Pyramid Coding and Subband Coding Predictive pyramids Transform pyramids Subband coding Perfect reconstruction filter banks Quadrature mirror filter banks Octave band splitting Transform coding as a special case of subband coding Thomas Wiegand: Digital Image Communication Pyramids and Subbands
Interpolation Error Coding, I Input picture Q - Reconstructed picture Subsampling Interpolator - Q Coder includes Decoder Subsampling Interpolator Sample encoded in current stage Previously coded sample Thomas Wiegand: Digital Image Communication Pyramids and Subbands 2
Interpolation Error Coding, II original image transmitted signals Thomas Wiegand: Digital Image Communication Pyramids and Subbands 3
Predictive Pyramid, I Input picture Q - Reconstructed picture Filtering Interpolator Subsampling - Q Coder includes Decoder Filtering Subsampling Interpolator Sample encoded in current stage Thomas Wiegand: Digital Image Communication Pyramids and Subbands 4
Predictive Pyramid, II Number of samples to be encoded = 2 N N... = N N Subsampling factor x number of original image samples Thomas Wiegand: Digital Image Communication Pyramids and Subbands 5
Predictive Pyramid, III original image transmitted signals transmitted signals Thomas Wiegand: Digital Image Communication Pyramids and Subbands 6
Comparison: Interpolation Error Coding vs. Pyramid, I Resolution layer # lowest resolution, interpolated to original size for display Interpolation Error Coding Pyramid Thomas Wiegand: Digital Image Communication Pyramids and Subbands 7 7
Comparison: Interpolation Error Coding vs. Pyramid, II Resolution layer #, interpolated to original size for display Interpolation Error Coding Pyramid Thomas Wiegand: Digital Image Communication Pyramids and Subbands 8
Comparison: Interpolation Error Coding vs. Pyramid, III Resolution layer #2, interpolated to original size for display Interpolation Error Coding Pyramid Thomas Wiegand: Digital Image Communication Pyramids and Subbands 9
Comparison: Interpolation Error Coding vs. Pyramid, IV Resolution layer #3 Interpolation Error Coding Pyramid = original Thomas Wiegand: Digital Image Communication Pyramids and Subbands
Subband Coding Transmitter Analysis filterbank Synthesis filterbank Receiver Input signal F k Q k G Reconstructed signal F F M k Q k k Q k M M G G M Number of degrees of freedom is preserved: Perfect reconstruction filterbank required K K... K M = Thomas Wiegand: Digital Image Communication Pyramids and Subbands
Thomas Wiegand: Digital Image Communication Pyramids and Subbands 2 Two Two-Channel Channel Filterbank Filterbank 2 2 2 2 F F G G X X Aliasing cancellation if : Aliasing ] [ 2 ] [ 2 ˆ π π π = X G F G F X G F G F X π π = = F G F G
Example : Two-Channel Filterbank with Perfect Reconstruction Analysis filter impulse responses: Lowpass band Highpass band Synthesis filter impulse responses: Lowpass band: Highpass band:, 2, 6, 2, 4, 2, 4, 2, 4, 2, 6, 2, 4 Frequency response 2 F G π 2 Frequency G F π Thomas Wiegand: Digital Image Communication Pyramids and Subbands 3
Quadrature Mirror Filters QMF QMFs achieve aliasing cancellation by choosing F = F π = G = G π Example: 6-tap QMF filterbank: Highpass band is the mirror image of the lowpass band in the frequency domain frequency Thomas Wiegand: Digital Image Communication Pyramids and Subbands 4
Cascaded Analysis / Synthesis Filterbanks Thomas Wiegand: Digital Image Communication Pyramids and Subbands 5
Octave Band Splitting Recursive application of a two-band filter bank to the lowpass band of the previous stage yields octave band splitting: frequency Same concept, but derived from wavelet theory: dyadic wavelet decomposition. Thomas Wiegand: Digital Image Communication Pyramids and Subbands 6
Separable 2D Filterbank,, I y y y x x x y y y x y x...etc x x y y x x Thomas Wiegand: Digital Image Communication Pyramids and Subbands 7
Separable 2D Filterbank,, II Thomas Wiegand: Digital Image Communication Pyramids and Subbands 8
Subband Coding vs. Transform Coding, I Transform coding is a special case of subband coding with: - Number of bands = order of transform N - Subsampling factor K = N - Length of impulse responses of analysis/synthesis filters N Filters used in subband coders are not in general orthogonal. Thomas Wiegand: Digital Image Communication Pyramids and Subbands 9
Subband Coding vs. Transform Coding, II Original image 8-channel Subband decomposition using DCT filters re-order 8x8 DCT Thomas Wiegand: Digital Image Communication Pyramids and Subbands 2
Summary: Pyramid Coding and Subband Coding Resolution pyramids with subsampling 2: horizontally and vertically Predictive pyramids: quantization error feedback closed loop Transform pyramids: no quantization error feedback open loop Pyramids: overcomplete representation of the image Application of pyramids: coarse-to-fine transmission, unequal error protection of resolution layers Subband coding: number of samples not increased Quadrature mirror filters: aliasing cancellation Transform coding is subband coding with non-overlapping impulse responses Thomas Wiegand: Digital Image Communication Pyramids and Subbands 2