Canny边缘检测C#版

Lecture Notes on the Gaussian Distribution

Hairong Qi

The Gaussian distribution is also referred to as the normal distribution or the

bell curve distribution for its bell-shaped density curve. There’s a saying that

within the image processing and computer vision area, you can answer all ques-

tions asked using a Gaussian. The Gaussian distribution is also the most popularly

used distribution model in the field of pattern recognition. So let’s take a closer

look at it.

1 The Definition

mean vector, calculated by

µ = E[x] =

and Σ is the d × d covariance matrix, calculated by

Σ = E[(x − µ)(x − µ

T

(x − µ)(x − µ)

T

p(x)dx

with the following form.

· · · · · · · · · · · ·

(2)

1

The covariance matrix is always symmetric and positive semidefinite, where pos-

itive semidefinite means that for all non-zero x ∈ R

, x

T

Σx ≥ 0. We normally

ij

i

j

. If the variables along

ij

= 0, and we would have a

diagonal covariance matrix,

(3)

If the covariances along each dimension is the same, then we’ll have an identify

matrix multiplied by a scalar,

2

With Eq. 4, the determinant of Σ becomes

1

σ

· · · 0

.

.

.

.

1

2

T

4

becomes

p(x

1

2

1

2πσ

) (7)

vector and the shape of the cluster is outlined by the covariance matrix. The

loci of points of constant density are hyperellipsoids for which the quadratic form

= (x − µ)

T

Σ

(x − µ) is constant. The principal axes of these hyperellipsoids

are given by the eigenvectors of Σ and the eigenvalues determine the lengths of

2

2