), we will need to pass an additional placeholder in the feed_dict {K.learning_phase(): 0}.


def predict(sess, image_file):
"""
Runs the graph stored in "sess" to predict boxes for "image_file". Prints and plots the preditions.

Arguments:
sess -- your tensorflow/Keras session containing the YOLO graph
image_file -- name of an image stored in the "images" folder.

Returns:
out_scores -- tensor of shape (None, ), scores of the predicted boxes
out_boxes -- tensor of shape (None, 4), coordinates of the predicted boxes
out_classes -- tensor of shape (None, ), class index of the predicted boxes

Note: "None" actually represents the number of predicted boxes, it varies between 0 and max_boxes.
"""

 # Preprocess your image

 # Run the session with the correct tensors and choose the correct placeholders in the
 # feed_dict. We'll need to use feed_dict={yolo_model.input: ... , K.learning_phase(): 0})

 # Print predictions info
print('Found {} boxes for {}'.format(len(out_boxes), image_file))
# Generate colors for drawing bounding boxes.
colors = generate_colors(class_names)
# Draw bounding boxes on the image file
draw_boxes(image, out_scores, out_boxes, out_classes, class_names, colors)
# Save the predicted bounding box on the image
image.save(os.path.join("out", image_file), quality=90)
# Display the results
output_image = scipy.misc.imread(os.path.join("out", image_file))
imshow(output_image)

 return out_scores, out_boxes, out_classes

Let’s Run the following cell on the following “test.jpg” image to verify that our function is correct.

Inputtest.jpg

out_scores, out_boxes, out_classes = predict(sess, “test.jpg”)

The following figure shows the output after car detection. Each of the bounding boxes have the name of the object detected on the top left along with the confidence value.

Output (with detected cars with YOLO)

Found 7 boxes for test.jpg
car 0.60 (925, 285) (1045, 374)
car 0.66 (706, 279) (786, 350)
bus 0.67 (5, 266) (220, 407)
car 0.70 (947, 324) (1280, 705)
car 0.74 (159, 303) (346, 440)
car 0.80 (761, 282) (942, 412)
car 0.89 (367, 300) (745, 648)

test


The following animation shows the output Images with detected objects (cars) using YOLO for a set of input images.

cars.gif



What we should remember:

  • YOLO is a state-of-the-art object detection model that is fast and accurate.
  • It runs an input image through a CNN which outputs a 19x19x5x85 dimensional volume.
  • The encoding can be seen as a grid where each of the 19×19 cells contains information about 5 boxes.
  • You filter through all the boxes using non-max suppression. Specifically:
    Score thresholding on the probability of detecting a class to keep only accurate (high probability) boxes.
  • Intersection over Union (IoU) thresholding to eliminate overlapping boxes.
  • Because training a YOLO model from randomly initialized weights is non-trivial and requires a large dataset as well as lot of computation, we used previously trained model parameters in this exercise.

 


References: The ideas presented in this notebook came primarily from the two YOLO papers. The implementation here also took significant inspiration and used many components from Allan Zelener’s github repository. The pretrained weights used in this exercise came from the official YOLO website.

  1. Joseph Redmon, Santosh Divvala, Ross Girshick, Ali Farhadi – You Only Look Once: Unified, Real-Time Object Detection (2015)
  2. Joseph Redmon, Ali Farhadi – YOLO9000: Better, Faster, Stronger (2016)
  3. Allan Zelener – YAD2K: Yet Another Darknet 2 Keras
  4. The official YOLO website .

Car detection dataset: Creative Commons License.

The Drive.ai Sample Dataset (provided by drive.ai) is licensed under a Creative Commons Attribution 4.0 International License.