## 2 A New VLSI Algorithm for Integer DCT
On the other hand, by taking all the coefficients into consideration when reconstructing the original image, the proposed transform in (4) achieves better results with a PSNR of 68 dB compared to 63 dB for the IntDCT from [10], as described by Equation (3). To obtain an efficient VLSI algorithm, we reformulated (4) by appropriately restructuring and permuting the input and output sequences. Thus, we obtained a parallel form of the algorithm that uses 5 quasi-cycle convolution structures that can operate concurrently. We introduce the following notations:  
\[ \begin{array}{l}x_{p}(i)=x(i)+x(N-1-i),i=\overline{0,N/2-1}\\ x_{m}(i)=x(i)-x(N-1-i),i=\overline{0,N/2-1}\\ x_{pp}(i)=x_{p}(i)+x_{p}(N/2-1-i),i=\overline{0,N/4-1}\\ x_{pm}(i)=x_{p}(i)-x_{p}(N/2-1-i),i=\overline{0,N/4-1}\end{array}  \tag{5}\]  
Thus, we can write Equation (4) in a matrix-vector product as follows:  
\[\left[ \begin{array}{c}Y_{a}(1)\\ Y_{a}(9)\\ Y_{a}(15)\\ Y_{a}(7)\end{array} \right]=\left[ \begin{array}{cccc}x_{m}(0)&x_{m}(4)&x_{m}( 7)&x_{m}(3)\\ x_{m}(3)&x_{m}(0)&-x_{m}(4)&x_{m}(7)\\ -x_{m}(7)&-x_{m}(3)&x_{m}(0)&x_{m}(4)\\ x_{m}(4)&-x_{m}(7)&x_{m}(3)&x_{m}(0)\end{array} \right]\cdot\left[ \begin{array} []{c}127\\ 81\\ 13\\ 100\end{array} \right] \tag{6}\]  
\[\left[ \begin{array}{c}Y_{b}(1)\\ Y_{b}(9)\\ Y_{b}(15)\\ Y_{b}(7)\end{array} \right]=\left[ \begin{array}{cccc}x_{m}(1)&x_{m}(2)&x_{m}( 6)&x_{m}(5)\\ -x_{m}(5)&-x_{m}(1)&-x_{m}(2)&x_{m}(6)\\ x_{m}(6)&-x_{m}(5)&-x_{m}(1)&x_{m}(2)\\ -x_{m}(2)&-x_{m}(6)&x_{m}(5)&-x_{m}(1)\end{array} \right]\cdot\left[ \begin{array} []{c}122\\ 113\\ 37\\ 60\end{array} \right] \tag{7}\]  
By scaled summing the results of the auxiliary sequences \(Y_{a}(k)\) and \(Y_{b}(k)\) with \(k\in\{1,9,15,7\}\) from Equations (6) and (7), a subset of the output sequence is obtained:  
\[\left[ \begin{array}{c}Y(1)\\ Y(9)\\ Y(15)\\ Y(7)\end{array} \right]=\frac{1}{128}\times\left[ \begin{array}{c}Y_{a}(1)\\ Y_{a}(9)\\ Y_{a}(15)\\ Y_{a}(7)\end{array} \right]+\left[ \begin{array}{c}Y_{b}(1)\\ Y_{b}(9)\\ Y_{b}(15)\\ Y_{b}(7)\end{array} \right] \tag{8}\]  
Figure 1: PSNR for the two Integer algorithms ([10] and proposed). \[\left[ \begin{array}{c}Y_{a}(3)\\ Y_{a}(5)\\ Y_{a}(13)\\ Y_{a}(11)\end{array} \right]=\left[ \begin{array}{cccc}-x_{m}(5)&x_{m}(1)&x_{m}(2 )&-x_{m}(6)\\ x_{m}(6)&x_{m}(5)&x_{m}(1)&-x_{m}(2)\\ x_{m}(2)&-x_{m}(6)&x_{m}(5)&-x_{m}(1)\\ -x_{m}(1)&x_{m}(2)&x_{m}(6)&x_{m}(5)\end{array} \right]\cdot\left[ \begin{array} []{c}127\\ 81\\ 13\\ 100\end{array} \right] \tag{9}\]  
\[\left[ \begin{array}{c}Y_{b}(5)\\ Y_{b}(13)\\ Y_{b}(11)\end{array} \right]=\left[ \begin{array}{cccc}x_{m}(0)&-x_{m}(4)&-x_{ m}(7)&-x_{m}(3)\\ -x_{m}(3)&x_{m}(0)&-x_{m}(4)&-x_{m}(7)\\ x_{m}(7)&-x_{m}(3)&x_{m}(0)&x_{m}(4)\\ -x_{m}(4)&-x_{m}(7)&x_{m}(3)&x_{m}(0)\end{array} \right]\cdot\left[ \begin{array} []{c}122\\ 113\\ 37\\ 60\end{array} \right] \tag{10}\]  
In a similar manner, another subset of the output sequence is obtained by combining the auxiliary results from Equations (9) and (10):  
\[\left[ \begin{array}{c}Y(3)\\ Y(5)\\ Y(13)\\ Y(11)\end{array} \right]=\frac{1}{128}\times\left[ \begin{array}{c}Y_{a}(3)\\ Y_{a}(5)\\ Y_{a}(13)\\ Y_{a}(11)\end{array} \right]+\left[ \begin{array}{c}Y_{b}(3)\\ Y_{b}(5)\\ Y_{b}(13)\\ Y_{b}(11)\end{array} \right] \tag{11}\]  
The subset \(Y(k)\) with \(k\in\{2,6,14,10\}\) of the output sequence is obtained directly from the matrix vector product below:  
\[\left[ \begin{array}{c}Y(2)\\ Y(6)\\ Y(14)\\ Y(10)\end{array} \right]=\frac{1}{128}\times\left[ \begin{array}{cccc}x_{pm}(0 )&x_{pm}(1)&x_{pm}(3)&x_{pm}(2)\\ -x_{pm}(2)&x_{pm}(0)&-x_{pm}(1)&-x_{pm}(3)\\ -x_{pm}(3)&x_{pm}(2)&x_{pm}(0)&-x_{pm}(1)\\ -x_{pm}(1)&x_{pm}(3)&x_{pm}(2)&x_{pm}(0)\end{array} \right]\cdot\left[ \begin{array} []{c}125\\ 106\\ 25\\ 71\end{array} \right] \tag{12}\]  
The particular structure of the matrices in Equations (6), (7), (9), (10) and (12), where all the elements along the main diagonal of the matrix or parallel to the main diagonal of the matrix are the same with the exception of the sign, is called a quasi-cycle convolution.