5.2 GTD Transform Coder for Optimizing Coding Gain
5.2.2 Generalized Triangular Decomposition Transform Coder
and is demonstrated in Fig. 5.4 forM = 4.Here we have used the MINLAB(I) structure [79]. The multipliersskmare the entries of the matrixL−11 .For example, whenM = 4,
L−11 =
1 0 0 0
s21 1 0 0 s31 s32 1 0 s41 s42 s43 1
.
The bit loading formula becomes
bi=b+1
2log2 R2ii
det(Rz)M1 =b+1
2log2 R2ii
det(Rx)M1 , (5.9)
where we have useddet(Rz) = det(PTRxP) = det(Rx).Note that the signal variance of the input to theith quantizer isR2ii.Again, by using the bit loading formula (5.9), the MSEs of the outputs of the quantizers are identical. This is the same property that the KLT and the PLT have, as introduced in Sec. 5.2.1.
The AM-MSE is invariant to the orthonormal matrixPat the decoder, therefore the AM-MSE is the same as the one for the PLT part for the transform coding ofz. As in eq. (5.6), the MSE is
EGT D=c2−2bdet(Rz)M1 =c2−2bdet(Rx)M1, (5.10)
which is the same as the MSE for KLT and PLT with optimal bit allocation. Note that this result is true because of the minimum noise structure for the PLT (which has unit noise gain).
P
Q2Q1
s21 s31 s41 !s41 !s31 !s21
P
x x ^
P T 2 Q3 s32 s42
s43 !s43
!s42 !s32
P
Q4
z z ^
Figure 5.4: The GTD transform coder implemented using MINLAB(I) structure.
We can regardPandPT as theprecoderandpostcoder, and the system in between as the PLT part as indicated in the figure. Since there are infinitely many GTD realizations [38], this framework includes many transform coders that achieve the maximized coding gain. Actually it contains both the KLT and the PLT as special cases:
1. Suppose in (5.7), the GTD{Q,R,P}is taken as the SVD ofD12LT:
D12L=VΣ12UT.
In this case, we actually have Rx = LDLT = UΣUT,thusP = U, which consists of the eigenvectors of the input covariance matrix. We also haveRz =UTRxU=Σ.In this case, the GTD-TC is reduced to the KLT. The PLT part in Fig. 5.4 is simply a series of scalar quantizers, and the optimal bit loading is according to the formula (5.2).
2. In (5.7), suppose{Q,R,P}is taken as the QR decomposition ofD12L.SinceD12Lis by itself an upper triangular matrix, we actually haveP=IandQ=I.In this case, the GTD-TC reduces to the original PLT-TC.
In the following, we will introduce three new transform coder schemes based on GTD theory.
Geometric Mean Decomposition – GMD
Suppose the GMD is used for the transform coder: in (5.7), Rhas all diagonal terms equal to
¯
σ= (QM
i=1σi)M1 . The bit loading formula becomes bi=b+1
2log2 σ¯2
det(Rx)M1 =b, (5.11)
becausedet(Rx) = ¯σ2M. The preceding equation says that all the quantizers are assigned the same number of bits. This is a consequence of the fact thatDiiin Eq. (5.5) are identical for alli.That is, the variances of the quantizer inputs are all identical, which means that the dynamic ranges of the signals being quantized are identical. This is a desirable property in practice.
Bi-Diagonal Transformation – Hessenberg Form
A matrixBis said to be bidiagonal if it has the form demonstrated below for the4×4case.
B=
b00 b01 0 0 0 b11 b12 0 0 0 b22 b23
0 0 0 b33
.
If the GTD form ofD12LT isQBPT,whereBis a bi-diagonal matrix, then we call it the bi-diagonal transform coder (BID-TC). It can be seen that
Rx=LDLT =PBTBPT,
whereBTBis a tri-diagonal matrix demonstrated below for size4×4:
BTB=
c00 c01 0 0 c10 c11 c12 0 0 c21 c22 c23
0 0 c32 c33
withcmk=ckm.This tri-diagonal formBTBis also known as the Hessenberg form [25] ofRx. The advantages of the BID-TC coder lie in its reduced computational complexity. To reduce a symmetric matrix to a tri-diagonal form by orthonormal transformation is computationally much less complex compared to eigenvalue decomposition [25]. The detail of reducing a symmetric matrix to the tri- diagonal form is discussed in [25], and requires only several Householder transformations. The LDU decomposition for a symmetric tri-diagonal matrix is also easy, which requires only O(M) operations now, instead ofO(M2)for general symmetric matrices. Therefore, the design cost for the BID-TC is less than KLT whereas the KLT requires iterative EVD computations. Also, due to the bi-diagonal structure ofB, the implementation cost for the inner PLT part is also reduced, which is only in the order ofO(M). This can be seen in Fig. 5.5, which shows the MINLAB(I) structure for the BID-TC encoder. Signal feedforward paths are only required for the adjacent data streams. The number of signal feedforward paths is much less than for the original PLT.
The detail comparison between the design and implementation costs for various GTD based coders are summarized in Table 5.1.
Combination of GMD and Progressive Transmission
Q Q1
S21
x
P T
Q2
S32
Q3
z
S43Q4
Figure 5.5: The BID Transform coder implemented using MINLAB(I) structure.
Table 5.1: Design and Implementation Costs of Transform Coders
Design cost Impl. cost (precoder, PLT)
KLT EVD, O(M3) O(M2),0
PLT LDU, O(M2) 0, O(M2)
GMD-TC EVD and GMD [38], O(M3) O(M2), O(M2)
BID-TC Hessenberg form O(M3) and easy LDU O(M) O(M2), O(M) General GTD-TC EVD and GTD [38], O(M3) O(M2), O(M2)
There are some applications where rapid transmission is required and a coarse signal approxi- mation is first produced [61]. When more bits are available, the system progressively enhances the performance by sending more information. Fig. 5.6 shows the example in which we divide the sig- nal data streams after the linear transformation into three groups. The first group is the significant group where theK1data streams contain a coarse approximation of the signal. The second group is the less significant group where theK2data streams contain detailed information about the signal.
The third group ofK3streams is the least significant group where the remainingM−K1−K2data streams contain components which are close to zero after the linear transformationPT.
y ^ y
T
. .
x
.PLT
...^ x
P
T
.
P
.
.
PLT
.... . .
. .
0
.Figure 5.6: Use of GTD-TC in the progressive transmission context.
Suppose we adopt the GTD form in (5.7). We are looking for a transformation such that the diagonal terms ofRhave the pattern
diag(R) = [ ¯σ1,· · · ,σ¯1,σ¯2,· · · ,σ¯2, σK1+K2+1,· · · , σM],
where
¯ σ12= (
K1
Y
i=1
σ2i)K11, σ¯22= (
K1+K2
Y
i=K1+1
σi2)K12.
Here[σ12,· · · , σ2M]are the eigenvalues ofRxwith non-increasing order. Pis the orthonormal ma- trix obtained from the GTD theory. Note that this decomposition exists for anyK1, K2combination, since the multiplicative majorization property holds. Because the eigenvalues are in non-increasing order, the firstK1substreams actually represent the firstK1principal components of the vectorx, and the nextK2substreams represent the nextK2principal components. Suppose for the signif- icant group the total bit budget isb1K1,for the less significant group the total bit budget isb2K2, and for the least significant group the average number of bits are zero. As shown in Fig. 5.6, for the first and the second group we use the local PLT for each of them. It can be seen that the bit loading formula under the high bit rate assumption will be
bi=b1+1
2log2 R2ii (QK1
i=1σi)K11
=b1
for the first group, and
bi=b2+1
2log2 R2ii (QK1+K2
i=K1+1σi)K12
=b2
for the second group. That is, uniform bit loading is used across the quantizers within each group.
The data streams in the third group are dropped (i.e., assigned zero bits).
It can be seen that the resulting AM-MSE of this transform coder is 1
M(K1c2−2b1σ¯1 +K2c2−2b2σ¯2 + ΣMi=K1+K2+1σi).
When we only have a very low bit budget, we can allocate the bits to the first group to get the coarse approximation of the signal. When we have more bits available, the information in the second group is exploited to get the detail information of the signal. Hence the progressive transmission scheme can be implemented when we are able to use uniform quantizers within each group. This
shows one example of the flexibility that our proposed GTD-TC scheme can have. One can have more groups of data streams where each group has a different bit budget.