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

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 observed. 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 1

Discrete feature space ( ) = T x x 1 ! , , x p { } Î x i A , G , C , T 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. 2

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. æ x ö æ x ö ç 1 ÷ ç 1 ÷ ç x ÷ ç x ÷ 2 2 ç ÷ ç ÷ . = Î Â p . = Î p - 2 x , or x S C . ç ÷ ç ÷ . . ç ÷ ç ÷ ç ÷ ç ÷ x x è ø è ø p p Shift and Scale Invariance = + + + Î u [ x iy , x iy ,..., x iy ] T C p , 1 1 2 2 p p . Im . = - Î p - 1 u u u C N u u = Î - z N CS p 2 u N . Re = r 1 q = i f ( z ) f ( ze ) = f ( z | A ) c ( A ) exp{ z * Az }, CB q = q q = f ( ze i | A ) c ( A ) exp{( ze i ) * A ( ze i )} CB - i q * i q = c ( A ) exp{ e z Aze } f ( z | A ) CB 3

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 our algorithm to be invariant to scale and in-plane rotations. × s = • Appearance-based recognition: brightness intensity should not affect recognition. × Norm = s × h × = s normalization 4

Norm and Var normalization × × T s h x × × s h y T Invariant × s x T y T × T to s y T x intensity Invariant to scale x ~ x = x Variance Norm normalization: normalization: x ~ S p-1 = Î p - 1 x x S . ~ = x i . 1 å = x i p x 2 p - 1 i i 1 5