1. MAP MAKING USING LINEAR ALGEBRA & STATISTICS
Project
MAP MAKING USING LINEAR ALGEBRA & STATISTICS

2. WHAT IS MY PROBLEM???????
I am asked to form a map of a city using some screen-shots.
So???????
What’s the problem there??
Yes. I have some problem. I have to join some mutually intersecting maps. In that case I have to fix the origin of one map to the another. For doing this I need some points which are present in both points. But while taking this intersecting points we are facing the problem which sounds a small but caused us to think a lot.& it’s a severe problem also, causing a great problem in every step.

3. What is my suggested solution then???
When we(more precisely I) face error the word for minimising it comes to my mind is “Least Square Method”.
But it’s applicable for linearly related variables.
So, where is the linearity??
After a lot of thinking, a “linear relation” came out.
Let us discuss what it is.

4. The Linearity:
I have lengths( value with some error) of the blue and red dotted lines. Now I want the black lines.
My claim is :
the relation between the blue dotted lines and the red dotted lines
is linear. Why??
Basically when we are taking 2 maps making them to join with 1 point, we give x coordinates a shift and y coordinates a shift.

5. So the relation between them must be linear & the relation should be of the form
x’ = x + b [b is the amount of shift]
But due to that error while locating the point the equation will have a scaling error 0.05 i.e, the relation will be
x’ =ax + b[b is the required shift, a belongs to (1-0.05,1+0.05) ]
I am neglecting this scaling for its too small to cause a great problem in calculation( by observation)
So my work is only with the shifting quantity.
But we can have a rough idea about how much our errors are from the scaling element.
Now we can use Least square method.
Now I apply least square method in the matrix form.
Here I am 1st fixing a map as the global map. Now I am going to shift all the maps such that they are in a single global coordinate system.

6. Then I formed matrices with 4 intersection points’ x coordinates in the 2nd map in one column and the 2nd column with only 1.
my matrix looked like:
A =
Then I am collecting the x coordinates of the same points in the 1st map and forming another matrix like:
B =

7. And my required matrix is : X = So, my matrix equation is : AX=B This is the matrix form of Least square method’s normal equations. So, Solving this matrix equation (with some special techniques) I can have the measurement of shift. I do the same thing for obtaining y shifting for the same map. Now, I can collect points from the screen-shots and giving them the proper shifting I can draw my own map for the city.

8. Process involved:
Take screen-shots( I deviced an algorithm of this process for ease of calculation).
Then join them one by one.

9. The diagram beside is my map now
The diagram beside is my map now. Now I am collecting points from these maps and after transformation I plot them. And with that I have made the road plan of Coochbehar (my hometown, on which I was working)

10. My map is as follows :

11. To make it user friendly
My Problems:
Time problem
My laziness
My helpers:
My friends
Dr. Arnab Chakraborty(ISI Professor)
Google
RStudio
My next mission:
To make it user friendly
To give some important points.(this field is almost done. some minor errors are there)

12. Thank You All……. By
Rounak Ray(BS1501)