Artificial Intelligence CS365 - Homework 3

A: Various Dimensionalities for the manifold of these images

Residual variance
Observations about data set based on the graph :
At Isomap Dimensionality =2, curve ceases to decrease significantly with added dimensions. Hence dimensionality of this manifold is best explained at 2

B : lower-dimensional 2-D embedding

2-D Isomap

C : Variation of theta1 and theta2 along map

It is shown in following file. matlab figure file
theta1 changes as we rotate around this torus figure in Isomap and theta2 varies in radial direction. It can be thought of as a projection of 3D torus on a plane where each point corresponds to unique (theta1,theta2).

D : Embedding in 3D space

3-D Isomap
View Angle = (45,45)

Codes for A,B,C,D part

Combined script for A,B,C,D
Isomap
theta 1 and theta2 on 2-D Isomap
3-D Isomap
Euclidean Distance
Load Image Data

E : Map Learning

From Isomap 2D parameters :


Code : Matlab Code- Map Learning

From Image :


As we can see from graph, approach 1 converges faster. This is due to large no. of data points(30k) in learning from Image.
Code : Matlab Code- Map Learning

Shortest path planning between 00001.png and 00161.png


Code :
e.m
dijkstra.m
Isomap.m
L2_distance.m
loadImageData.m

F


Code :
f.m
dijkstra.m
Isomap.m
L2_distance.m
loadImageData.m

G


Difference :

G has distorted torus because of absence of some part of it due to collision. And in this isomap only collision free nodes are shown.
Code :
g.m
dijkstra.m
Isomap.m
L2_distance.m
loadImageData.m