# Giving eyes to a micro controller: DCMI interface on an STM32F4

It has been another long hiatus between posts. But, I have managed to learn and do quite a bit of stuff in these last few months and it has been rewarding to say the least.

Recently, I have had to work on an embedded platform for image processing. It was quite a big deal as I had never worked with any sort of embedded platform before and the kind of work is quite different from what I have done before. So, my first task was to interface a camera with a microcontroller. After consulting with my friends, Shrenik and Vinod, I decided to use a microcontroller which provides a hardware camera interface instead of writing the complete firmware from scratch for an ATMEGA as I had planned on doing earlier.

After some research, I ended up selecting the well known STM32F4 series of microcontrollers. The STM32F407 is a high powered μC with an ARM Cortex M4 processor running at 168 MHz. The development board available has 1 Mb of onchip flash and 192 kB of SRAM. After playing with GBs of RAM, it sure was tough to be excited about a few kB of SRAM, but it was a different challenge to solve the problem using as few resources as possible. The STM32F407 features a a hardware camera interface known as DCMI ( Digital Camera Interface). It is compatible with a huge range of camera modules on the market.

I had also decided to use the OV2640 camera module as it features an on-board JPEG encoder and is quite well documented. After a few days of familiarizing with the basic concepts and fiddling around with the standard peripherals library from  STM, I came across this amazing implementation, OpenMV. I found it extremely helpful to understand the intricacies of image processing on embedded systems.

The primary issues that I faced while working on this project were:

• Understanding DCMI and DMA interfaces of the STM32 controller.
• Understanding the communication and synchronization between the camera and the controller.
• Clocks and STM’s unique proposition of allowing us to turn off peripheral clocks when required for low power usage.

I am going to be blogging about my experiences regarding my foray into the world of micro-controllers and camera control soon. Until then, here are a few images that I captured with my setup.

# Back to Basics: Sparse Coding?

A good introduction to Sparse Coding. Hope to do some stuff regarding this in the future.

Originally posted on the Serious Computer Vision Blog:

by Gooly (Li Yang Ku)

It’s always good to go back to the reason that lured you into computer vision once in a while. Mine was to understand the brain after I astonishingly realized that computers have no intelligence while I was studying EE in undergrad. In fact if they use the translation “computer” instead of  “electrical brain” in my mother language, I would probably be better off.

Anyway, I am currently revisiting some of the first few computer vision papers I read, and to tell the truth I still learn a lot from reading stuffs I read several times before, which you can also interpret it as I never actually understood a paper.

So back to the papers,

Simoncelli, Eero P., and Bruno A. Olshausen. “Natural image statistics and neural representation.” Annual review of neuroscience 24.1 (2001): 1193-1216.

Olshausen, Bruno A., and David J. Field. “Sparse coding with an…

View original 237 more words

# VLAD- An extension of Bag of Words

Recently, I was a participant at TagMe- an image categorization competition conducted by Microsoft and Indian Institute of Science, Bangalore. The problem statement was to classify a set of given images into five classes: faces, shoes, flowers, buildings and vehicles. As it goes, it is not a trivial problem to solve. So, I decided to attempt my existing bag-of-words algorithm on that. It worked to an extent, I got an accuracy of 86% approximately with SIFT features and an RBF SVM for classification. In order to improve my score though, I decided to look at better methods of feature quantization. I had been looking at VLAD (Vector of Locally Aggregated Descriptors): A first order extension to BoW for my Leaf Recognition project.

So, I decided to attempt to use VLAD using OpenCV and implemented a small function based on the BoW API currently in OpenCV for VLAD. The results showed remarkable improvement with an accuracy of 96.5 % using SURF descriptors on teh validation dataset provided by the organizers.

Recalling BoW, it involved simply counting the no. of descriptors associated with each cluster in a codebook(vocabulary) and creating a histogram for each set of descriptors from an image, thus representing the information in a an image in a compact vector. VLAD is an extension of this concept. We accumulate the residual of each descriptor with respect to its assigned cluster. In simpler terms, we match a descriptor to its closest cluster, then for each cluster, we store the sum of the differences of the descriptors assigned to the cluster and the centroid of the cluster. Let us have a look at the math behind VLAD..

### Mathematical Formulation

As with bag of words, we first train a codebook from the descriptors from our training dataset, as $C=\{c_1,c_2,...c_k\}$ where $k$ is the no. of clusters in K-means. We then associate each $d$-dimensional local descriptor, $x$ from an image with its nearest neighbour in the codebook.

The idea behind VLAD feature quantization is that, for each cluster centroid, $c_i$, we accumulate the difference $x-c_i$ where for each $x$, $c_i = NN(x)$

Representing the VLAD vector for each image by $v$, we have,

$v_{ij} =\sum_{x|x=NN(c_i)} {(x_j - c_{ij})}$

where $i=1,2,3...k$ and $j=1,2,3..d$

The vector $v$ is subsequently normalized with its $L_2$ norm as $v=\frac{v}{\|v\|_2}$

### Comparison with BoW

The primary advantage of VLAD over BoW is that we add more discriminative property in our feature vector by adding the difference of each descriptor from the mean in its voronoi cell. This first order statistic adds more information in our feature vector and hence gives us better discrimination for our classifier. This also points us to other improvements we can   adding higher order statistics to our feature encoding as well as looking at soft assignment,i,e. assigning each descriptor multiple centroids weighed by their distance from the descriptor.

### Experiments

Here are a few of my results on the TagMe dataset.

There are several extension possible for VLAD, primarily various normalization options. Arandjelov and Zissermann in their paper, All about VLAD, propose several normalization techniques, including intra normalization and power normalization alonging with a spatial extension – MultiVLAD. Delhumeau et al, propose several different normalization techniques as well as a modification to the VLAD pipeline to show improvements to almost state of the art.

Other references also stress on spatial pooling i.e. dividing your image into regions to get multiple VLAD vectors for each tile to better represent local features and spatial structure. A few also advise soft assignment, which refers to assignment of descriptors to multiple clusters, weighed by their distance from the cluster.

### Code:

Here is a link to my code for TagMe. It was a quick has job for testing so it is not very clean though I am going to clean it up soon.

also, a few references for those who want to read the papers I referred:

Well, that concludes this post. Be on the lookout for more on image retrieval – improvements to VLAD and Fisher Vectors.

# SuperPixels: SLIC and more

Hey guys,

I recently came across a very cool computer vision concept – SuperPixels. Superpixels are what you would call aggregations of pixels with common features – for example, color, illumination, absorbance, spatial location etc. It is perhaps the precursor to full object segmentation. Here, I am going to discuss a specific kind of superpixels called SLIC.

SLIC stands for Simple Linear Iterative Clustering. The name defines it all as it is just a simple representation of clustering using k-means. This was first proposed in the following paper:

Radhakrishna Achanta, Appu Shaji, Kevin Smith, Aure-lien Lucchi, Pascal Fua, and Sabine Susstrunk, SLIC Superpixels, EPFL Technical Report 149300, June 2010.

The basic idea of superpixels is that they contain a lot more high level information as compared to just pixels. Pixels give extremely low level and local information which many-a-times cannot be used without heavy duty processing. Instead, superpixels provide less localized information which makes more sense to have considering problems like segmentation of objects. A simple analogy would be to step back a bit and look at a slightly bigger picture.

To start of with, lets look at a few example images given in the paper:

As you can see, pixels are clustered according to their location and color. The different slices represent different sizes of superpixels. One of the major advantages of using SLIC  superpixels is that they show low undersegmentation and high boundary recall. It is one of the most efficient methods in the methods for getting superpixels.

The algorithm is very intuitive. We start off by converting the image to L*a*b* color-space. We then create feature points from each pixel consisting of the L*,a*,b* values and the x and y co-ordinates. These features are then clustered using k-means clustering

Now, one of the brilliant things of SLIC is that it takes the x-y co-ordinates into the picture while clustering. So, you get uniform and well defines superpixels. This can then be further used for segmentation. The distance function used for clustering is given by

$d_{lab}&space;=&space;\sqrt{&space;(l_{k}&space;-&space;l_{i})^{2}&space;+&space;(a_{k}&space;-&space;a_{i})^{2}&space;+(b_{k}&space;-b_{i})^{2}&space;}$

$d_{xy}&space;=&space;\sqrt{&space;(x_{k}&space;-&space;x_{i})^{2}&space;+&space;(y_{k}&space;-&space;y_{i})^{2}&space;}$

$D_{S}&space;=&space;d_{lab}&space;+&space;\frac{m}{S}d_{xy}$

So, we start off with K equally spaced clusters all across the image. These clusters are centred around the lowest gradient position in a 3×3 neighbourhood around the equally spaced points. We do this to avoid putting them at edges and to reduce the probability of choosing a noisy pixel. We calculate the image gradients as follows:

where $I (x,y)$ is the lab vector corresponding to the pixel and $\|.\|$ is the L2 norm.
Each pixel in the image is then associated with the nearest cluster centre in the $labxy$ feature space. After all pixels are associated with one or the other cluster, we again calculate a new cluster centre as the average value of the $labxy$ features in each cluster. We then repeat the process of clustering and recalculating until we achieve convergence.

As a result, we will have nicely clustered groups of pixels, both in color as well as spatial domain. Now, there might be a few stray pixels that may have got clustered in the wrong clusters. Though this is pretty rare according to the authors of the paper and my experiments, we enforce a connectivity constraint on them in order to remove these outliers. The paper doesnot mention this in any great detail, but connectivity is an important condition to impose, especially if we are going to use SLIC superpixels for segmentation.

Thats it for the explanation of an amazing technique, folks! I will post my OpenCV code link in a few days as well as will discuss a few issues with the code and segmentation techniques using these in the next few posts.

P.S.:By the way, we will also be looking at Bag of Words in a little more detail in a few more days. Be on the look out for a barrage of posts.

# Bag of Words – Object Recognition

Hey guys,

Its been a really long time since my last post. But this series of posts is going to be a really cool I hope.

Today we are going to discuss one of the most important problems in Computer Vision- Object Recognition. We humans tend to trivially recognize objects without consciously paying attention to the fact or even wondering how exactly do we achieve this. You look at a baseball flying towards your face, you recognize it as a baseball about to break your nose, and you duck! All in a matter of a few microseconds.

But the process that your brain undertakes in those few microseconds has eluded perfect implementation in computation for several years now. Object recognition is perhaps, rightly considered the primary problem in computer vision. But recent research advances have made strides in this matter.

I recently undertook a project in which I had to classify leaves into species they come from. And as it sounds, it’s not really a trivial problem. It took me a few days to figure out the first steps to such a process. And to start of with I decided to use the Bag-of Words model, a highly cited method for scene and object classification for the above problem.

To begin with, I found a really nice dataset to work with here: http://flavia.sourceforge.net/ . The dataset contains images for 32 species of leaves on plain white backgrounds which simplified my experiment. I am really grateful to them for providing such a comprehensive dataset for free on the web. (Kinda all for Open Access now.).

Bag of words is a basically a simplified representation of an image. Its actually a concept taken form Natural Language Processing where you represent documents as an unordered collection of words disregarding grammar. Translating this into CV jargon, it means that we simplify images by picking out features from an image and representing it as a collection of features. A good explanation of what features are can be found at my friend, Siddharth’s blog here.

To get more technical about the BoW- we construct a vocabulary of features. We then use this vocabulary to create histograms from features for each image and then use a simple machine learning algorithm like SVM or Naive Bayes for classification.

This is the algorithm I followed for BoW. I got a lot of help from Roy’s blog here.

1. We pick out features from all the images in our training dataset. I used SIFT (Scale Invariant Feature Transform).

2. We cluster these features using any clustering algorithm. I used K-Means. (Pretty fast in OpenCV)

3. We use the cluster as a vocabulary to construct histograms. We simply count the no. of features from each image belonging to each cluster. Then we normalize the histograms by dividing it with the no. of features. Therefore, each image in the dataset is represented by one histogram.

4. Then these histograms are passed to an SVM for training. I currently use a Radial Basis function multi-class SVM in OpenCV. Using OpenCV’s CvSVM::train_auto() function, we get parameters for the SVM using cross validation.

Now why does Bag of Words work? Why use it rather than simple feature matching? The answer to that question is simple: features provide just local information. Using the bag-of-words model we create a global representation of an object. Thus, we take a group of features, create a representation of the image in a simpler form and classify it.

That was for the pros of the algorithm. But there are a few cons associated with this model.

1. As evident, we cannot localize an object in an image using this model. That is to say, the problem of finding where the object of interest lies is still open and needs other methods.

2. We neglect grammar. In CV terms, it means we neglect the position of features relative to each other. Thus the concept of a global shape maybe lost.

As for our Leaf Recognizer, we are still working on improving the accuracy. We are almost at our goal! The following are some of the images we got as a result of the above algorithm:

# The Traveling Salesman Problem – Using Ant Algorithms

Hey guys,

As I had promised, this post will be about using the Ant algorithms I had discussed in the previous post to solve a complex computational problem. But, before we go on, let us have a look again at Ant Colony optimization.

Ant Colony Optimization is basically a group of algorithms used to find optimum paths in a graph. It does so by mimicking the foraging behavior of an ant colony.  As I had said before, ants use pheromones to communicate and thus the amount of pheromone on each path decides the optimality of the path. Thus, an ant is more likely to take a path with more pheromones on it. We use this concept and basically create ‘Agents’ that we call ‘Ants’. Each Ant is placed at a node and is the allowed to traverse the graph. At the end of it’s tour, we calculate the amount of pheromone on each edge of the path and then assign a probability value to it based on the amount of pheromone and the ‘visibility’ of the next node. Thus, by using a no. of ants and allowing them to traverse the graph repeatedly, we get the optimum path. Thus, we actually mimic the co-operative nature of ants in a colony to solve optimization problems.
Now, lets move on to the problem at hand. As the title suggests, today, we are going to solve one of the most famous an computationally complex problems in computer science – The Traveling Salesman Problem. The Traveling Salesman problem is modeled in the following way:

There are N cities randomly placed on a map. A salesman has to visit each city and return to the starting point such that he visits each city only once. The problem is to find the shortest path given pairwise distances between the cities.

This problem was first formulated in 1930 and is one of the most intensively studied problems in computer science and optimization. The problem is NP hard in nature. We will discuss NP hard and other such classifications in a future post. It has got several exact and heuristic solutions, so problems with large numbers of cities can be solved. Ant colony Algorithm is one of the methods of finding a solution.

The following source code has been mostly sourced from M.Tim Jones’ AI Application Programming (1st Edition) by Dreamtech. I have made slight modifications to the code and also written a python script for visualizing the problem.

Let us start with the code: ( The code is an adaptation the sourcecode found in Ch. 4 –  Ant Algorithms of AI Application Programming, 1st edition by M.Tim Jones. )

Let us first define the parameters required for the Ant Colony Algorithms.

#define MAX_CITIES 30
#define MAX_DIST 100
#define MAX_TOUR (MAX_CITIES * MAX_DIST)
#define MAX_ANTS 30


MAX_CITIES refers to the maximum number of cities that we have on the map. We also define the maximum pairwise distance between the cities to be 100. Thus, worst case scenario, modeled by MAX_TOUR represents the tour in which the ant has to travel through MAX_CITIES*MAX_DIST. Here, we also define the no. of ants to be 30 i.e. exactly equal to the no. of cities on the map. It has been determined experimentally that having the no. of ants equal to the no. of cities gives the optimum solution in less time than having more ants in the system.

Now, we create the environment fro the Traveling Salesman Problem (TSP).

//Initial Definiton of the problem
struct cityType{
int x,y;
};

struct antType{

int curCity, nextCity, pathIndex;
int tabu[MAX_CITIES];
int path[MAX_CITIES];
double tourLength;
};



Here, we have defined a struct cityType which contains the members – x and y. These represent the co-ordinates of the location of the city. We follow this by defining the struct antType which contains the members necessary for an ant to function. The integer values of curCity and nextCity represent the indices of the current city and the next city to be visited on the graph. The array, tabu refers to the list of cities already traversed by an Ant whereas path[] stores the path through which the Ant has traveled. The value tourLength is the distance through which the ant has traveled as of the tour.

Now that we have set up the basic setup of the problem as well as created the ants required for simulation, we will move on to defining the parameters by which the algorithm will converge to a solution.

#define ALPHA 1.0
#define BETA 5.0 //This parameter raises the weight of distance over pheromone
#define RHO 0.5 //Evapouration rate
#define QVAL 100
#define MAX_TOURS 20
#define MAX_TIME (MAX_TOURS * MAX_CITIES)
#define INIT_PHER (1.0/MAX_CITIES)



The parameters, ALPHA and BETA are the exponents of the heuristic functions defining the pheromone deposition on the edges. ALPHA represents the importance of the trail whereas BETA defines the relative importance of visibility. Visibility here is modeled by the inverse of distance between two nodes. RHO represents the factor which defines the evaporation rate of the pheromones on the edges. QVAL is the constant used in the formula

We also declare the no. of maximum tours that the ant undertakes as 20. We also have declared a maximum time constraint in case the algorithm is not able to converge to a solution. We then weigh the edges with pheromones with the value of (1/MAX_CITIES).

//runtime Structures and global variables

cityType cities[MAX_CITIES];
antType ants[MAX_ANTS];

double dist[MAX_CITIES][MAX_CITIES];

double phero[MAX_CITIES][MAX_CITIES];

double best=(double)MAX_TOUR;
int bestIndex;



Now, we declare an array of cityType structs which will contain be defining the nodes of our graph. We also create an array of Ants with the size equal to that of cities. The double dimensional array dist records the pairwise distances between the cities whereas the array phero records the pheromone levels on each edge between the cities. The variable best is used as a control variable for recording the best tour after every iteration.

void init()
{
int from,to,ant;

//creating cities

for(from = 0; from < MAX_CITIES; from++)
{
//randomly place cities

cities[from].x = rand()%MAX_DIST;

cities[from].y = rand()%MAX_DIST;
//printf("\n %d %d",cities[from].x, cities[from].y);
for(to=0;to<MAX_CITIES;to++)
{
dist[from][to] = 0.0;
phero[from][to] = INIT_PHER;
}
}

//computing distance

for(from = 0; from < MAX_CITIES; from++)
{
for( to =0; to < MAX_CITIES; to++)
{
if(to!=from && dist[from][to]==0.0)
{
int xd = pow( abs(cities[from].x - cities[to].x), 2);
int yd = pow( abs(cities[from].y - cities[to].y), 2);

dist[from][to] = sqrt(xd + yd);
dist[to][from] = dist[from][to];

}
}
}

//initializing the ANTs

to = 0;
for( ant = 0; ant < MAX_ANTS; ant++)
{
if(to == MAX_CITIES)
to=0;

ants[ant].curCity = to++;

for(from = 0; from < MAX_CITIES; from++)
{
ants[ant].tabu[from] = 0;
ants[ant].path[from] = -1;
}

ants[ant].pathIndex = 1;
ants[ant].path[0] = ants[ant].curCity;
ants[ant].nextCity = -1;
ants[ant].tourLength = 0;

ants[ant].tabu[ants[ant].curCity] =1;

}
}


The above snippet of code is used to initialize the entire graph and the Ants that we are going to use to traverse the graph. We basically allot random co-ordinates to the cities and then compute the pairwise distances between them. Also, we initialize the ants with an Ant on each node of the graph. We also initialize the rest of the properties of each Ant including the tabu list and the path. The next part of the algorithm is to restart the ants after each traversal by returning them to their original positions.

//reinitialize all ants and redistribute them
void restartAnts()
{
int ant,i,to=0;

for(ant = 0; ant<MAX_ANTS; ant++)
{
if(ants[ant].tourLength < best)
{
best = ants[ant].tourLength;
bestIndex = ant;
}

ants[ant].nextCity = -1;
ants[ant].tourLength = 0.0;

for(i=0;i<MAX_CITIES;i++)
{
ants[ant].tabu[i] = 0;
ants[ant].path[i] = -1;
}

if(to == MAX_CITIES)
to=0;

ants[ant].curCity = to++;

ants[ant].pathIndex = 1;
ants[ant].path[0] = ants[ant].curCity;

ants[ant].tabu[ants[ant].curCity] = 1;
}
}



We also store the best path and the value of the best path in the global variables that we have defined.

double antProduct(int from, int to)
{
return(( pow( phero[from][to], ALPHA) * pow( (1.0/ dist[from][to]), BETA)));
}

int selectNextCity( int ant )
{
int from, to;
double denom = 0.0;

from=ants[ant].curCity;

for(to=0;to<MAX_CITIES;to++) 	{ 		if(ants[ant].tabu[to] == 0) 		{ 			denom += antProduct( from, to ); 		} 	} 	 	assert(denom != 0.0); 	 	do 	{ 		double p; 		to++; 		 		if(to >= MAX_CITIES)
to=0;
if(ants[ant].tabu[to] == 0)
{
p = antProduct(from,to)/denom;

//printf("\n%lf %lf", (double)rand()/RAND_MAX,p);
double x = ((double)rand()/RAND_MAX);
if(x < p)
{
//printf("%lf %lf Yo!",p,x);

break;
}
}
}while(1);

}


This set of functions is used in the process of selecting the next edge to traverse. Each ant in the environment selects the next edge in accordance with the equation discussed in the last post. Here we have a simple reciprocal function of distance as a measure of visibility as well as the actual value of the pheromone itself as the function to represent pheromone. There has been extensive research in this domain such that these functions may be replaced by other suitable heuristic functions. For the simple TSP that we are going to solve, the functions thatwe use prove to be a good enough measure.

Now comes the actual process of simulating an ant colony. The following function is an example of simulating the ant colony to solve the TSP.


int simulateAnts()
{
int k;
int moving = 0;

for(k=0; k<MAX_ANTS; k++)
{
//checking if there are any more cities to visit

if( ants[k].pathIndex < MAX_CITIES ) 		{ 			ants[k].nextCity = selectNextCity(k); 			ants[k].tabu[ants[k].nextCity] = 1; 			ants[k].path[ants[k].pathIndex++] = ants[k].nextCity; 			 			ants[k].tourLength += dist[ants[k].curCity][ants[k].nextCity]; 			 			//handle last case->last city to first

if(ants[k].pathIndex == MAX_CITIES)
{
ants[k].tourLength += dist[ants[k].path[MAX_CITIES -1]][ants[k].path[0]];
}

ants[k].curCity = ants[k].nextCity;
moving++;

}
}

return moving;
}

//Updating trails

void updateTrails()
{
int from,to,i,ant;

//Pheromone Evaporation

for(from=0; from<MAX_CITIES;from++)
{
for(to=0;to<MAX_CITIES;to++)
{
if(from!=to)
{
phero[from][to] *=( 1.0 - RHO);

if(phero[from][to]<0.0)
{
phero[from][to] = INIT_PHER;
}
}
}
}

//Add new pheromone to the trails

for(ant=0;ant<MAX_ANTS;ant++)
{
for(i=0;i<MAX_CITIES;i++)
{
if( i < MAX_CITIES-1 )
{
from = ants[ant].path[i];
to = ants[ant].path[i+1];
}
else
{
from = ants[ant].path[i];
to = ants[ant].path[0];
}

phero[from][to] += (QVAL/ ants[ant].tourLength);
phero[to][from] = phero[from][to];

}
}

for (from=0; from < MAX_CITIES;from++)
{
for( to=0; to<MAX_CITIES; to++)
{
phero[from][to] *= RHO;
}
}

}



The next snippet of code is simply a helper function which emits all the data into atext file so as a python script can present it in a better and easily understandable form.

void emitDataFile(int bestIndex)
{
ofstream f1;
f1.open("Data.txt");
antType antBest;
antBest = ants[bestIndex];
//f1<<antBest.curCity<<" "<<antBest.tourLength<<"\n";
int i;
for(i=0;i<MAX_CITIES;i++)
{
f1<<antBest.path[i]<<" ";
}

f1.close();

f1.open("city_data.txt");
for(i=0;i<MAX_CITIES;i++)
{
f1<<cities[i].x<<" "<<cities[i].y<<"\n";
}
f1.close();

}


Now let us move on to the main()

int main()
{
int curTime = 0;
cout<<"MaxTime="<<MAX_TIME;

srand(time(NULL));

init();

while( curTime++ < MAX_TIME)
{
if( simulateAnts() == 0)
{
updateTrails();

if(curTime != MAX_TIME)
restartAnts();

cout<<"\nTime is "<<curTime<<"("<<best<<")";

}
}

cout<<"\nBest tour = "<<best<<endl;

emitDataFile(bestIndex);

return 0;
}


Here we simply run the simulations and keep on updating trails and restarting the simulation with the updated trails until the no. of iterations does not exceed MAX_TIME. This ensures thatthe simulation terminates regardless of whether the ANT agents converge onto a solution or not after a set no. of iterations.

This is the primary code for the Ant Algorithm solution for the Traveling Salesman Problem in C++. Now, I have written a small python script which plots the data using matplotlib and creates a beautiful graph for us to view.

from numpy import *
import matplotlib.pyplot as plt

print order
print x
print y
'''order1[]
x1[]
y1[]'''
x1=[]
y1=[]
for i in order:
x1.append(x[i])
y1.append(y[i])
x1.append(x[order[0]])
y1.append(y[order[0]])
plt.plot(x1,y1, marker='x', linestyle = '--', color = 'r')
for i in range(len(order)):
plt.text(x1[i],y1[i],order[i])
plt.show()

def main():

if __name__=="__main__":
main()



The script outputs an easy to read graph which pinpoints the cities on a graph and highlights the optimum path as outputted by the ant algorithm. Here are a few output images:

A solved TSP

ANTs plot out a path

Output for a randomly generated city map

P.S.: You can find the above code as well as several other implementations at my git:

https://github.com/ameya005/AntColonyAlgorithms

# Ant Colony Algorithms

Hey guys, Thanks for the great response on my previous article.

Today, I will be writing about a brilliant algorithm called Ant Colony Algorithm. Ant Colony Algorithm or Ant Algorithm as it is popularly known as, is basically an optimization algorithm based on swarm intelligence. It mimics the behavior of an ant colony foraging for food. This algorithm was first proposed by Marco Dorigo in 1992 as his PhD. thesis. It is a part of the amazing world of swarm intelligence wherein a swarm of low ability agents are used to solve complex problems.

Ants are perhaps one of the most amazing creatures in this world. They are nearly blind but still mange to navigate through difficult terrain and find their way to food and back. Not only that, they can lift up to10 times their body weight and perhaps one of the sturdiest creatures of the insect world. The major reason of their success in the battle for survival is the fact that they co=operate with each other. The survival of the ant colony is foremost rather than survival of a single ant. It is this co-operation and communication that we attempt to model through ant algorithms.

Now, ants have an amazing method of navigation. They use pheromones to communicate the optimal path to any point to other ants. This works in the following way.

1. An ant finds a food source.
2. It lays down a trail of pheromones which can be detected by other ants upon the path from the food source to the storage area
3. Other ants also reach the food source through various paths.
4. Now, based on the time taken to complete each trip to and fro, the shortest path will have the most pheromone as it has been traversed the most.
5. As the amount of pheromone on each path dictates the probability of that path being taken, more and more ants will be traveling on the optimal path. Thus, using several ants, the optimal path to a food source is obtained.

We attempt to mimic this behavior by using agents that we call ants. To solve a problem using Ant algorithms, we model it as a fully connected bidirectional graph. You can read up more on graphs and graph theory here. On each node, we place an ant and then ask it to traverse the other nodes on the graph thus completing a trip. The constraint on the ant is that it must visit each node only once. As it travels, we calculate the pheromone levels on that path based on its length. Upon the completion of each trip( known as a Tour), we replace the ant on a node and send it on its way again. But each time, it reaches a node, the probability of it taking the next edge is given by the amount of pheromone on that edge. Thus, in the end, through traversing the graph, we find out the most optimal path.

For example, lets take a situation with 2 ants and a food source. Now Ant1,  who will call Han takes a path with 2L distance to the food source whereas Ant2, Luke, takes a path with distance L. Now both lay down small amounts of pheromone as they pass. Now Luke completes the trip to the food and back in time T, while Han takes 2T, assuming both travel at the same speed. He takes the food and returns back. And then proceeds on his second trip. While he starts of his second lap, Han is still just at the food. So, when Han returns back, Luke will have completed two trips. Thus, there is approximately twice the mount of pheromone on the Luke’s path as compared to Han’s path. So, next time, Han will be more probable to taking Luke’s path based on simply the amount of pheromones.
So, how do we calculate the probability of an ant taking the path? Before that, we also add another factor to the situation- that the pheromone evaporates as time goes on. Thus, paths which are poor are discarded more easily. Now, we need to describe the probability equation governing the movement of ants.

While an ant has not completed a tour i.e. it has not visited all the nodes, the probability of the next edge to take is given by the following equation:

Here $\tau(r,u)$ is the intensity of pheromone on the edge between the nodes r and u. $\tau(r,u)$ is actually a heuristic function which represents the inverse of the distance between the nodes, $\alpha$ is the weight for the pheromone, and $\beta$ is the weight for the distance of the edge which can be modeled after visibility. $k$ is the no. of edges not traversed yet. This formula can be said to be the ratio of the weight of an edge to the sum of the weights of all untraveled edges. Thus, this will give us a probability based on the amount of pheromone deposited and the distance of the edge. Both of these values will help us in converging to an optimal value.

Now, we calculate the pheromone left on each edge only at the end of each ant tour. We do this using the following formula:

Here $Q$ is a constant and $L^{k}(t)$ is the total length of the tour. We then use this value to calculate the total pheromone value due to a single ant after a tour.

We have to note that this operation is carried out only after the tour is completed as the pheromone levels are inversely proportional to the length of the tour. Constant $\rho$ is a constant between 0 and 1.

As I have mentioned before, we consider pheromone evaporation as a way to discard poor paths. The evaporation is modeled by a simple equation

Here, we are using the inverse coefficient of the weight of the change in pheromone levels on a n edge to model evaporation. This proves to be an efficient way to discard edges which are not optimal.

As each ant completes it’s tour, we update the edges based on the equations above. Then, we restart the entire process with the ants being placed on nodes once more and then allowed to move according to the updates edges. hence, we can see that more and more ants will converge to a single path. The no. of iterations can be controlled so as the algorithm ends when there is no change in the value of the shortest length for a few iterations.

Thus, we see how a complex natural system is modeled and used to solve problems. A few applications of Ant Algorithms include

• Network Routing
• Graph Coloring
• Vehicle Routing
• Path Optimization
• Swarm Robotics

This concludes this post. In the next post, we will use the discussed Ant algorithm to solve a famous computational problem – The Traveling Salesman Problem.