1 Wavelets and compression Dr Mike Spann

2 Contents Scale and image compression Signal (image) approximation/prediction – simple wavelet construction Statistical dependencies in wavelet coefficients – why wavelet compression works State-of-the-art wavelet compression algorithms

3 Image at different scales

4 Correlation between features at different scales

5 Wavelet construction – a simplified approach Traditional approaches to wavelets have used a filterbank interpretation Fourier techniques required to get synthesis (reconstruction) filters from analysis filters Not easy to generalize

6 3 steps Split Predict (P step) Update (U step) Wavelet construction – lifting

7 Example – the Haar wavelet S step Splits the signal into odd and even samples even samples odd samples

8 For the Haar wavelet, the prediction for the odd sample is the previous even sample : l Example – the Haar wavelet P step Predict the odd samples from the even samples

9 Example – the Haar wavelet l l Detail signal :

10 The signal average is maintained : Example – the Haar wavelet U step Update the even samples to produce the next coarser scale approximation

11 ….. Summary of the Haar wavelet decomposition Can be computed ‘in place’ : ….. 1/2 P step U step

12 Then merge even and odd samples Merge Inverse Haar wavelet transform Simply run the forward Haar wavelet transform backwards!

13 General lifting stage of wavelet decomposition - Split P U +

14 Multi-level wavelet decomposition lift … We can produce a multi-level decomposition by cascading lifting stages

15 General lifting stage of inverse wavelet synthesis - Merge P U +

16 lift …... lift We can produce a multi-level inverse wavelet synthesis by cascading lifting stages Multi-level inverse wavelet synthesis

17 Advantages of the lifting implementation Inverse transform  Inverse transform is trivial – just run the code backwards  No need for Fourier techniques  Generality  The design of the transform is performed without reference to particular forms for the predict and update operators  Can even include non-linearities (for integer wavelets)

18 Example 2 – the linear spline wavelet A more sophisticated wavelet – uses slightly more complex P and U operators Uses linear prediction to determine odd samples from even samples

19 Linear prediction at odd samples Original signal Detail signal (prediction error at odd samples) The linear spline wavelet P-step – linear prediction

20 The linear spline wavelet The prediction for the odd samples is based on the two even samples either side :

21 The linear spline wavelet The U step – use current and previous detail signal sample

22 Preserves signal average and first-order moment (signal position) : The linear spline wavelet

23 1/4 -1/2 P step U step The linear spline wavelet Can still implement ‘in place’

24 Summary of linear spline wavelet decomposition Computing the inverse is trivial : The even and odd samples are then merged as before

25 Wavelet decomposition applied to a 2D image detail approx lift approx

26 approx detail approx detail approx detail lift Wavelet decomposition applied to a 2D image

27 Why is wavelet-based compression effective? Allows for intra-scale prediction (like many other compression methods) – equivalently the wavelet transform is a decorrelating transform just like the DCT as used by JPEG Allows for inter-scale (coarse-fine scale) prediction

28 1 level Haar 1 level linear spline 2 level Haar Original Why is wavelet-based compression effective?

29 Why is wavelet-based compression effective? Wavelet coefficient histogram

30 Entropy Original image level Haar wavelet level linear spline wavelet level Haar wavelet level linear spline wavelet4.57 Why is wavelet-based compression effective? Coefficient entropies

31 X Why is wavelet-based compression effective? Wavelet coefficient dependencies

32 Why is wavelet-based compression effective? Lets define sets S (small) and L (large) wavelet coefficients The following two probabilities describe interscale dependancies

33 Why is wavelet-based compression effective? Without interscale dependancies

Why is wavelet-based compression effective? Measured dependancies from Lena

35 X X1X1 X8X8 Why is wavelet-based compression effective? Intra-scale dependencies

Why is wavelet-based compression effective? Measured dependancies from Lena

37 Why is wavelet-based compression effective? Have to use a causal neighbourhood for spatial prediction

38 We will look at 3 state of the art algorithms Set partitioning in hierarchical sets (SPIHT) Significance linked connected components analysis (SLCCA) Embedded block coding with optimal truncation (EBCOT) which is the basis of JPEG2000 Example image compression algorithms

39 lsb msb … xx … xxxx … xxxxxxx … xxxxxxxxxxxxx1… xxxxx x xxxxxxxx… Coeff. number …… The SPIHT algorithm Coefficients transmitted in partial order 0

40 2 components to the algorithm Sorting pass Sorting information is transmitted on the basis of the most significant bit-plane Refinement pass Bits in bit-planes lower than the most significant bit plane are transmitted The SPIHT algorithm

41 N= msb of (max(abs(wavelet coefficient))) for (bit-plane-counter)=N downto 1 transmit significance/insignificance wrt bit-plane counter transmit refinement bits of all coefficients that are already significant The SPIHT algorithm

42 The SPIHT algorithm Insignificant coefficients (with respect to current bitplane counter) organised into zerotrees

43 The SPIHT algorithm Groups of coefficients made into zerotrees by set paritioning

44 … ……… …. bitstream The SPIHT algorithm SPIHT produces an embedded bitstream

45 The SLCCA algorithm Bit-plane encode significant coefficients Wavelet transform Quantise coefficients Cluster and transmit significance map

46 The SLCCA algorithm The significance map is grouped into clusters

47 Seed Significant coeff Insignificant coeff The SLCCA algorithm Clusters grown out from a seed

48 Significance link The SLCCA algorithm Significance link symbol

49 Image compression results Evaluation Mean squared error Human visual-based metrics Subjective evaluation

50 Usually expressed as peak-signal-to-noise (in dB) Image compression results Mean-squared error

51 Image compression results

52 Image compression results

53 SPIHT 0.2 bits/pixelJPEG 0.2 bits/pixel Image compression results

54 SPIHT JPEG Image compression results

55 EBCOT, JPEG2000 JPEG2000, based on embedded block coding and optimal truncation is the state-of-the-art compression standard Wavelet-based It addresses the key issue of scalability SPIHT is distortion scalable as we have already seen JPEG2000 introduces both resolution and spatial scalability also An excellent reference to JPEG2000 and compression in general is “JPEG2000” by D.Taubman and M. Marcellin

56 Resolution scalability is the ability to extract from the bitstream the sub-bands representing any resolution level … ……… …. bitstream EBCOT, JPEG2000

57 Spatial scalability is the ability to extract from the bitstream the sub-bands representing specific regions in the image Very useful if we want to selectively decompress certain regions of massive images … ……… …. bitstream EBCOT, JPEG2000

58 Introduction to EBCOT JPEG2000 is able to implement this general scalability by implementing the EBCOT paradigm In EBCOT, the unit of compression is the codeblock which is a partition of a wavelet sub-band Typically, following the wavelet transform,each sub-band is partitioned into small blocks (typically 32x32)

59 Introduction to EBCOT Codeblocks – partitions of wavelet sub- bands codeblock

60 Introduction to EBCOT A simple bit stream organisation could comprise concatenated code block bit streams …… Length of next code-block stream

61 Introduction to EBCOT This simple bit stream structure is resolution and spatially scalable but not distortion scalable Complete scalability is obtained by introducing quality layers Each code block bitstream is individually (optimally) truncated in each quality layer Loss of parent-child redundancy more than compensated by ability to individually optimise separate code block bitstreams

62 Introduction to EBCOT Each code block bit stream partitioned into a set of quality layers … … …

63 EBCOT advantages Multiple scalability Distortion, spatial and resolution scalability Efficient compression This results from independent optimal truncation of each code block bit stream Local processing Independent processing of each code block allows for efficient parallel implementations as well as hardware implementations

64 EBCOT advantages Error resilience Again this results from independent code block processing which limits the influence of errors

65 Performance comparison A performance comparison with other wavelet-based coders is not straightforward as it would depend on the target bit rates which the bit streams were truncated for With SPIHT, we simply truncate the bit stream when the target bit rate has been reached However, we only have distortion scalability with SPIHT Even so, we still get favourable PSNR (dB) results when comparing EBCOT (JPEG200) with SPIHT

66 Performance comparison We can understand this more fully by looking at graphs of distortion (D) against rate (R) (bitstream length) R D R-D curve for continuously modulated quantisation step size Truncation points

67 Performance comparison Truncating the bit stream to some arbitrary rate will yield sub-optimal performance R D

68 Performance comparison

69 Performance comparison Comparable PSNR (dB) results between EBCOT and SPIHT even though: Results for EBCOT are for 5 quality layers (5 optimal bit rates) Intermediate bit rates sub-optimal We have resolution, spatial, distortion scalability in EBCOT but only distortion scalability in SPIHT