Subband Coding

Abdou Youssef

1. Motivation

Problems with DCT-based compression
- Blocking artifacts, especially at low bitrate
- The methods for reducing blocking artifacts, such as overlapped transforms, are costly and complicated
- Applying DCT on the whole image, rather than on small blocks, ignores the significant differences in frequency contents in various regions of the image, thus leading to less quality-bitrate performance
Advantages of wavelets/subband coding
- They operate on the whole image as one single block
- Thus avoiding blocking artifacts
- While dynamically adjusting the spatial/frequency resolution to the appropriate level in various regions of the image
In practice, wavelets/subband coding performs as well as DCT and sometimes better, especially at low bitrate

2. Linear Filters

An LPF eliminates the high-frequency contents of any input signal, and preserves the low-frequency contents
An ideal LPF must then have its Fourier Transform as a nonzero constant in a frequency range , and zero in the remaining range $[a,\pi]$
Applications: Noise removal and image smoothing

A HPF eliminates the low-frequency contents of any input signal, and preserves the high-frequency contents
An ideal HPF must then have its Fourier Transform equal to zero in a frequency range , and equal to a nonzero constant in the remaining range $[a,\pi]$
Applications: Sharpening and edge detection

Ideal LPF's and HPF's are not realizable in practice, but many realizable filters are good approximations of ideal filters
A filter is called a finite-impulse-response (FIR) filter if has a finite number of taps; otherwise, the filter is called an infinite-impulse response (IIR) filter

Original Lena:

A one-level transform of Lena (Lena 1):

Lena reconstructed from just the low-frequency subband of Lena 1 (CR=4):

A two-level transform of Lena:

Lena reconstructed from just the low-frequency subband of Lena 2 (CR=16):

Classical filter design techniques for LPF's and HPF's
- Least Mean Square technique
- Butterworth technique
- Chebychev technique
Those techniques are for designing single filters, rather than a bank of four filters working together
The four filters for a subband coding system must have the perfection reconstruction property
- the output signal is identical to the input signal if no quantization takes place

The z-transform of a sequence is $X(z)=\sum_nx_nz^n$
If is the output of a linear filter given input , then
Therefore, for the subband coding scheme
- ${\hat X}(z)=P(z){\hat U}(z)+Q(z){\hat V}(z)$
- ${\hat U}(z)=\sum_n{\hat u}_nz^n=\sum_{n\ even}u_{n\over 2}z^n =\sum_nu_nz^{2n}=\sum_n{\overline u}_{2n}z^{2n}$
  $={1\over 2}\left[\sum_n{\overline u}_nz^n+\sum_n{\overline u}_nz^{-n}\right]$
  $={1\over 2}\left[{\overline U}(z)+{\overline U}(-z)\right]= {1\over 2}\left[G(z)X(z)+G(-z)X(-z)\right]$
- Similarly, ${\hat V}(z)= {1\over 2}\left[H(z)X(z)+H(-z)X(-z)\right]$
- Thus, ${\hat X}(z)={1\over 2}P(z)\left[G(z)X(z)+G(-z)X(-z)\right]+$
  ${1\over 2}Q(z)\left[H(z)X(z)+H(-z)X(-z)\right]$
- By regrouping we get
  
  ${\hat X}(z)={1\over 2}\left[G(z)P(z)+H(z)Q(z)\right]X(z)+$
  
  ${1\over 2}\left[G(-z)P(z)+H(-z)Q(z)\right]X(-z)$
To have ${\hat x}=x$ , we must have ${\hat X}(z)=X(z)$ , leading to the following perfect reconstruction (PR) condition:

$\begin{displaymath}G(z)P(z)+H(z)Q(z)=2\end{displaymath}$

$\begin{displaymath}G(-z)P(z)+H(-z)Q(z)=0\end{displaymath}$
Consequently, to get a subband filter bank (of four filters), one has to solve the two equations above, subject to the constraints that
- For HPF's
  - $H(-1)\neq 0$
  - $Q(-1)\neq 0$
- For LPF's
  - $H(1)\neq 0$
  - $Q(1)\neq 0$

Quantization approaches of the subbands
- Uniform scalar quantization
- Non-uniform scalar quantization
- Vector quantization
- One quantizer for all the subbands, or
- Different quantizers for different subbands
Prospects for optimal Max-LLoyd scalar quantization of high-frequency subbands
- Probability distribution of the pixel values in HF subbands: The generalized Gaussian distribution
  
  $\begin{displaymath}p(x)=ae^{-\vert bx\vert^r}\end{displaymath}$
  
  where
  
  $\begin{displaymath}b={1\over\sigma}\left({{\Gamma({3\over r})}\over{\Gamma({1\over r})}}\right)^{1\over 2}\end{displaymath}$
  
  $\begin{displaymath}a={{br}\over{2\Gamma({1\over r})}}\end{displaymath}$
  
  and $\sigma$ is the standard deviation of the underlying data
- Experimentation has shown that is about
Therefore, the sender need not send the decision levels and reconstruction levels of the Max-Lloyd quantizer; rather, only the standard deviation $\sigma$ need be sent.

Is VQ needed for high-frequency subbands?
- Answer: It depends on how good the filters are
- Under ideal filters, the high frequency coefficients are completely decorrelated, making VQ unnecessary (and rather undesirable)
- In practice, the farther the filters are from ideal, the more correlation ``leaks'' into the high-frequency subbands, thus opening the door for VQ
- With the commonly used filters, there is some correlation leakage; but there is still the tradeoff between the slight improvement brought by VQ and the high time overhead associated with VQ
Design Issues for VQ in subband coding
- One VQ table for all subbands, or
- One VQ table per subband, or
- One VQ table for the subbands of a whole class of images?
- How large should the vector size be?

Questions about the tree shape:
- What is the best shape?
- Is there a best shape for all images, or at least one best shape per class of images?
- If not, is there an efficient way of deciding the shape of the tree on-line?

Intuitively, the best filter set for a given signal is the one whose corresponding wavelet best resembles the signal in shape (i.e., in plot)
The data in the subbands have different plots than the original data, suggesting the use for different filters than the ones applied on the original data
For better understanding of this issue, one has to draw on the insight provided by wavelet theory, which is the subject matter of next lecture