Assignment 3
There is a sharp elbow in this graph at dimensionality 2, so we can deduce that the remaining dimensions are not important. Also, Residual error doesn't fall off significantly for dimensions > 2.
In this 2d embedding,the manifold observed is a torus. Here, theta 1 varies along the circumference of the torus and theta 2 along the radius. Theta 2's variation can also be observed from part b above, ie. the points close to each other but having different distance from centre have same values of theta1 and different values of theta 2.
Variation of theta 1
Variation of theta 2 - Zoomed in
A bit Zoomed out version
Still zoomed out
PART D
Manifold observed is a torus. Here I am showing this from different angles.
The distribution between training, validation and test is of 2100,450,450 samples ie. 70,15,15 percentages.
The training from embedding of (v1,v2) to the control parameters is faster than the training from embedding of images to the control parameters because images contain a lot of unnecessary data whereas (v1,v2) are intrinsic parameters of the data.
From Isomap 2d parameters to Sines-Cosines
From Images Sines-Cosines
Left one corresponds to Isomap 2d parameters and the right one corresponds to images.
This is the path planning without any obstacles.
- First the pixels of the obstacles are identified. This is solely done by identifying the green(0,255,0) pixels from the image containing the obstacle.
- Then those images' indices are noted in which the pixels pertaining to the above are having any color other than black(0,0,0). They are stored in intersecting array.
- Then in isomap the distances pertaining to these intersecting indices are made infinite for a copy of the distance matrix on which we plan the shortest path.
- This path is plotted on the coordinates computed by the original distance matrix.
The obstacle 2 affects only 81 nodes out of 3000, ie. most of the nodes are still intact. In F obstacle was so near to the object that shortest path between 1 and 161 had to go through the other side. Here the obstacle can be avoided even after going through the same side. Thus position of obstacle alters the paths by a great deal in part f and g.
REFERENCES
Isomap.m this link.loadImageData from this link.