Math Review CS474/674 – Prof. Bebis

Math Review Complex numbers Sine and Cosine Functions Sinc function Vector Basis Function Basis

Complex Numbers A complex number x is of the form: α: real part, b: imaginary part Addition: Multiplication:

Complex Numbers (cont’d) Magnitude-Phase (i.e., vector) representation: Magnitude: Phase: φ Magnitude-Phase notation:

Complex Numbers (cont’d) Multiplication using magnitude-phase representation Complex conjugate Properties

Complex Numbers (cont’d) Euler’s formula Properties j

Sine and Cosine Functions Periodic functions General form of sine and cosine functions: y(t)=Asin(αt+b) y(t)=Acos(αt+b)

Sine and Cosine Functions (cont’d) Special case: A=1, b=0, α=1 period=2π π 3π/2 π/2 π π/2 3π/2

Sine and Cosine Functions (cont’d) Changing the phase shift b: Note: cosine is a shifted sine function:

Sine and Cosine Functions (cont’d) Changing the amplitude A:

Sine and Cosine Functions (cont’d) Changing the period T=2π/|α|: Asssume A=1, b=0: y=cos(αt) α =4 period 2π/4=π/2 shorter period higher frequency (i.e., oscillates faster) frequency is defined as f=1/T Alternative notation: cos(αt) or cos(2πt/T) or cos(t/T) or cos(2πft) or cos(ft)

Sinc function The sinc function is defined as: In image processing, we use the normalized sinc function: The normalization causes the definite integral of the function over the real numbers to equal 1 (whereas the same integral of the un-normalized sinc function has a value of π). Also, it crosses the x-axis at integer locations.

Vectors An n-dimensional vector v is denoted as follows: The transpose vT is denoted as follows:

Dot product Given vT = (x1, x2, . . . , xn) and wT = (y1, y2, . . . , yn), their dot product defined as follows: (scalar) or

Linear combinations of vectors A vector v is a linear combination of the vectors v1, ..., vk: where c1, ..., ck are scalars Example: any vector in R3 can be expressed as a linear combinations of the unit vectors i = (1, 0, 0), j = (0, 1, 0), and k = (0, 0, 1)

Linear dependence A set of vectors v1, ..., vk are linearly dependent if at least one of them is a linear combination of the others. (i.e., vj does not appear at the right side)

Linear independence A set of vectors v1, ..., vk is linearly independent if implies Example:

Space spanning A set of vectors S=(v1, v2, . . . , vk ) span some space W if every vector in W can be written as a linear combination of the vectors in S Example: the vectors i, j, and k span R3 w

Vector basis A set of vectors (v1, ..., vk) is said to be a basis for a vector space W if (1) (v1, ..., vk) are linearly independent (2) (v1, ..., vk) span W Standard bases: R2 R3 Rn

Uniqueness of Vector Expansion Suppose v1, v2, . . . , vn represents a base in W, then any v є W has a unique vector expansion in this base: The coefficients of the expansion can be computed as follows:

Orthogonal Basis A basis with orthogonal/orthonormal basis vectors. Any set of basis vectors (x1, x2, . . . , xn) can be transformed to an orthogonal basis (o1, o2, . . . , on) using the Gram-Schmidt orthogonalization. k

Basis Functions We can compose arbitrary functions x(t) in “function space S” as a linear combination of simpler functions: The set of functions φi(t) are called the expansion set of S. If the expansion is unique, the set φi(t) is a basis.

Basis Functions (cont’d) The basis functions φi(t) are orthogonal in some interval [t1,t2] if: For complex valued basis sets:

Basis Functions (cont’d) The coefficients of the expansion can be computed as: Example: polynomial basis functions φi(t) = ti Taylor Series expansion