6Mathematical Background: Sine and Cosine Functions Periodic functionsGeneral form of sine and cosine functions:
7Mathematical Background: Sine and Cosine Functions Special case: A=1, b=0, α=1π3π/2π/2ππ/23π/2
8Mathematical Background: Sine and Cosine Functions (cont’d) Shifting or translating the sine function by a const bNote: cosine is a shifted sine function:
9Mathematical Background: Sine and Cosine Functions (cont’d) Changing the amplitude A
10Mathematical Background: Sine and Cosine Functions (cont’d) Changing the period T=2π/|α|consider A=1, b=0: y=cos(αt)α =4period 2π/4=π/2shorter periodhigher frequency(i.e., oscillates faster)Frequency is defined as f=1/TAlternative notation: cos(αt)=cos(2πt/T)=cos(2πft)
11Basis FunctionsGiven a vector space of functions, S, then if any f(t) ϵ S can be expressed asthe set of functions φk(t) are called the expansion set of S.If the expansion is unique, the set φk(t) is a basis.
12Image TransformsMany times, image processing tasks are best performed in a domain other than the spatial domain.Key steps:(1) Transform the image(2) Carry the task(s) in the transformed domain.(3) Apply inverse transform to return to the spatial domain.
18Continuous Fourier Transform (FT) Transforms a signal (i.e., function) from the spatial (x) domain to the frequency (u) domain.where
19Why is FT Useful? Easier to remove undesirable frequencies. Faster perform certain operations in the frequency domain than in the spatial domain.
20Example: Removing undesirable frequencies noisy signalTo remove certainfrequencies, set theircorresponding F(u)coefficients to zero!remove highfrequenciesreconstructedsignal
21How do frequencies show up in an image? Low frequencies correspond to slowly varying information (e.g., continuous surface).High frequencies correspond to quickly varying information (e.g., edges)Original ImageLow-passed
22Example of noise reduction using FT Input imageSpectrumBand-passfilterOutput image
23Frequency Filtering Steps 1. Take the FT of f(x):2. Remove undesired frequencies:3. Convert back to a signal:We’ll talk more about these steps later .....
24Definitions F(u) is a complex function: Magnitude of FT (spectrum): Phase of FT:Magnitude-Phase representation:Power of f(x): P(u)=|F(u)|2=
53DFT Properties: (8) Average value F(u,v) at u=0, v=0:So:
54Magnitude and Phase of DFT What is more important?Hint: use the inverse DFT to reconstruct the input image using magnitude or phase only informationmagnitudephase
55Magnitude and Phase of DFT (cont’d) Reconstructed image usingmagnitude only(i.e., magnitude determines thestrength of each component!)Reconstructed image usingphase only(i.e., phase determinesthe phase of each component!)