Pyramid Coding and Subband Coding
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1 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
2 Interpolation Error Coding, II original image transmitted signals Thomas Wiegand: Digital Image Communication Pyramids and Subbands 3 Predictive Pyramid, I Input picture Filtering Q - Interpolator Reconstructed picture Subsampling - Q Coder includes Decoder Filtering Subsampling Interpolator Sample encoded in current stage Thomas Wiegand: Digital Image Communication Pyramids and Subbands 4
3 Predictive Pyramid, II Number of samples to be encoded = (...) = N 2 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
4 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
5 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
6 Subband Coding Transmitter Analysis filterbank Synthesis filterbank Receiver Input signal F ( k Q k ω) G ( ω) Reconstructed signal F ( ω) F ( M ω) k Q k k Q k M M G ( ω) G M ( ω) Number of degrees of freedom is preserved: Perfect reconstruction filterbank required... = K K K M Thomas Wiegand: Digital Image Communication Pyramids and Subbands Two-Channel Filterbank X(ω) F ( ω) 2 2 G ( ω) X( ω) F ( ω) 2 2 G ( ω) Xˆ( ω ) = [ F ( ω ) G ( ω ) F ( ω ) G ( ω )] X ( ω ) 2 [ F ( ω π ) G ( ω ) F ( ω π ) G ( ω )] X ( ω π ) 2 Aliasing Aliasing cancellation if : G ( ω) = F( ω π) G ( ω) = F ( ω π) Thomas Wiegand: Digital Image Communication Pyramids and Subbands 2
7 Example : Two-Channel Filterbank with Perfect Reconstruction Analysis filter impulse responses: Lowpass band Highpass band (, 2, 6, 2, ) 4 (, 2, ) 4 Synthesis filter impulse responses: Frequency response 2 F G G F Lowpass band: Highpass band: (, 2, ) 4 (, 2, 6, 2, ) 4 π 2 Frequency π Thomas Wiegand: Digital Image Communication Pyramids and Subbands 3 Quadrature Mirror Filters (QMF) QMFs achieve aliasing cancellation by choosing F( ω) = F ( ω π) = G ( ω) = G ( ω π) Example: 6-tap QMF filterbank: Highpass band is the mirror image of the lowpass band in the frequency domain frequency ω Thomas Wiegand: Digital Image Communication Pyramids and Subbands 4
8 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.
9 Separable 2D Filterbank,, I...etc Thomas Wiegand: Digital Image Communication Pyramids and Subbands 7 Separable 2D Filterbank,, II Thomas Wiegand: Digital Image Communication Pyramids and Subbands 8
10 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
11 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
Pyramid Coding and Subband Coding
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