Model-Based Segmentation, Reconstruction and Analysis of the Vertebral Body from Spinal CT - Spinal Imaging and Image Analysis - page 418

Fig. 40 An example representation of signed distance function. a A vertebral body shape.
b Signed distance function as an image intensity representation. c Level set representation of SDF

over PCA. First, it is easier to evaluate G accurately since its size using 2D-PCA is
much smaller. Second,

less time is required to determine the corresponding

eigenvectors [ 43 ].
2D-PCA projects an image matrix X, which is an m × n matrix onto a vector, b,

which is an n × 1 vector, by the linear transformation.

y ¼ Xb : ð 45 Þ

Suppose that there are M training images, the ith training image is denoted by Xi; i ;
(i = 1,2, … ,M) and the average image of all training samples is denoted by

M P i¼1 X i . Then, let us de

1

X ¼ ne the image covariance matrix G [ 43 ]:

M X

M

1

X i X

t X i X

G ¼ ð Þ ð Þ: ð 46 Þ

i¼1

It is clear that, the matrix G is n × n nonnegative de nite matrix.

find a projection axis that maximizes
b t Gb. The optimal K projection axes b k , where k = 1,2,

Similar to PCA, the goal of 2D-PCA is to

,K, that maximize the
above criterion are the eigenvectors of G corresponding to the largest K eigenvalues.
For an image X, we can use its reconstruction X de

…

ned below to approximate it.

X

K

X ¼ X þ

y k b t k ;

ð 47 Þ

k¼1

X where y k ¼ b k is called the kth principal component vector of the sample

image X. The principal component vectors obtained are used to form an m

ð X Þ

× K ma-

trix Y =[y 1 ,y 2 , … ,y K ] and let B =[b 1 ,b 2 , … ,b K ], then we can rewrite ( 47 ) as:

X

¼ X YB t :

þ ð 48 Þ