Multiplication Symmetric Convolution Property for Discrete Trigonometric Transforms

Abstract

The symmetric-convolution multiplication (SCM) property of discrete trigonometric transforms (DTTs) based on unitary transform matrices is developed. Then as the reciprocity of this property, the novel multiplication symmetric-convolution (MSC) property of discrete trigonometric transforms, is developed.

1. Introduction

Shen et al. [1] developed fast DCT-domain convolution for time/spatial-domain multiplication using DCT type-2. They exploited symmetry and orthogonality for the fast algorithm. Logo-keying operation in the spatial domain can be done in the DCT domain for compressed image/video editing. However, they have not derived the convolution from the symmetric-convolution multiplication (SCM) property of discrete trigonometric transforms (DTTs), which will be the focus of this paper.

Time-domain symmetric convolution for a linear phase filtering application has been developed by Martucci [2] and is called the symmetric-convolution multiplication property of DTTs. Based on the SCM property, Zou et al. [3] have developed a symmetric convolution for linear phase FIR filtering. Filter coefficients consist of symmetric and asymmetric parts.

Based on those earlier works, Reju et al. [4] have finally developed fast circular convolution using DTTs. The input sequences to be convolved need not be symmetric or asymmetric. Thus fast DCTs and DSTs can be used instead of the FFTs for FIR filtering. Generalized fast convolution using numerous transforms as well as the DFT and DTTs has been studied by Korohoda et al. in [5].

In this paper we show that swapping the forward and inverse transforms in the SCM property [2] yields a multiplication-convolution (MSC) property. That is, the convolution of transformed sequences gives the same results as the forward transform after element-by-element multiplication of the data sequences. The necessary scaling factor M for these new properties has been described in the third line below Eq. (20) in [2], saying “it is possible to swap the usage of the forward and inverse transform; in that case, an extra scaling factor may be required”. Here M is the size of the generalized DFT (GDFT) when a DTT is derived from an M-point GDFT. We can get 40 types of MSCs corresponding to 40 types of SCMs in Tables VI, VII of [2].

In (1) of [6], the formulation of the SCM for the unitary matrices is presented, with only the exception of some of the 40 options mentioned by Martucci [2], because they cannot be expressed with the assumed tools. Here, we fill that gap. Consider the following matrix relationship between convolutional/unnormalized DTTs (denoted with lower-case subscript like C_1e) and unitary DTTs (denoted with capital-letter subscript like C_IE).

$$\begin{array}{cc}{\mathrm{C}}_{1e}=\sqrt{2N}{{\mathrm{A}}_1}^{-1}{\mathrm{C}}_{1E}{\mathrm{A}}_1\hfill & {\mathrm{C}}_{2e}=\sqrt{2N}{{\mathrm{A}}_2}^{-1}{\mathrm{C}}_{\mathrm{I}\mathrm{I}E}\hfill \end{array}$$

(1)

$$\begin{array}{cc}{\mathrm{C}}_{3e}=\sqrt{2N}{\mathrm{C}}_{\mathrm{I}\mathrm{I}\mathrm{I}E}{\mathrm{A}}_2\hfill & {\mathrm{C}}_{4e}=\sqrt{2N}{\mathrm{C}}_{\mathrm{I}\mathrm{V}E}\hfill \end{array}$$

(2)

$$\begin{array}{cc}{\mathrm{S}}_{1e}=\sqrt{2N}{\mathrm{S}}_{\mathrm{I}E}\hfill & {\mathrm{S}}_{2e=}\sqrt{2N}{{\mathrm{A}}_3}^{-1}{\mathrm{S}}_{\mathrm{I}\mathrm{I}E}\hfill \end{array}$$

(3)

$$\begin{array}{cc}{\mathrm{S}}_{3e}=\sqrt{2N}{\mathrm{S}}_{\mathrm{I}\mathrm{I}\mathrm{I}E}{\mathrm{A}}_3\hfill & {\mathrm{S}}_{4e}=\sqrt{2N}{\mathrm{S}}_{\mathrm{I}\mathrm{V}E}\hfill \end{array}$$

(4)

$$\begin{array}{cc}{\mathrm{C}}_{1o}=\sqrt{2N-1}{{\mathrm{A}}_2}^{-1}{\mathrm{C}}_{\mathrm{I}O}{\mathrm{A}}_2\hfill & {\mathrm{C}}_{2o}=\sqrt{2N-1}{{\mathrm{A}}_2}^{-1}{\mathrm{C}}_{\mathrm{I}\mathrm{I}O}{\mathrm{A}}_4\hfill \end{array}$$

(5)

$$\begin{array}{cc}{\mathrm{C}}_{3o}=\sqrt{2N-1}{{\mathrm{A}}_4}^{-1}{\mathrm{C}}_{\mathrm{I}\mathrm{I}\mathrm{I}O}{\mathrm{A}}_2\hfill & {\mathrm{C}}_{4o}=\sqrt{2N-1}{\mathrm{C}}_{\mathrm{I}\mathrm{V}O}\hfill \end{array}$$

(6)

$$\begin{array}{cc}{\mathrm{S}}_{1o}=\sqrt{2N-1}{\mathrm{S}}_{\mathrm{I}O}\hfill & {\mathrm{S}}_{2o}=\sqrt{2N-1}{\mathrm{S}}_{\mathrm{I}\mathrm{I}O}\hfill \end{array}$$

(7)

$$\begin{array}{cc}{\mathrm{S}}_{3o}=\sqrt{2N-1}{\mathrm{S}}_{\mathrm{I}\mathrm{I}\mathrm{I}O}\hfill & {\mathrm{S}}_{4o}=\sqrt{2N-1}{{\mathrm{A}}_4}^{-1}{\mathrm{S}}_{\mathrm{I}\mathrm{V}O}{\mathrm{A}}_4\hfill \end{array}$$

(8)

Here the fact that scalars commute with matrices is used for C_3e and others.

$$\begin{array}{cc}{\mathrm{A}}_1={\left[\sqrt{k_n}\right]}_{N+1}={\left[\sqrt{k_m}\right]}_{N+1}=\mathrm{diag}(\frac{1}{\sqrt{2}},1,1,\dots ,1,\frac{1}{\sqrt{2}}),\hfill & \text{where} m,n=0,1,\dots ,N\hfill \end{array}$$

(9)

$$\begin{array}{cc}{\mathrm{A}}_2={\left[\sqrt{k_n}\right]}_N={\left[\sqrt{k_m}\right]}_N=\mathrm{diag}(\frac{1}{\sqrt{2}},1,1,\dots ,1),\hfill & \text{where} m,n=0,1,\dots ,N-1\hfill \end{array}$$

(10)

$$\begin{array}{cc}{\mathrm{A}}_3={\left[\sqrt{k_n}\right]}_N={\left[\sqrt{k_m}\right]}_N=\mathrm{diag}(1,1,\dots ,1,\frac{1}{\sqrt{2}}),\hfill & \text{where} m,n=1,2,\dots ,N\hfill \end{array}$$

(11)

$$\begin{array}{cc}{\mathrm{A}}_4={\left[\sqrt{l_n}\right]}_N={\left[\sqrt{l_m}\right]}_N=\mathrm{diag}(1,1,\dots ,1,\frac{1}{\sqrt{2}}),\hfill & \text{where} m,n=0,1,\dots ,N-1\hfill \end{array}$$

(12)

$$\begin{array}{ccc}\hfill k_p& =1/2\hfill & p=0,N\hfill \\ \hfill & =1\hfill & p=1,2,\dots ,N-1\hfill \end{array}$$

(13)

$$\begin{array}{ccc}\hfill l_p& =1\hfill & p=0,1,\dots ,N-2\hfill \\ \hfill & =1/2\hfill & p=N-1\hfill \end{array}$$

(14)

where diag(a₁₁, a₁₁, …, a_NN) implies a diagonal matrix with the diagonal elements as (a₁₁, a₁₁, …, a_NN). I_N is the identity matrix of size N × N.

Let matrices J, K, Q, and R be defined as:

$$\begin{array}{cccc}\mathrm{J}=\left(\begin{array}{c}{I}_N\\ O_{1\times N}\end{array}\right)& \mathrm{K}=\left(\begin{array}{c}{O}_{1\times N}\\ I_N\end{array}\right)& \mathrm{Q}=\left(\begin{array}{c}{I}_{N-1}\\ O_{1\times \left(N-1\right)}\end{array}\right)& \mathrm{R}=\left(\begin{array}{c}{O}_{1\times \left(N-1\right)}\\ I_{N-1}\end{array}\right)\\ (N+1)\times N& (N+1)\times N& N\times (N-1)& N\times (N-1)\end{array}$$

(15)

where O_1×N is the zero row-vector, with N elements, whose entries are all zero. Multiplying one of those matrices with an input vector appends a single zero on the top of the first or under the last element of the vector. Multiplying the transpose of one of those matrices with an input vector discards the first or last element of the vector.

2. Symmetric-Convolution Multiplication Property

One of the forty cases of symmetric convolution is proven as an example.

Property.

y = A₁⁻¹C_IE⁻¹ J $\mathscr{H}$_C2e C_IIE x        [(A8) in

Table 3. 20 of 40 types of SCM and MSC properties for the DTTs, M = 2N.

SCM (Symmetric-Conv. Mult.)	MSC (Mult. Symmetric-Conv.)
y = A1−1CIE−1$\mathscr{H}$C1e CIE (A1 x)	Y = A1−1CIE hC1e CIE −1(A1 X)	(A1)
y = SIE−1 (RT JT $\mathscr{H}$C1e) SIE x	Y = SIE (RT JT $\mathscr{H}$C1e) SIE−1 X	(A2)
y = SIE−1 $\mathscr{H}$S1e RT JT CIE (A1 x)	Y = SIE$\mathscr{H}$S1e RT JT CIE−1 (A1 X)
y = − A1−1CIE−1 J R $\mathscr{H}$S1e SIE x	Y = − A1−1CIE J RhS1e SIE −1 X	(A3)
y = CIIE−1$\mathscr{H}$C1e CIIE x	Y = CIIIE hC1e CIIIE −1 X	(A4)
y = CIIE −1$\mathscr{H}$C2e JT CIE(A1 x)	Y = CIIIE hC3e JT CIE−1(A1 X)
y = SIIE−1$\mathscr{H}$S1e CIIE x	Y = SIIIE hS1e CIIIE−1 X	(A5)
y = SIIE−1$\mathscr{H}$C2e SIE x	Y = SIIIE hC3e SIE−1 X
y = SIIE−1$\mathscr{H}$C1e SIIE x	Y = SIIIE hC1e SIIIE−1 X	(A6)
y = SIIE−1$\mathscr{H}$S2e KT CIE (A1 x)	Y = SIIE hS3e KT CIE−1(A1 X)
y = −CIIE−1 R $\mathscr{H}$S1e QT SIIE x	Y = −CIIIE R hS1e QT SIIIE−1 X	(A7)
y = −CIIE−1R (QT $\mathscr{H}$S2e) SIE x	Y = −CIIIE R (QT hS3e) SIE−1 X
y = A1−1CIE−1 J $\mathscr{H}$C2e CIIE x	Y = A1−1CIE J h C3e CIIIE −1 X	(A8)
y = SIE−1 (QT $\mathscr{H}$S2e) RT CIIE x	Y = SIE (QT h S3e) RT CIIIE−1 X	(A9)
y = SIE−1 (RT $\mathscr{H}$C2e) Q T SIIE x	Y = SIE (RT hC3e) Q T SIIIE−1 X
y = − A1−1 CIE−1 K $\mathscr{H}$S2e SIIE x	Y = − A1−1 CIE K h S3e SIIIE −1 X	(A10)
y = A2−1 CIIIE−1$\mathscr{H}$C3e CIIIE(A2 x)	Y = A2−1 CIIE h C2e CIIE−1(A2 X)	(A11)
y = A3−1 SIIIE−1$\mathscr{H}$S3e CIIIE(A2 x)	Y = A3−1 SIIE h S2e CIIE −1(A2 X)	(A12)
y = A3−1 SIIIE−1$\mathscr{H}$C3e SIIIE(A3 x)	Y = A3−1 SIIE h S2e SIIE −1(A3 X)
y = A2−1 CIIIE−1$\mathscr{H}$S3e SIIIE(A3 x)	Y = A2−1 CIIE h S2e SIIE −1(A3 X)	(A13)
y = CIVE−1$\mathscr{H}$C3e CIVE x	Y = CIVE h C2e CIVE−1 X
y = CIVE−1$\mathscr{H}$C4e CIIIE(A2 x)	Y = CIVE h C4e CIIE−1(A2 X)	(A14)
y = SIVE−1$\mathscr{H}$S3e CIVE x	Y = SIVEh S2e CIVE−1X	(A15)
y = SIVE−1$\mathscr{H}$C4e SIIIE(A3 x)	Y = SIVEh C4e SIIE−1(A3 X)
y = SIVE−1$\mathscr{H}$C3e SIVE x	Y = SIVE h C2e SIVE−1 X	(A16)
y = SIVE−1$\mathscr{H}$S4e CIIIE(A2 x)	Y = SIVE h S4e CIIE−1(A2 X)
y = −CIVE−1$\mathscr{H}$S3e SIVE x	Y = −CIVE h S2e SIVE−1 X	(A17)
y = −CIVE−1$\mathscr{H}$S4e SIIIE(A3 x)	Y = −CIVE h S4e SIIE−1 (A3 X)
y = A2−1 CIIIE−1$\mathscr{H}$C4e CIVE x	Y = A2−1 CIIE h $\mathscr{H}$C4e CIVE −1 X	(A18)
y = A3−1 SIIIE−1$\mathscr{H}$S4e CIVE x	Y = A3−1 SIIE h S4e CIVE−1 X	(A19)
y = A3−1 SIIIE−1$\mathscr{H}$C4e SIVE x	Y = A3−1 SIIE h C4e SIVE−1 X
y = − A2−1 CIIIE−1$\mathscr{H}$S4e SIVE x	Y = − A2−1 CIIE h S4e SIVE−1 X	(A20)

]

Proof.

Equation (A.1) in the appendix of [2] can be rewritten as:

$${\mathrm{C}}_{1e}=2{\widehat{C}}_{1e}[k_n{]}_{N+1}$$

(16)

where ${\widehat{C}}_{1e}$ is the kernel.

$${\mathrm{C}}_{\mathrm{I}E}=\sqrt{2/N}\left[\sqrt{k_m}{]}_{N+1}{\widehat{C}}_{1e}\right[\sqrt{k_n}{]}_{N+1}$$

(17)

$$\left(\right[\sqrt{k_n}{]}_{N+1}{)}^{-1}=[\sqrt{1/k_n}{]}_{N+1}$$

(18)

From (16), (17) and (18):

$${\mathrm{C}}_{1e}=\left[\sqrt{2N/k_m}{]}_{N+1}{\mathrm{C}}_{1E}\right[\sqrt{k_n}{]}_{N+1}$$

(19)

$${{\mathrm{C}}_{1e}}^{-1}=\left[\sqrt{1/k_n}{]}_{N+1}{{\mathrm{C}}_{\mathrm{I}E}}^{-1}\right[\sqrt{k_m/\left(2N\right)}{]}_{N+1}$$

(20)

From (1) and (10):

$${{\mathrm{C}}_{2e}}^{-1}={{\mathrm{C}}_{\mathrm{I}\mathrm{I}E}}^{-1}[\sqrt{k_m/\left(2N\right)}{]}_N$$

(21)

Let x and w be input column vectors in time domain. Let y be an output column vector in time domain. Let $\mathscr{H}$_C2e be a diagonal matrix defined as $\mathscr{H}$_C2e = diag([C_2e w]^T), where superscript T denotes the transpose operator. We rewrite the 8th property in Table VI of [2] as:

$$\begin{array}{cc}\mathrm{y}\hfill & ={{\mathrm{C}}_{1\mathrm{e}}}^{-1}\left\{{\mathrm{\mathscr{H}}}_{\mathrm{C}2e}{\mathrm{C}}_{2\mathrm{e}}\mathrm{x}\right\}\hfill \\ & ={{\mathrm{C}}_{1\mathrm{e}}}^{-1}\left\{{\mathrm{\mathscr{H}}}_{\mathrm{C}2e}\right[\sqrt{2N/k_m}{]}_N{\mathrm{C}}_{\mathrm{I}\mathrm{I}E}\mathrm{x}\}\hfill \\ & =\left[\sqrt{1/k_n}{]}_{N+1}{{\mathrm{C}}_{\mathrm{I}E}}^{-1}\right[\sqrt{k_m}{]}_{N+1}\mathrm{J}\left\{{\mathrm{\mathscr{H}}}_{\mathrm{C}2e}\right[\sqrt{1/k_m}{]}_N{\mathrm{C}}_{\mathrm{I}\mathrm{I}E}\mathrm{x}\}\hfill \end{array}$$

(22)

The matrix J shows up in the last line of (22) for zero padding. Since $\mathscr{H}$_C2e and [$\sqrt{1/k_m}$]_N are diagonal matrices, they can commute. Thus:

$$\begin{array}{cc}\mathrm{y}\hfill & =\left[\sqrt{1/k_n}{]}_{N+1}{{\mathrm{C}}_{\mathrm{I}E}}^{-1}\mathrm{J}\right\{{\mathrm{\mathscr{H}}}_{\mathrm{C}2e}{\mathrm{C}}_{\mathrm{I}\mathrm{I}E}\mathrm{x}\}\hfill \\ & =\left[\sqrt{1/k_n}{]}_{N+1}{{\mathrm{C}}_{\mathrm{I}E}}^{-1}\mathrm{J}\right\{{\mathrm{\mathscr{H}}}_{\mathrm{C}2e}{\mathrm{C}}_{\mathrm{I}\mathrm{I}E}\mathrm{x}\}\hfill \end{array}$$

(23)

Another example is shown in (11) of [6]. Equation (12) of [6] is expanded to full for our derivation and only results are listed in the first column of Table 3 and

Table 4. 20 of 40 types of SCM and MSC properties for the DTTs, M = 2N − 1.

SCM (Symmetric-Conv. Mult.)	MSC (Mult. Symmetric-Conv.)
y = A2−1 CIO−1$\mathscr{H}$C1o CIO (A2 x)	Y = A2−1 CIO h C1o CIO −1(A2 X)	(A21)
y = SIO−1 (RT$\mathscr{H}$C1o) SIO x	Y = SIO (RT h C1o) SIO−1 X	(A22)
y = SIO−1 $\mathscr{H}$S1o RT CIO (A2 x)	Y = SIO h S1o RT CIO−1(A2 X)
y = − A2−1 CIO−1 R$\mathscr{H}$S1o SIO x	Y = − A2−1 CIO R h S1o SIO −1 X	(A23)
y = A4−1 CIIO−1$\mathscr{H}$C1o CIIO (A4 x)	Y = A4−1 CIIIO h C1o CIIIO −1 (A4 X)	(A24)
y = A4−1 CIIO−1 $\mathscr{H}$C2o CIO (A2 x)	Y = A4−1 CIIIO h C3o CIO−1(A2 X)
y = SIIO−1$\mathscr{H}$S1o RT CIIO (A4 x)	Y = SIIIO h S1o RT CIIIO −1 (A4 X)	(A25)
y = SIIO−1 (RT$\mathscr{H}$C2o) SIO x	Y = SIIIO (R T h C3o) SIO−1 X
y = SIIO−1 (RT$\mathscr{H}$C1o) SIIo x	Y = SIIIO (RT h C1o) SIIIO−1 X	(A26)
y = SIIO−1$\mathscr{H}$S2o RT CIO (A2 x)	Y = SIIIO h S3o RT CIO−1 (A2 X)
y = − A4−1 CIIO−1 R $\mathscr{H}$S1o SIIO x	Y = − A4−1 CIIIO R h S3o SIO−1 X	(A27)
y = − A4−1 CIIO−1 R $\mathscr{H}$S2o SIO x	Y = − A4−1 CIIIO R h S1o SIIIO−1 X
y = A2−1 CIO−1 $\mathscr{H}$C2o CIIO (A4 x)	Y = A2−1 CIOh C3o CIIIO−1 (A4 X)	(A28)
y = SIO−1 $\mathscr{H}$S2o RT CIIO (A4 x)	Y = SIO h S3o RT CIIIO−1 (A4 X)	(A29)
y = SIO−1 (RT$\mathscr{H}$C2o) SIIO x	Y = SIO−1 (RT h C3o) SIIIO X
y = − A2−1 CIO−1 R $\mathscr{H}$S2o SIIO x	Y = − A2−1 CIO R h S3o SIIIO−1 X	(A30)
y = A2−1 CIIIO−1 $\mathscr{H}$C3o CIIIO(A2 x)	Y = A2−1 CIIO h C2o CIIO−1(A2 X)	(A31)
y = SIIIO−1 $\mathscr{H}$S3o QT CIIIO (A2 x)	Y = SIIO h S2o QT CIIO−1(A2 X)	(A32)
y = SIIIO−1 QT $\mathscr{H}$C3o Q SIIIO x	Y = SIIO QT h S2o Q SIIO−1 X
y = − A2−1 CIIIO−1 Q $\mathscr{H}$S3o SIIIO x	Y = − A2−1 CIIO Q h S2o SIIO−1 X	(A33)
y = CIVO−1 (QT $\mathscr{H}$C3o) CIVO x	Y = CIVO (QT h C2o) CIVO−1 X	(A34)
y = CIVO−1$\mathscr{H}$C4o QT CIIIO (A2 x)	Y = CIVO h C4o QT CIIO−1(A2 X)
y = A4−1 SIVO−1 Q $\mathscr{H}$S3o CIVO x	Y = A4−1 SIVO Q h S2o CIVO−1 X	(A35)
y = A4−1 SIVO−1 Q $\mathscr{H}$C4o SIIIO x	Y = A4−1 SIVOQ h C4o SIIO−1 X
y = A4−1 SIVO−1$\mathscr{H}$C3o SIVO (A4 x)	Y = A4−1 SIVO h C2o SIVO−1 (A4 X)	(A36)
y = A4−1 SIVOE−1$\mathscr{H}$S4o CIIIO (A2 x)	Y = A4−1 SIVOh S4o CIIO−1 (A2 X)
y = − CIVO−1$\mathscr{H}$S3o QT SIVO (A4 x)	Y = − CIVOh S2o QT SIVO−1 (A4 X)	(A37)
y = − CIVO−1 (QT$\mathscr{H}$S4o) SIIIO x	Y = − CIVO (QT h S4o) SIIIO−1 X
y = A2−1 CIIIO−1 Q $\mathscr{H}$C4o CIVO x	Y = A2−1 CIIO Q h C4o CIVO−1 X	(A38)
y = SIIIO−1 (QT$\mathscr{H}$S4o) CIVO x	Y = SIIO (QT h S4o) CIVO−1 x	(A39)
y = SIIIO−1$\mathscr{H}$C4o QT SIVO (A4 x)	Y = SIIO−1 h C4o QT SIVO−1 (A4 X)
y = − A2−1 CIIIO−1$\mathscr{H}$S4o SIVO (A4 x)	Y = − A2−1 CIIO h S4o SIVO−1 (A4 x)	(A40)

in the Appendix.

3. Multiplication Symmetric-Convolution Property

Let X and Y be transformed input and output data vectors. Since there are one-to-one correspondences between unitary discrete trigonometric transforms (DTTs), we can exchange the forward transform for inverse one and vice versa as follows. In other words, a pair has the same matrix but has different names. Define h _C3e as:

h _C3e = diag( [(C^3e) ⁻¹ W]^T )

(24)

where C^3e will be defined in (27). That is, $\mathscr{H}$_C2e and h _C3e are the same matrix with different names (thus names of DTTs need to change).

Then from (23):

$$\mathrm{Y}=[\sqrt{1/k_m}{]}_{N+1}{\mathrm{C}}_{\mathrm{I}E} \mathrm{J} h_{\mathrm{C}3e} {{\mathrm{C}}_{\mathrm{I}\mathrm{I}\mathrm{I}E}}^{-1} \mathrm{X}$$

(25

Notice the forward transform matrix C_IIE is replaced by the inverse transform matrix C_IIIE ⁻¹ since they are the same matrix, and vice versa. Now this equation represents a SCM of DTTs. A key point of this new property is that we need to redefine convolution forms of DCTs and DSTs. The factor of M is divided for the inverse DCT of the convolution form in [2] whereas it is divided for the forward DCT of the new convolution form. M is 2N for even and 2N – 1 for odd. Now new convolution form for DCT 2 is denoted as C^2e:

$$\begin{array}{cccc}& \hfill \text{Forward}\hfill & & \hfill \text{Inverse}\hfill \\ \hfill \left(\text{old}\right)\hfill & {\mathrm{C}}_{2e}\hfill & \hfill \iff \hfill & {{\mathrm{C}}_{2e}}^{-1}=\frac{1}{M}{\mathrm{C}}_{3e}\hfill \end{array}$$

(26)

$$\begin{array}{cccc}\hfill \left(\text{new}\right)\hfill & {\mathrm{C}}^{2e}=\frac{1}{M}{\mathrm{C}}_{2e}\hfill & \hfill \iff \hfill & {\left({\mathrm{C}}^{2e}\right)}^{-1}={\mathrm{C}}_{3e} \text{or} {\left({\mathrm{C}}^{3e}\right)}^{-1}={\mathrm{C}}_{2e}\hfill \end{array}$$

(27)

The rest of DTTs can be readily obtained from the appendix of [2].

MSC properties can be described in terms of convolutional / unnormalized DTTs to obtain similar results:

$$\begin{array}{cc}\left(11\mathrm{th}\text{ SCM in Table IV of }\right[2\left]\right)\hfill & {{\mathrm{C}}_{3e}}^{-1}({\mathrm{C}}_{3e}\times {\mathrm{C}}_{3e})\hfill \end{array}$$

(28)

$$\begin{array}{cc}(\text{ours}, \text{MSC})\hfill & M{\mathrm{C}}^{2e}({\left({\mathrm{C}}^{2e}\right)}^{-1}\times {\left({\mathrm{C}}^{2e}\right)}^{-1})\hfill \end{array}$$

(29)

Only results are listed. Matrices J, K, Q, and R are required for different index ranges between operands. Link between the index range-based maneuvers described by Martucci [2] and the introduction of the J, K, Q and R matrices is presented in

Table 1. Link between the index range-based maneuvers described by Martucci [2] and the introduction of the J, K, Q and R matrices, M = 2N.

SCM (Symmetric-Conv. Mult.)	Forward transform			$\mathscr{H}$			Inverse transform
	Input index range	Output index range		Input index range	Output index range		Input index range	Output index range
y = SIE−1 (RT JT$\mathscr{H}$C1e) SIE x				0→N	1→N−1	RT JT
y = SIE−1 $\mathscr{H}$S1e RT JT CIE (A1 x)	0→N	1→N−1	RT JT
y = − A1−1CIE−1 J R$\mathscr{H}$S1e SIE x	1→N−1	0→N	J R
y = CIIE−1$\mathscr{H}$C2e JT CIE(A1 x)	0→N	0→N−1	JT
y = SIIE−1$\mathscr{H}$S2e KT CIE (A1 x)	0→N	1→N	KT
y = −CIIE−1 R $\mathscr{H}$S1e QT SIIE x	1→N	1→N−1	QT				1→N−1	0→N−1	R
y = −CIIE−1R (QT $\mathscr{H}$S2e) SIE x				1→N	1→N−1	QT	1→N−1	0→N−1	R
y = A1−1CIE−1 J $\mathscr{H}$C2e CIIE x							0→N−1	0→N	J
y = SIE−1 (QT$\mathscr{H}$S2e) RT CIIE x	0→N−1	1→N−1	RT	1→N	1→N−1	QT
y = SIE−1 (RT$\mathscr{H}$C2e) Q T SIIE x	1→N	1→N−1	QT	0→N−1	1→N−1	RT
y = − A1−1 CIE−1 K $\mathscr{H}$S2e SIIE x							1→N	0→N	K

and

Table 2. Link between the index range-based maneuvers described by Martucci [2] and the introduction of the J, K, Q and R matrices, M = 2N − 1.

SCM (Symmetric-Conv. Mult.)	Forward transform			$\mathscr{H}$			Inverse transform
	Input index range	Output index range		Input index range	Output index range		Input index range	Output index range
y = SIO−1 (RT$\mathscr{H}$C1o) SIO x				0→N−1	1→N−1	RT
y = SIO−1 $\mathscr{H}$S1o RT CIO (A2 x)	0→N−1	1→N−1	R T
y = − A2−1 CIO−1 R $\mathscr{H}$S1o SIO x							1→N−1	0→N−1	R
y = SIIO−1$\mathscr{H}$S1o RT CIIO (A4 x)	0→N−1	1→N−1	RT
y = SIIO−1 (RT$\mathscr{H}$C2o) SIO x				0→N−1	1→N−1	RT
y = SIIO−1 (RT$\mathscr{H}$C1o) SIIo x				0→N−1	1→N−1	RT
y = SIIO−1$\mathscr{H}$S2o RT CIO (A2 x)	0→N−1	1→N−1	RT
y = − A4−1 CIIO−1 R $\mathscr{H}$S1o SIIO x							1→N−1	0→N−1	R
y = − A4−1 CIIO−1 R $\mathscr{H}$S2o SIO x							1→N−1	0→N−1	R
y = SIO−1 $\mathscr{H}$S2o RT CIIO (A4 x)	0→N−1	1→N−1	RT
y = SIO−1 (RT$\mathscr{H}$C2o) SIIO x				0→N−1	1→N−1	RT
y = − A2−1 CIO−1 R $\mathscr{H}$S2o SIIO x							1→N−1	0→N−1	R
y = SIIIO−1 $\mathscr{H}$S3o QT CIIIO (A2 x)	0→N−1	0→N−2	Q T
y = SIIIO−1 QT$\mathscr{H}$C3o Q SIIIO x	0→N−2	0→N−1	Q				0→N−1	0→N−2	QT
y = − A2−1 CIIIO−1 Q $\mathscr{H}$S3o SIIIO x							0→N−2	0→N−1	Q
y = CIVO−1 (QT$\mathscr{H}$C3o) CIVO x				0→N−1	0→N−2	Q T
y = CIVO−1$\mathscr{H}$C4o QT CIIIO (A2 x)	0→N−1	0→N−2	Q T
y = A4−1 SIVO−1 Q $\mathscr{H}$S3o CIVO x							0→N−2	0→N−1	Q
y = A4−1 SIVO−1 Q $\mathscr{H}$C4o SIIIO x							0→N−2	0→N−1	Q
y = −CIVO−1$\mathscr{H}$S3o QT SIVO (A4 x)	0→N−1	0→N−2	QT
y = −CIVO−1 (QT$\mathscr{H}$S4o) SIIIO x				0→N−1	0→N−2	QT
y = A2−1 CIIIO−1 Q $\mathscr{H}$C4o CIVO x							0→N−2	0→N−1	Q
y = SIIIO−1 (QT$\mathscr{H}$S4o) CIVO x				0→N−1	0→N−2	Q T
y = SIIIO−1$\mathscr{H}$C4o QT SIVO (A4 x)	0→N−1	0→N−2	QT

for M = 2N and M = 2N − 1, respectively.

4. Applications

For an image resizing (filter) application, one of 40 MSC properties is used. The definition of a normalizing parameter F(k) in [7] needs a minor change as:

$$\begin{array}{ccc}F\left(\mathrm{k}\right)=\hfill & \sqrt{2}\hfill & k=0\hfill \\ & 1\hfill & 1\le k\le N-1\hfill \end{array}$$

Then we can derive one of MSC properties, which is (4) in [7], from our equation as follows:

$$\begin{array}{cc}{\mathrm{C}}_{\mathrm{I}\mathrm{I}}\left\{x\right(n)\times w(n\left)\right\}\hfill & ={\mathrm{C}}_{\mathrm{I}\mathrm{I}}\left\{x\right(n\left)\right\}\otimes C^{2e}\left\{w\right(n\left)\right\}\hfill \\ & ={\mathrm{C}}_{\mathrm{I}\mathrm{I}}\left\{x\right(n\left)\right\}\otimes \frac{1}{2N}{C}_{2e}\left\{w\right(n\left)\right\}\hfill \end{array}$$

where the symbol ⊗ denotes symmetric convolution. Since C_2e = $\sqrt{2N}$ A₂⁻¹C_IIE in (1),

$$\begin{array}{cc}{\mathrm{C}}_{\mathrm{I}\mathrm{I}}\left\{x\right(n)\times w(n\left)\right\}\hfill & ={\mathrm{C}}_{\mathrm{I}\mathrm{I}}\left\{x\right(n\left)\right\}\otimes \frac{1}{\sqrt{2N}}{{\mathrm{A}}_2}^{-1}{\mathrm{C}}_{\mathrm{I}\mathrm{I}}\left\{w\right(n\left)\right\}\hfill \end{array}$$

By the associativity of (continuous and discrete) convolution:

$$\begin{array}{cc}{\mathrm{C}}_{\mathrm{I}\mathrm{I}}\left\{x\right(n)\times w(n\left)\right\}\hfill & =\frac{1}{\sqrt{2N}}{\mathrm{C}}_{\mathrm{I}\mathrm{I}}\left\{x\right(n\left)\right\}\otimes {{\mathrm{A}}_2}^{-1}{\mathrm{C}}_{\mathrm{I}\mathrm{I}}\left\{w\right(n\left)\right\}\hfill \\ & =\frac{1}{\sqrt{2N}}{\mathrm{A}}_2\left[{{\mathrm{A}}_2}^{-1}{\mathrm{C}}_{\mathrm{I}\mathrm{I}}\right\{x\left(n\right)\}\otimes {{\mathrm{A}}_2}^{-1}{\mathrm{C}}_{\mathrm{I}\mathrm{I}}\{w\left(n\right)\left\}\right]\hfill \end{array}$$

(30)

This is shown in block diagram format in Fig. 1(b). Since G(k) and F(k) are A₂⁻¹ and $\frac{1}{\sqrt{2N}}$ A₂ in matrix form:

C_II{x(n) × w(n)} = G(k) [ F(k) C_II{x(n)} ⊗ F(k) C_II{w(n)} ]

Convolution has the property of associativity with scalar multiplication. Let F and G be any real sequences. Then:

a(F ⊗ G) = (aF) ⊗ G = F ⊗ (aG)

for any real (or complex) number a.

For a numerical example, let:

w = (1, 2, 3, 4) ^T, x = (1, 0, 3, 2) ^T and N = 4

Then the time domain element-by-element multiplication of the two vectors is:

w × x = (1, 0, 9, 8) ^T

W = A₂⁻¹ C_II w = (a₀, a₁, a₂, a₃) ^T

       = (7.071, − 2.230, 0, − 0.159) ^T

X = A₂⁻¹ C_II x
         = (4.243, −1.465, 0, 1.689) ^T

Y = W ⊗ X = (W_t + W_h) X
   = (36, − 19.823, 0, 11.272) ^T

(31)

where:

$$\mathrm{W}=({\mathrm{W}}_t+{\mathrm{W}}_h)\left(\begin{array}{cccc}{a}_0& a_1& a_2& a_3\\ a_1& a_0& a_1& a_2\\ a_2& a_1& a_0& a_1\\ a_3& a_2& a_1& a_0\end{array}\right)+\left(\begin{array}{cccc}0& a_1& a_2& a_3\\ 0& a_2& a_3& 0\\ 0& a_3& 0& -a_3\\ 0& 0& -a_3& -a_2\end{array}\right)$$

(32)

Equation (32) is $\left[y_{\left(N\right),t}^{3e}\right]+\left[y_{\left(N\right),h}^{3e}\right]$ in [8, p. 2635], and W_t and W_h are a symmetric Toeplitz matrix and a Hankel matrix [9]. Symmetric convolution is represented in matrix multiplication form of (31) using (32). Since the expression inside the square brackets of (30) is the matrix Y defined in (31):

$$\mathrm{Y}=\sqrt{2N}{{\mathrm{A}}_2}^{-1}{\mathrm{C}}_{\mathrm{I}\mathrm{I}} (\mathrm{w}\times \mathrm{x})$$

(33)

$${\mathrm{C}}_{\mathrm{I}\mathrm{I}} (\mathrm{w}\times \mathrm{x})=\frac{1}{\sqrt{2N}}{\mathrm{A}}_2\mathrm{Y}$$

(34)

Equation (34) corresponds to (30). Thus Y can be computed by using either (33) or (31). In other words, the symmetric convolution of DCT coefficients, Figure 1(b) is an alternative method to computing the DCT of multiplication of two time sequences, Figure 1(a).

Figure 1. For compressed image / video editing, logo-keying operation (alpha blending) can be done in (a) the spatial and (b) transform domains [1]. The symbol × denotes the element-by-element multiplication of the two vectors and ⊗ denotes the symmetric convolution of the two vectors.

Figure 1. For compressed image / video editing, logo-keying operation (alpha blending) can be done in (a) the spatial and (b) transform domains [1]. The symbol × denotes the element-by-element multiplication of the two vectors and ⊗ denotes the symmetric convolution of the two vectors.

5. Conclusions

For logo-keying operation or alpha blending, one image is scaled by α and the other is scaled by (1 − α), where α is a value between zero and one, and then two images are added up. For compressed image / video editing, multiplication operation in logo-keying operation in the spatial can be done in the DTT domain by convolving the unitary DTT coefficients of the two images. For those applications, we have developed 40 types of the MSC properties for the unitary DTTs (DCTs and DSTs [11,12,13]).