Home Assignment # 4

Discovering Manifolds in Images

Codes used : Code

1. Isomap

a) Residual error v/s Dimensions:-

b) Discussion: The given set of images of the 2D planar robot arm, have a basic fixed structure (the two hinged arms) with the variation occurring only along two dimensions (theta1 and theta2). Thus, from the above graph we can conclude that Isomap is a pretty good estimation of the dimensionality of the input 16k dimension space.

c) Plotting boundary thetas:-

Discussion: The boundary thetas are observed to lie at the boundary points in the y1-y2 graph as well. This implies that this set of bases (y1-y2) has some correlation with the (theta1-theta2) bases.

d) Graph for randomMotion1K.zip :-

Discussion: It could be observed that the residual for the set of 1000 images is lesser compared to that of the set of 100 images. This result is according to our expectations because the greater the size of observation set, lesser would be the error.

e) Correlation between (theta1, theta2) & (y1,y2)

Theta1	Theta2	Y1	Y2
21.70828	23.289759	-969	1705
20.963717	26.445587	-321	1917
25.773035	19.94295	2059	486
26.921942	10.453383	2059	-723
28.73904	16.162862	-408	-723
24.460459	12.614105	3223	-826
27.325196	7.879244	-1683	-224
23.942938	21.592861	-837	-1008
28.450352	13.065413	931	846
27.155386	4.504699	1804	-733
24.558251	7.640685	-1865	-1372
25.45228	12.361451	-2989	-706
25.685854	23.495475	-1018	-390
28.677191	4.586599	3064	1104
22.755089	12.452133	-1106	-1420
29.240948	9.477505	-3374	196
27.894608	4.657732	1752	-1258
20.822057	27.990783	-1224	-1395
20.255749	12.828469	221	1995
23.821824	9.27291	-4930	431

Discussion: From the above correlation plots, we can observe that theta1 varies linearly with y1 while theta2 shows some non-linear trend. If we are able to find the transformation matrices, we could transform the coordinates from one bases to other.

2. Linear Mapping and Reconstruction:-

a) 2-D graph using PCA:-

Eigen values:-

9520000

403000

333000

102000

720000

2260

163

127

Discussion: It could be observed that the first two eigen values are much more larger than the remaining values

b) Reconstruction of the image using PCA:-

Discussion: The above image is much unclear compared to the original image. Thus, we can conclude that PCA is not effective in contracting information contained in the image.

3. Non Linear Mapping & Reconstruction:-

a) 2-D representation using LLE

b) Reconstruction using LLE:-

c) Reconstruction using Isomap:-

Discussion: It could be observed that the reconstruction of the mapping produced by LLE has the maximum clarity than PCA or Isomap, PCA being the least clear. This proves that the data does not suffer much loss of information when we use LLE or Isomap to do multidimensional scaling. Since the input images had variations only along two directions (theta1, theta2) , we can say that LLE and Isomap actually captured the theta1, theta2 information and thus were able to reconstruct the image in an efficient manner.

LLE and Isomap (non-linear mapping) use geodesic manifold distances while PCA(linear mapping) uses Euclidean distances. Hence the former methods are well capable of preserving the information about the local neighborhood of each point on the manifold than PCA.

4. Complete Motion_2K:-

a) 2-D representation using Isomap:-

b) 2-D representation using PCA:-