# Point Distribution

This is a model for representing the mean geometry of a shape and some statistical modes of geometric variation inferred from a training set of shapes.Point distribution models rely on landmark points.A landmark is an annotating point posed by an anatomist onto a given locus for every shape instance across the training set population.The same landmark will designate the tip of the index in a training set of 2D hands outlines.

A point within a network where the cable or fiber terminates,this point provides a point of entry to terminate or test the networks.

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Degenerate Distribution:

In mathematics a degenerate distribution is the probability distribution of a discrete random variable whose support consists of only one value.

Constant random variable:

In probability theory a cnstant random variable is a discrete random variable that takes a constant value regardless of any event that occurs.

Pr(X=c) =1,

For practical proposes the distinction between x being constant or almost unimportant.

Point distribution model:

A set of training images are manually landmarked with enough corresponding landmarks to sufficiently approximate the geometry of the original shapes.These landmarks are aligned using the generalized procrustes analysis which minimises the least squared error between the points.

The probability distribution is the major part in the probability theory and the statistics. The probability distribution is used to determine the number of possibility for the occurrence of an event. The most commonly used probability distributions are the binomial distribution, geometric distribution, normal distribution and the gamma distribution. These above mentioned distributions are included in the discrete and continuous probability distribution. The major type of the probability distribution is the discrete probability distribution and the continuous probability distribution. This article has the study about the point distribution model.

k aligned landmarks in two dimensions are given as

X = (x1,y1,………xk,yk)

It’s important to note that the landmark i ‘epsi’ {1,………k} should represent the same anatomical location.

The shape outlines are reduced to sequences of k landmarks,so that a given training shape is defined as the vector X ‘epsi’ R2k.

Assuming the scattering is gaussian in this space,the matrix of the top d eigen vector is given as P ‘epsi’ R2k*d and each eigen vector describe a principal mode of variation along the set.

Limitations of the linear point distribution model:

1) A good deformable model should be accurate,specific and compact.

2) An accurate model includes all valid shapes.

3) A specific model excludes all invalid shapes.

4) A compact model uses the smallest number of parameters possible to describe a shape.

The linear point distribution model assumes that the set of all valid shapes forms a gaussian distribution about some mean point in the shape space.

The linear point distribution model is forced to non-linear deformations by the combination of two or more linear deformations.Such models are not compact because the dimensionality is increased and not specific because invalid shapes can be produced by an invalid combination of linear deformations.