CS 450: Other transform provides the fourier transform; discrete consine transform (DCT); different transform; hotelling (karhunen - leuve) transform; co-joint repretations; energy compaction.
CS 450 Other Transforms Other Transforms A transform is a change in the numeric representation of a signal that preserves all of the signal’s information Transforms can be thought of as a change of coordinates into some coordinate system (basis set) They all have the same basic form: choose your basis functions get the weights using inner product of signal and basis functions reconstruct by adding weighted basis functions CS 450 Other Transforms The Fourier Transform Basis functions: complex harmonics ei2πst or ei2πsn/N Transform (calculating the weights of each basis function): ∞ F (s) = f (t) e−i2πst dt −∞ Inverse Transform (putting together the weights): ∞ f (t) = −∞ F (s) ei2πst ds CS 450 Other Transforms Discrete Cosine Transform (DCT) Basis functions: real-valued cosines (2n + 1)sπ fu [n] = α(s) cos 2N where α(u) = N for s = N for s = 1, 2, , N − CS 450 Other Transforms Discrete Cosine Transform (DCT) Transform: C[s] = f [n] α(s) cos (2n + 1)sπ 2N C[s] α(s) cos (2n + 1)sπ 2N n Inverse Transform: f [n] = s Treats signal as alternating-periodic Real-valued transform! CS 450 Other Transforms Different Transforms Transform Basis Functions Good for Fourier Sines and Cosines Frequency analysis, Convolution Cosine Cosines Frequency analysis (but not convolution) Haar Square pulses of different widths and offsets Binary data Slant Ramp signals of different slopes and offsets First-order changes Wavelets Various Time/frequency analysis CS 450 Other Transforms Hotelling (Karhunen-Leuve) Transform Basis functions: eigenvectors of covariance matrix Idea: • Measure statistical properties of the relationship between pixels • Find the “optimal” relationships (eigenvectors) • Use these as basis functions Signal/image specific! CS 450 Other Transforms Wavelets Basis functions: • scaled (resized) copies of the same function • functions must have finite extent Stretching = “frequency” Limited extent = spatial localization CS 450 Other Transforms Co-joint Representations Signals are pure time/space domain—no frequency part Fourier Transforms are pure frequency domain—no spatial part Wavelets and other co-joint representations are • somewhat localized in space • somewhat localized in frequency Accuracy in the spatial domain is inversely proportional to accuracy in the frequency domain CS 450 Other Transforms Energy Compaction All of these transforms produce a more compact representation than the original image “Energy compaction” means large part of information content in small part of representation Representation Compaction And/But Image Poor Easily interpreted Fourier Good Convolution Theorem Cosine Better Fast Hotelling Best Basis functions are signal-specific Wavelets Good Some spatial representation as well .. .CS 450 Other Transforms The Fourier Transform Basis functions: complex harmonics ei2πst or ei2πsn/N Transform (calculating the weights of each basis... N for s = 1, 2, , N − CS 450 Other Transforms Discrete Cosine Transform (DCT) Transform: C[s] = f [n] α(s) cos (2n + 1)sπ 2N C[s] α(s) cos (2n + 1)sπ 2N n Inverse Transform: f [n] = s Treats... Transform: f [n] = s Treats signal as alternating-periodic Real-valued transform! CS 450 Other Transforms Different Transforms Transform Basis Functions Good for Fourier Sines and Cosines Frequency