ijing 2008 
The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part Bl. Beijing 2008 
strie and doesn’t 
ns of A. In this 
A. to 3D object 
ICA. To the best 
;d to appearance- 
estimation. The 
>n of the IPCA - 
Dject recognition, 
ether IPCA-ICA 
: the appearance- 
g an input image 
ig the resulting 
order to find the 
s of n images and 
takes input image 
hich is passed as 
components will 
>m the previous 
IPCA returns the 
:nts subspaces of 
/ vector, FastICA 
eq(3) where the 
iximize the non- 
i 
ion 
ling non-Gaussian 
stor x with a 
atrix C satisfies 
A.x = C.x (2) 
By replacing C with the sample covariance matrix 
n 
and using v= A.X we will get the nth 
/=l 
eigenvector i.e. v(n) for the n images of the database. 
Then, this vector will be the initial direction in the FastICA 
algorithm. 
w=v(l) (3) 
v(l) is the first principal component. 
The FastICA[16] algorithms will repeat until convergence the 
following rule: 
Wnew=E[v(l).g(w T .v(l))]-E[g’(w T v(l))].w (4) 
Where g’(x) is the derivative of the function g(x) (6). It should be 
noted that this algorithm uses an approximation of negentropy in 
order to assure the non-Gaussianity of the independent vectors. 
Before starting the calculation of negentropy, a non-quadratic 
function G should be chosen, for example, 
G(u)=-exp(-u 2 /2 (5) 
And its derivative: 
g(u)=u.exp(-u 2 /2) (6) 
In general, the corresponding non-Gaussian vector w, for the 
estimated eigenvector v(l), will be estimated using the following 
repeated rule: 
Wnew=E[v(l).g(w r .v(l))]-E[g’(w r v(l))].w (7) 
The previous discussion only estimates the first non-Gaussian 
vector. One way to compute the other higher order vectors is to 
start with a set of orthonormalized vectors, update them using the 
suggested iteration step and recover the orthogonality. Further, 
the non-Gaussian vectors should be orthogonal to each other in 
order to ensure the independency. So, it helps to generate 
“observations” only in a complementary space for the 
computation of the higher order eigenvectors. After convergence, 
the non-Gaussian vector will also be enforced to be orthogonal, 
since they are estimated in complementary spaces. As a result, all 
the estimated vectors w k will be: Non-Gaussian according to the 
learning rule in the algorithm. . Independent according to the 
complementary spaces introduced in the algorithm. 
The nearest neighbor algorithm is used to evaluate the object 
recognition technique. Each Object Database is truncated into 
two sets. The training set that contains images used to calculate 
the independent non- Gaussian vectors and come up with the 
appropriate basis and, the test set that contains images to be 
tested by the Object recognition algorithm in order to evaluate 
the performance of the proposed method. The whole set of 
training images (rows in the image data matrix) are projected into 
the basis found in order to calculate the coordinates of each 
image with respect to the basis v^n Each new testing image v tes t 
is compared to whole set of training images v^ in order to come 
up with nearest one that corresponds to the maximum k in (8). 
[k] =nearest_match (V test v train ) (8) 
k th image gives the index of the object that is recognized from 
the database. 
3. RESULTS 
Object recognition and pose estimation experiments were 
performed by using Matlab7.1. The object set is COIL-20 
(Columbia Object Image Library) database [14]. The images are 
stored at every 5° of pose angle, from 0° to 360°. Hence 72 
images of each object, and 1440 total number of images. The size 
of the images is rescaled to 64x64.The 0° pose angle views are 
shown in Fig.3 the maximum pixel value is 255. 
Fig 3 COIL database of 20 objects 
To construct the non Gaussian space of the object, a few of the 
images were chosen as the training images. The representations 
of images make a manifold with the pose angle as the parameter