Feature Space Aleix M. Martinez aleix@ece.osu.edu Feature Space - - PDF document

1

Machine Learning & Pattern Recognition

Feature Space

Aleix M. Martinez aleix@ece.osu.edu

Feature Space

• Many problems in science and engineering

can be formulated as a PR one.

• For this, we need to define a feature space.
• A feature space is a collection of features

related to some properties of the object or event under study.

• Feature: An individually measurable

property of the phenomenon being

• bserved.

Example: DNA sequencing

• DNA sequencing – Nucleotide order:

– adenine (A), – guanine (G), – cytosine (C), – thymine (T).

• Each sequence is preceded by a 5’ marker

and ends with a 3’.

• E.g.: 5'-TAATGTCG-3'.

Discrete Features => Discrete feature space

2 ( )

T p

x x , ,

1 !

= x { }

T C G A xi , , , Î

Discrete feature space The four bases are detected using different fluorescent labels. These are detected and represented as 'peaks' of different colors. The Human Genome Project was funded at many laboratories around the U.S. by the Department of Energy (DOE), and the National Institutes of Health (NIH).

Continuous Feature Space

Example: Faces (appearance-based)

• In computer vision (and image processing)

it may be convenient to represent images as a sequence of pixel intensities.

3

Example: Shape Analysis

• In many applications of shape analysis, such

as morphometrics, biology, psychology, and image processing, 2D shapes are represented as feature vectors in the complex domain.

. . .

• r

, . .

2 2 1 2 1

• Î

= ÷ ÷ ÷ ÷ ÷ ÷ ø ö ç ç ç ç ç ç è æ Â Î = ÷ ÷ ÷ ÷ ÷ ÷ ø ö ç ç ç ç ç ç è æ

p p p p

S x x x x x x C x x

.

Shift and Scale Invariance

, ] ,..., , [

2 2 1 1 p T p p

C iy x iy x iy x u Î + + + =

.

u

.

1

• Î
• =

p N

C u u u

2

• Î

=

p N N

CS u u z

Im Re

1 = r

) ( ) (

q i

ze f z f =

) | ( } exp{ ) ( )} ( ) exp{( ) ( ) | (

* *

A z f Aze z e A c ze A ze A c A ze f

i i CB i i CB i

= = =

• q

q q q q

}, exp{ ) ( ) | (

*Az

z A c A z f

CB

=

4

Spherical Feature Spaces

• In many cases, the data describing our

“object” is spherical (e.g., circular).

Human Evolution

Spherical Feature Spaces

• Shape-based object recognition: we would like
• ur algorithm to be invariant to scale and in-plane

rotations.

=

• Appearance-based recognition: brightness

intensity should not affect recognition.

= =

Norm normalization

× s × s × ×h s

5

Norm and Var normalization

Invariant to intensity Invariant to scale

x x x = ~

T

x

T

y

T

h s x × ×

T

s x ×

T

s y ×

T

h s y × × . ~

1 2 1 1 å =

• =

p i i p i i

x x x

Variance normalization: Norm normalization:

. ~

1

• Î

=

p

S x x x Sp-1