High compression rate, based on the RLS adaptive algorithm in progressive image transmission

Abstract

The purpose of this paper is to develop a novel method based on recursive least squares (RLS) adaptive algorithm for progressive image transmission (PIT). The image is divided into non-overlapping blocks. Having an agreed vector sequence between the transmitter and the receiver, each block is related to a regressive model. Meanwhile, at the transmitter the blocks are estimated using the RLS algorithm. The high correlation between error vectors, regarding to the RLS execution, causes a very high compression rate in their transmission. The error vectors at the receiver are used to run the RLS algorithm and to estimate the image in a same manner. The method is easy to implement with a low computational complexity and achieves high quality, compared to other well-known methods. In comparison with its counterparts, simulation results show how efficient the proposed method is.

Appendices

Proof of Equation (7)

According to the matrix inversion lemma, the right-hand side of Eq. (6) is equal to $(Q^{-1}_{n-1}+X_nX^T_n)^{-1}$ and therefore

$$\begin{aligned} Q^{-1}_n=Q^{-1}_{n-1}+X_nX^T_n. \end{aligned}$$

(A.1)

Starting from $Q^{-1}_0=\alpha ^{-1}I$, for $n\geqslant 1$ we will have

$$\begin{aligned} Q^{-1}_n=(\alpha ^{-1}I+\sum _{l=1}^{n}X_lX^T_l)^{-1}. \end{aligned}$$

(A.2)

On the other hand from (3), (4) and (5), we have

$$\begin{aligned} W_n^{(1)}(k)=W_{n-1}^{(1)}(k)+Q_nX_nd_{n}^{(1)}(k)-Q_nX_nX^T_nW_{n-1}^{(1)}(k), \end{aligned}$$

(A.3)

or equivalently

$$\begin{aligned} W_n^{(1)}(k)=Q_n(Q^{-1}_n-X_nX^T_n)W_{n-1}^{(1)}(k)+Q_nX_nX^T_nd_{n}^{(1)}(k). \end{aligned}$$

(A.4)

From (A.1) and (A.4)

$$\begin{aligned} W_n^{(1)}(k)=Q_nQ^{-1}_{n-1}W_{n-1}^{(1)}(k)+Q_nX_nd_{n}^{(1)}(k), \end{aligned}$$

(A.5)

which means

$$\begin{aligned} Q^{-1}_nW_n^{(1)}(k)=Q^{-1}_{n-1}W_{n-1}^{(1)}(k)+X_nd_{n}^{(1)}(k). \end{aligned}$$

(A.6)

Since $W_{0}^{(1)}(k)=[0,\ldots ,0]^T$, from recursion (A.6), we will have

$$\begin{aligned} Q^{-1}_nW_n^{(1)}(k)=\sum ^{n}_{l=1}X_ld_{l}^{(1)}(k). \end{aligned}$$

(A.7)

Using (2) and (A.2) in (A.7) results in

$$\begin{aligned} W_n^{(1)}(k)=(\alpha ^{-1}I+\sum _{l=1}^{n}X_lX^T_l)^{-1}(\sum ^{n}_{l=1}X_lX^T_l)W^{(1)}(k). \end{aligned}$$

(A.8)

For $n=N$, since $\{X_l\}$ is a white sequence $\sum ^{n}_{l=1}X_lX^T_l$ is invertible and because $\alpha ^{-1}$ is a very small number,

$(\alpha ^{-1}I+\sum _{l=1}^{n}X_lX^T_l)^{-1}(\sum ^{n}_{l=1}X_lX^T_l)\approx I$. Hence, $W_N^{(1)}(k)\approx W^{(1)}(k)$.

Proof of Equation (9)

From Eqs. (2) and (3), we have

$$\begin{aligned} e_n^{(1)}(k)=X_n^T(W^{(1)}(k)-W_{n-1}^{(1)}(k)). \end{aligned}$$

(B.1)

Using (B.1) in (4) and subtracting $W^{(1)}(k)$ from both sides of the resulted formula leads to

$$\begin{aligned} W^{(1)}(k)-W_{n}^{(1)}(k)=(I-Q_nX_nX^T_n)(W^{(1)}(k)-W_{n-1}^{(1)}(k)). \end{aligned}$$

(B.2)

From recursion (B.2) and because $W_{0}^{(1)}(k)=[0,\ldots ,0]^T$, we will have:

$$\begin{aligned} W^{(1)}(k)-W_{n}^{(1)}(k)=G_nW^{(1)}(k), \end{aligned}$$

(B.3)

where $G_n$ is given from (10). Equation (B.4) is the result of using (B.3) in (B.1) for $n=0,1,\ldots ,N-1$

$$\begin{aligned} e_n^{(1)}(k)=X_n^TG_nW^{(1)}(k), \end{aligned}$$

(B.4)

which is equivalent to (9).