1. Math Review
CS474/674 – Prof. Bebis

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

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

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

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

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

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

8. 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

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

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

11. 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)

12. 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.

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

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

15. 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)

16. 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)

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

18. 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

19. 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

20. 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:

21. 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

22. 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.

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

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