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Extensions on Low-Complexity DCT Approximations for Larger Blocklengths Based on Minimal Angle Similarity

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Abstract

The discrete cosine transform (DCT) is a central tool for image and video coding because it can be related to the Karhunen-Loéve transform (KLT), which is the optimal transform in terms of retained transform coefficients and data decorrelation. In this paper, we introduce 16-, 32-, and 64-point low-complexity DCT approximations by minimizing individually the angle between the rows of the exact DCT matrix and the matrix induced by the approximate transforms. According to some classical figures of merit, the proposed transforms outperformed the approximations for the DCT already known in the literature. Fast algorithms were also developed for the low-complexity transforms, asserting a good balance between the performance and its computational cost. Practical applications in image encoding showed the relevance of the transforms in this context. In fact, the experiments showed that the proposed transforms had better results than the known approximations in the literature for the cases of 16, 32, and 64 blocklength.

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Acknowledgements

We gratefully acknowledge partial financial support from Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES), Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), and Fundação de Amparo a Ciência e Tecnologia de Pernambuco (FACEPE), Brazil.

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Authors and Affiliations

  1. Programa de Pós-Graduação em Estatística, Universidade Federal de Pernambuco, Recife, Brazil

    Anabeth P. Radünz & Luan Portella

  2. Programa de Pós-Graduação em Engenharia Elétrica, Universidade Federal de Pernambuco (UFPE), Recife, Brazil

    R. S. Oliveira

  3. Departamento de Estatística and LACESM, Universidade Federal de Santa Maria, Santa Maria, Brazil

    Fábio M. Bayer

  4. Signal Processing Group, Departamento de Estatística, Universidade Federal de Pernambuco, Recife, Brazil

    Renato J. Cintra

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  1. Anabeth P. Radünz
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  2. Luan Portella
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  3. R. S. Oliveira
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  5. Renato J. Cintra
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Contributions

Anabeth P. Radünz: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Writing - original draft, Visualization. Luan Portella: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Writing - original draft, Visualization. Raiza S. Oliveira: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Visualization. Fábio M. Bayer: Conceptualization, Methodology, Validation, Investigation, Project administration, Supervision, Writing - review & editing. Renato J. Cintra: Conceptualization, Methodology, Validation, Investigation, Project administration, Supervision, Writing - review & editing.

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Appendix. Matrix Factorization

Appendix. Matrix Factorization

The matrices used in the Section 5.4 are detailed here. First, consider the following butterfly-structure:

$$\begin{aligned} \mathbf {B}_{N}= \begin{bmatrix} {\mathbf {I}}_{\frac{N}{2}}&{}\bar{\mathbf {I}}_{\frac{N}{2}}\\ -\bar{\mathbf {I}}_{\frac{N}{2}}&{}{\mathbf {I}}_{\frac{N}{2}} \end{bmatrix}. \end{aligned}$$

For the sake of brevity, cycle notation  [111,112,113] is used to present the permutation matrices. The resulting permutation matrix is obtained by permuting the columns of the identity matrix following the cycle mapping zero indexed.

1.1 Matrix Factorization for \(N=16\)

The low-complexity matrix \(\mathbf {T}_{16,5}\) can be represented as:

$$\begin{aligned} \mathbf {T}_{16,5}=\mathbf {P}_{16}\cdot \mathbf {M}_{16} \cdot \left[ \begin{array}{cc} \mathbf {B}_{2}&{}\\ &{}{\mathbf {I}}_{14} \end{array} \right] \cdot \left[ \begin{array}{cc} \mathbf {B}_{4}&{}\\ &{}{\mathbf {I}}_{12} \end{array} \right] \cdot \left[ \begin{array}{cc} \mathbf {B}_{8}&{}\\ &{}{\mathbf {I}}_{8} \end{array}\right] \cdot \mathbf {B}_{16} \end{aligned}$$

where

$$\begin{aligned} {\mathbf {P}}_{16}= {(1\;8)\kern 0.10em(2\;4)\kern 0.10em(3\;12\;9)\kern 0.10em(5\;6\;10)\kern 0.10em(7\;14\;13\;11)}, \end{aligned}$$
$$\begin{aligned} {\mathbf {M}}_{16}= \left[ \begin{array}{rrrrrrrrrrrrrrrr} 1 &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} \\ &{} -1 &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} \\ &{} &{} -1 &{} -2 &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} \\ &{} &{} 2 &{} -1 &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} \\ &{} &{} &{} &{} -\frac{1}{2} &{} -1 &{} -2 &{} -2 &{} &{} &{} &{} &{} &{} &{} &{} \\ &{} &{} &{} &{} 1 &{} 2 &{} \frac{1}{2} &{} -2 &{} &{} &{} &{} &{} &{} &{} &{} \\ &{} &{} &{} &{} -2 &{} -\frac{1}{2} &{} 2 &{} -1 &{} &{} &{} &{} &{} &{} &{} &{} \\ &{} &{} &{} &{} 2 &{} -2 &{} 1 &{} -\frac{1}{2} &{} &{} &{} &{} &{} &{} &{} &{} \\ &{} &{} &{} &{} &{} &{} &{} &{} -\frac{1}{4} &{} -\frac{1}{2} &{} -1 &{} -1 &{} -2 &{} -2 &{} -2 &{} -2 \\ &{} &{} &{} &{} &{} &{} &{} &{} \frac{1}{2} &{} 2 &{} 2 &{} 2 &{} 1 &{} -\frac{1}{4} &{} -1 &{} -2 \\ &{} &{} &{} &{} &{} &{} &{} &{} -1 &{} -2 &{} -1 &{} \frac{1}{2} &{} 2 &{} 2 &{} -\frac{1}{4} &{} -2 \\ &{} &{} &{} &{} &{} &{} &{} &{} 1 &{} 2 &{} -\frac{1}{2} &{} -2 &{} -\frac{1}{4} &{} 2 &{} 1 &{} -2 \\ &{} &{} &{} &{} &{} &{} &{} &{} -2 &{} -1 &{} 2 &{} \frac{1}{4} &{} -2 &{} \frac{1}{2} &{} 2 &{} -1 \\ &{} &{} &{} &{} &{} &{} &{} &{} 2 &{} -\frac{1}{4} &{} -2 &{} 2 &{} -\frac{1}{2} &{} -1 &{} 2 &{} -1 \\ &{} &{} &{} &{} &{} &{} &{} &{} -2 &{} 1 &{} -\frac{1}{4} &{} -1 &{} 2 &{} -2 &{} 2 &{} -\frac{1}{2} \\ &{} &{} &{} &{} &{} &{} &{} &{} 2 &{} -2 &{} 2 &{} -2 &{} 1 &{} -1 &{} \frac{1}{2} &{} -\frac{1}{4} \end{array}\right] . \end{aligned}$$

1.2 Matrix Factorization for \(N=32\)

The low-complexity matrix \(\mathbf {T}_{32,2}\) can be represented as:

$$\begin{aligned} \mathbf {T}_{32,2}=\mathbf {P}_{32}\cdot \left[ \begin{array}{cc} \mathbf {L}_{1}&{}\\ &{}{\mathbf {L}}_{2} \end{array} \right] \cdot \left[ \begin{array}{cc} \mathbf {B}_{2}&{}\\ &{}{\mathbf {I}}_{30} \end{array} \right] \cdot \left[ \begin{array}{cc} \mathbf {B}_{4}&{}\\ &{}{\mathbf {I}}_{28} \end{array} \right] \cdot \left[ \begin{array}{cc} \mathbf {B}_{8}&{}\\ &{}{\mathbf {I}}_{24} \end{array} \right] \\ \cdot \left[ \begin{array}{cc} \mathbf {B}_{16}&{}\\ &{}{\mathbf {I}}_{16} \end{array} \right] \cdot \mathbf {B}_{32} \end{aligned}$$

where

$$\begin{aligned} {\mathbf {P}}_{32}= {(1\;16\;5\;20\;13\;26\;25\;23\;19\;11\;18\;9\;10\;14\;30)\kern 0.10em(2\;8\;6\;28\;29\;31\;3\;24\;21\;15)\kern 0.10em(4\;12\;22\;17\;7)}, \end{aligned}$$
$$\begin{aligned} {\mathbf {L}}_1= \left[ \begin{array}{rrrrrrrrrrrrrrrr} 1 &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} \\ &{} -1 &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} \\ &{} &{} -\frac{1}{2} &{} -1 &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} \\ &{} &{} 1 &{} -\frac{1}{2} &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} \\ &{} &{} &{} &{} \frac{1}{2} &{} 1 &{} &{} -1 &{} &{} &{} &{} &{} &{} &{} &{} \\ &{} &{} &{} &{} -1 &{} &{} 1 &{} -\frac{1}{2} &{} &{} &{} &{} &{} &{} &{} &{} \\ &{} &{} &{} &{} 1 &{} -1 &{} \frac{1}{2} &{} &{} &{} &{} &{} &{} &{} &{} &{} \\ &{} &{} &{} &{} &{} -\frac{1}{2} &{} -1 &{} -1 &{} &{} &{} &{} &{} &{} &{} &{} \\ &{} &{} &{} &{} &{} &{} &{} &{} \frac{1}{2} &{} 1 &{} 1 &{} 1 &{} \frac{1}{2} &{} &{} -\frac{1}{2} &{} -1 \\ &{} &{} &{} &{} &{} &{} &{} &{} -\frac{1}{2} &{} -1 &{} -\frac{1}{2} &{} \frac{1}{2} &{} 1 &{} 1 &{} &{} -1 \\ &{} &{} &{} &{} &{} &{} &{} &{} \frac{1}{2} &{} 1 &{} -\frac{1}{2} &{} -1 &{} &{} 1 &{} \frac{1}{2} &{} -1 \\ &{} &{} &{} &{} &{} &{} &{} &{} -1 &{} -\frac{1}{2} &{} 1 &{} &{} -1 &{} \frac{1}{2} &{} 1 &{} -\frac{1}{2} \\ &{} &{} &{} &{} &{} &{} &{} &{} 1 &{} &{} -1 &{} 1 &{} -\frac{1}{2} &{} -\frac{1}{2} &{} 1 &{} -\frac{1}{2} \\ &{} &{} &{} &{} &{} &{} &{} &{} -1 &{} \frac{1}{2} &{} &{} -\frac{1}{2} &{} 1 &{} -1 &{} 1 &{} -\frac{1}{2} \\ &{} &{} &{} &{} &{} &{} &{} &{} 1 &{} -1 &{} 1 &{} -1 &{} \frac{1}{2} &{} -\frac{1}{2} &{} \frac{1}{2} &{} \\ &{} &{} &{} &{} &{} &{} &{} &{} &{} -\frac{1}{2} &{} -\frac{1}{2} &{} -\frac{1}{2} &{} -1 &{} -1 &{} -1 &{} -1 \\ \end{array}\right] \end{aligned}$$
$$\begin{aligned} {\mathbf {L}}_2= \left[ \begin{array}{rrrrrrrrrrrrrrrr} -\frac{1}{2} &{} -1 &{} -1 &{} -1 &{} -1 &{} -\frac{1}{2} &{} &{} \frac{1}{2} &{} 1 &{} 1 &{} 1 &{} \frac{1}{2} &{} &{} -\frac{1}{2} &{} -1 &{} -1 \\ \frac{1}{2} &{} 1 &{} 1 &{} 1 &{} &{} -\frac{1}{2} &{} -1 &{} -1 &{} -\frac{1}{2} &{} \frac{1}{2} &{} 1 &{} 1 &{} 1 &{} &{} -\frac{1}{2} &{} -1 \\ -\frac{1}{2} &{} -1 &{} -1 &{} &{} 1 &{} 1 &{} \frac{1}{2} &{} -\frac{1}{2} &{} -1 &{} -1 &{} &{} 1 &{} 1 &{} \frac{1}{2} &{} -\frac{1}{2} &{} -1 \\ \frac{1}{2} &{} 1 &{} \frac{1}{2} &{} -\frac{1}{2} &{} -1 &{} -\frac{1}{2} &{} 1 &{} 1 &{} \frac{1}{2} &{} -1 &{} -1 &{} &{} 1 &{} 1 &{} &{} -1 \\ -\frac{1}{2} &{} -1 &{} &{} 1 &{} \frac{1}{2} &{} -1 &{} -1 &{} &{} 1 &{} \frac{1}{2} &{} -1 &{} -1 &{} \frac{1}{2} &{} 1 &{} \frac{1}{2} &{} -1 \\ 1 &{} 1 &{} -\frac{1}{2} &{} -1 &{} \frac{1}{2} &{} 1 &{} &{} -1 &{} &{} 1 &{} \frac{1}{2} &{} -1 &{} -\frac{1}{2} &{} 1 &{} \frac{1}{2} &{} -1 \\ -1 &{} -\frac{1}{2} &{} 1 &{} \frac{1}{2} &{} -1 &{} -\frac{1}{2} &{} 1 &{} &{} -1 &{} &{} 1 &{} -\frac{1}{2} &{} -1 &{} \frac{1}{2} &{} 1 &{} -1 \\ 1 &{} \frac{1}{2} &{} -1 &{} \frac{1}{2} &{} 1 &{} -1 &{} -\frac{1}{2} &{} 1 &{} &{} -1 &{} 1 &{} \frac{1}{2} &{} -1 &{} &{} 1 &{} -\frac{1}{2} \\ -1 &{} &{} 1 &{} -1 &{} &{} 1 &{} -1 &{} -\frac{1}{2} &{} 1 &{} -1 &{} -\frac{1}{2} &{} 1 &{} -\frac{1}{2} &{} -\frac{1}{2} &{} 1 &{} -\frac{1}{2} \\ 1 &{} -\frac{1}{2} &{} -\frac{1}{2} &{} 1 &{} -1 &{} &{} 1 &{} -1 &{} \frac{1}{2} &{} \frac{1}{2} &{} -1 &{} 1 &{} &{} -1 &{} 1 &{} -\frac{1}{2} \\ -1 &{} \frac{1}{2} &{} &{} -1 &{} 1 &{} -1 &{} \frac{1}{2} &{} \frac{1}{2} &{} -1 &{} 1 &{} -\frac{1}{2} &{} &{} 1 &{} -1 &{} 1 &{} -\frac{1}{2} \\ 1 &{} -1 &{} \frac{1}{2} &{} &{} -\frac{1}{2} &{} 1 &{} -1 &{} 1 &{} -\frac{1}{2} &{} &{} \frac{1}{2} &{} -1 &{} 1 &{} -1 &{} 1 &{} -\frac{1}{2} \\ -1 &{} 1 &{} -1 &{} \frac{1}{2} &{} -\frac{1}{2} &{} &{} \frac{1}{2} &{} -\frac{1}{2} &{} 1 &{} -1 &{} 1 &{} -1 &{} 1 &{} -1 &{} \frac{1}{2} &{} \\ 1 &{} -1 &{} 1 &{} -1 &{} 1 &{} -1 &{} 1 &{} -1 &{} 1 &{} -\frac{1}{2} &{} \frac{1}{2} &{} -\frac{1}{2} &{} \frac{1}{2} &{} -\frac{1}{2} &{} &{} \\ &{} &{} -\frac{1}{2} &{} -\frac{1}{2} &{} -\frac{1}{2} &{} -\frac{1}{2} &{} -\frac{1}{2} &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 \\ &{} \frac{1}{2} &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} \frac{1}{2} &{} \frac{1}{2} &{} &{} -\frac{1}{2} &{} -\frac{1}{2} &{} -1 &{} -1 &{} -1 \\ \end{array}\right] . \end{aligned}$$

1.3 Matrix Factorization for \(N=64\)

The low-complexity matrix \(\mathbf {T}_{64,1}\) can be represented as:

$$\begin{aligned} \mathbf {T}_{64,1}=\mathbf {P}_{64} \cdot \left[ \begin{array}{ccc} \mathbf {Z}_{1}&{}&{}\\ &{}{\mathbf {Z}}_{2}&{}\\ &{}&{}{\mathbf {Z}}_{3} \end{array} \right] \cdot \left[ \begin{array}{cc} \mathbf {B}_{2}&{}\\ &{}{\mathbf {I}}_{62} \end{array} \right] \cdot \left[ \begin{array}{cc} \mathbf {B}_{4}&{}\\ &{}{\mathbf {I}}_{60} \end{array} \right] \cdot \left[ \begin{array}{cc} \mathbf {B}_{8}&{}\\ &{}{\mathbf {I}}_{56} \end{array} \right] \\ \cdot \left[ \begin{array}{cc} \mathbf {B}_{16}&{}\\ &{}{\mathbf {I}}_{48} \end{array} \right] \cdot \left[ \begin{array}{cc} \mathbf {B}_{32}&{}\\ &{}{\mathbf {I}}_{32} \end{array} \right] \cdot \mathbf {B}_{64} \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} {\mathbf {P}}_{64}=&\ {(1\;32\;17\;22\;42\;37\;27\;62\;5\;40\;33\;19\;30\;14\;4\;24\;50\;53\;59\;9\;28\;2\;16\;18\;26\;58\;7\;8\;20\;34\;21\;38\;29\;6\;56)} \\&\ {(3\;48\;49\;51\;55\;63\;13\;60\;11\;44\;41\;35\;23\;46\;45\;43\;39\;31\;10\;36\;25\;54\;61\;15\;12\;52\;57)}, \end{aligned} \end{aligned}$$
$$\begin{aligned} {\mathbf {Z}}_1= \left[ \begin{array}{rrrrrrrrrrrrrrrr} 1 &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} \\ &{}-1&{} &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} \\ &{} &{}-1&{}-1&{} &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} \\ &{} &{} 1&{} &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} \\ &{} &{} &{} &{} 1&{}1&{} &{} -1&{} &{} &{} &{} &{} &{} &{} &{} \\ &{} &{} &{} &{}-1&{} &{} 1&{} -1&{} &{} &{} &{} &{} &{} &{} &{} \\ &{} &{} &{} &{} 1&{}-1&{}1&{} &{} &{} &{} &{} &{} &{} &{} &{} \\ &{} &{} &{} &{} &{}-1&{}-1&{}-1&{} &{} &{} &{} &{} &{} &{} &{} \\ &{} &{} &{} &{} &{} &{} &{} &{} -1 &{} -1 &{} -1 &{} &{} 1 &{} 1 &{} &{} -1\\ &{} &{} &{} &{} &{} &{} &{} &{} 1 &{} 1 &{} &{} -1 &{} &{} 1 &{} 1 &{} -1 \\ &{} &{} &{} &{} &{} &{} &{} &{} -1 &{} -1 &{} 1 &{} &{} -1 &{} &{} 1 &{} -1\\ &{} &{} &{} &{} &{} &{} &{} &{} 1 &{} &{} -1 &{} 1 &{} &{} -1 &{} 1 &{} -1 \\ &{} &{} &{} &{} &{} &{} &{} &{} -1 &{} 1 &{} &{} -1 &{} 1 &{} -1 &{} 1 &{} \\ &{} &{} &{} &{} &{} &{} &{} &{} 1 &{} -1 &{} 1 &{} -1 &{} 1 &{} -1 &{} &{} \\ &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} -1&{}-1 &{} -1&{} -1&{} -1&{} -1 \\ &{} &{} &{} &{} &{} &{} &{} &{} &{} 1 &{} 1 &{} 1 &{} 1 &{} &{} -1 &{} -1 \\ \end{array}\right] , \end{aligned}$$
$$\begin{aligned} {\mathbf {Z}}_2= \left[ \begin{array}{rrrrrrrrrrrrrrrr} -1 &{} -1 &{} -1 &{} &{} 1 &{} 1 &{} 1 &{} &{} -1 &{} -1 &{} &{} 1 &{} 1 &{} 1 &{} &{} -1\\ 1 &{} 1 &{} 1 &{} -1 &{} -1 &{} &{} 1 &{} 1 &{} &{} -1 &{} -1 &{} &{} 1 &{} 1 &{} &{} -1\\ -1 &{} -1 &{} &{} 1 &{} 1 &{} -1 &{} -1 &{} &{} 1 &{} 1 &{} -1 &{} -1 &{} &{} 1 &{} &{} -1\\ 1 &{} 1 &{} -1 &{} -1 &{} &{} 1 &{} &{} -1 &{} &{} 1 &{} &{} -1 &{} -1 &{} 1 &{} 1 &{} -1\\ -1 &{} -1 &{} 1 &{} 1 &{} -1 &{} &{} 1 &{} &{} -1 &{} &{} 1 &{} &{} -1 &{} 1 &{} 1 &{} -1 \\ 1 &{} &{} -1 &{} &{} 1 &{} -1 &{} -1 &{} 1 &{} &{} -1 &{} 1 &{} 1 &{} -1 &{} &{} 1 &{} -1\\ -1 &{} &{} 1 &{} -1 &{} &{} 1 &{} -1 &{} &{} 1 &{} -1 &{} &{} 1 &{} -1 &{} -1 &{} 1 &{} -1\\ 1 &{} &{} -1 &{} 1 &{} -1 &{} &{} 1 &{} -1 &{} &{} 1 &{} -1 &{} 1 &{} &{} -1 &{} 1 &{} -1\\ -1 &{} 1 &{} &{} -1 &{} 1 &{} -1 &{} &{} 1 &{} -1 &{} 1 &{} -1 &{} &{} 1 &{} -1 &{} 1 &{} \\ 1 &{} -1 &{} &{} &{} -1 &{} 1 &{} -1 &{} 1 &{} -1 &{} &{} 1 &{} -1 &{} 1 &{} -1 &{} 1 &{} \\ -1 &{} 1 &{} -1 &{} 1 &{} &{} &{} &{} -1 &{} 1 &{} -1 &{} 1 &{} -1 &{} 1 &{} -1 &{} 1 &{} \\ 1 &{} -1 &{} 1 &{} -1 &{} 1 &{} -1 &{} 1 &{} -1 &{} 1 &{} -1 &{} 1 &{} -1 &{} &{} &{} &{} \\ &{} &{} &{} &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 &{} -1\\ &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} &{} &{} &{} -1 &{} -1 &{} -1 &{} -1\\ &{} 1 &{} 1 &{} 1 &{} &{} -1 &{} -1 &{} -1 &{} -1 &{} &{} 1 &{} 1 &{} 1 &{} &{} -1 &{} -1\\ &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 &{} &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} &{} &{} -1 &{} -1\\ \end{array}\right],\ {\mathbf {Z}}_{3}= \left[ \begin{array}{ccc} \mathbf {Q}_{1}&{}\mathbf {Q}_{2}\\ \mathbf {Q}_{3}&{}{\mathbf {Q}}_{4} \end{array} \right] , \end{aligned}$$

and

$$\begin{aligned} {\mathbf {Q}}_{1}= \left[ \begin{array}{rrrrrrrrrrrrrrrr} -1 &{} -1 &{} -1 &{} &{} 1 &{} 1 &{} 1 &{} &{} -1 &{} -1 &{} -1 &{} &{} 1 &{} 1 &{} 1 &{} \\ 1 &{} 1 &{} 1 &{} &{} -1 &{} -1 &{} &{} 1 &{} 1 &{} 1 &{} &{} -1 &{} -1 &{} &{} 1 &{} 1 \\ -1 &{} -1 &{} -1 &{} 1 &{} 1 &{} 1 &{} -1 &{} -1 &{} -1 &{} &{} 1 &{} 1 &{} &{} -1 &{} -1 &{} \\ 1 &{} 1 &{} &{} -1 &{} -1 &{} &{} 1 &{} 1 &{} &{} -1 &{} -1 &{} 1 &{} 1 &{} 1 &{} -1 &{} -1 \\ -1 &{} -1 &{} &{} 1 &{} 1 &{} -1 &{} -1 &{} &{} 1 &{} 1 &{} &{} -1 &{} &{} 1 &{} 1 &{} \\ 1 &{} 1 &{} &{} -1 &{} &{} 1 &{} 1 &{} -1 &{} -1 &{} &{} 1 &{} 1 &{} -1 &{} -1 &{} &{} 1 \\ -1 &{} -1 &{} 1 &{} 1 &{} &{} -1 &{} &{} 1 &{} 1 &{} -1 &{} -1 &{} 1 &{} 1 &{} &{} -1 &{} \\ 1 &{} 1 &{} -1 &{} -1 &{} 1 &{} 1 &{} -1 &{} -1 &{} &{} 1 &{} &{} -1 &{} &{} 1 &{} &{} -1 \\ -1 &{} -1 &{} 1 &{} 1 &{} -1 &{} -1 &{} 1 &{} 1 &{} -1 &{} &{} 1 &{} &{} -1 &{} &{} 1 &{} \\ 1 &{} 1 &{} -1 &{} &{} 1 &{} &{} -1 &{} &{} 1 &{} -1 &{} -1 &{} 1 &{} 1 &{} -1 &{} &{} 1 \\ -1 &{} -1 &{} 1 &{} &{} -1 &{} 1 &{} 1 &{} -1 &{} &{} 1 &{} &{} -1 &{} 1 &{} 1 &{} -1 &{} \\ 1 &{} &{} -1 &{} 1 &{} 1 &{} -1 &{} &{} 1 &{} -1 &{} -1 &{} 1 &{} &{} -1 &{} 1 &{} 1 &{} -1 \\ -1 &{} &{} 1 &{} -1 &{} &{} 1 &{} -1 &{} -1 &{} 1 &{} &{} -1 &{} 1 &{} &{} -1 &{} 1 &{} \\ 1 &{} &{} -1 &{} 1 &{} &{} -1 &{} 1 &{} &{} -1 &{} 1 &{} &{} -1 &{} 1 &{} &{} -1 &{} 1 \\ -1 &{} &{} 1 &{} -1 &{} 1 &{} 1 &{} -1 &{} 1 &{} &{} -1 &{} 1 &{} &{} -1 &{} 1 &{} -1 &{} \\ 1 &{} &{} -1 &{} 1 &{} -1 &{} &{} 1 &{} -1 &{} 1 &{} &{} -1 &{} 1 &{} -1 &{} &{} 1 &{} -1 \\ \end{array}\right] , \end{aligned}$$
$$\begin{aligned} {\mathbf {Q}}_{2}= \left[ \begin{array}{rrrrrrrrrrrrrrrr} -1 &{} -1 &{} &{} 1 &{} 1 &{} 1 &{} &{} -1 &{} -1 &{} -1 &{} &{} 1 &{} 1 &{} 1 &{} &{} -1 \\ &{} -1 &{} -1 &{} -1 &{} &{} 1 &{} 1 &{} &{} -1 &{} -1 &{} -1 &{} 1 &{} 1 &{} 1 &{} &{} -1 \\ 1 &{} 1 &{} &{} -1 &{} -1 &{} &{} 1 &{} 1 &{} &{} -1 &{} -1 &{} &{} 1 &{} 1 &{} &{} -1 \\ &{} 1 &{} 1 &{} &{} -1 &{} -1 &{} &{} 1 &{} 1 &{} -1 &{} -1 &{} &{} 1 &{} 1 &{} &{} -1 \\ -1 &{} -1 &{} 1 &{} 1 &{} &{} -1 &{} -1 &{} 1 &{} 1 &{} &{} -1 &{} -1 &{} 1 &{} 1 &{} &{} -1 \\ &{} -1 &{} -1 &{} 1 &{} 1 &{} &{} -1 &{} &{} 1 &{} 1 &{} -1 &{} -1 &{} &{} 1 &{} 1 &{} -1 \\ 1 &{} &{} -1 &{} -1 &{} 1 &{} 1 &{} -1 &{} -1 &{} &{} 1 &{} &{} -1 &{} &{} 1 &{} 1 &{} -1 \\ &{} 1 &{} &{} -1 &{} &{} 1 &{} &{} -1 &{} -1 &{} 1 &{} 1 &{} -1 &{} -1 &{} 1 &{} 1 &{} -1 \\ -1 &{} &{} 1 &{} &{} -1 &{} &{} 1 &{} &{} -1 &{} 1 &{} 1 &{} -1 &{} -1 &{} 1 &{} 1 &{} -1 \\ &{} -1 &{} &{} 1 &{} -1 &{} -1 &{} 1 &{} 1 &{} -1 &{} &{} 1 &{} &{} -1 &{} 1 &{} 1 &{} -1 \\ 1 &{} &{} -1 &{} 1 &{} 1 &{} -1 &{} &{} 1 &{} -1 &{} -1 &{} 1 &{} &{} -1 &{} &{} 1 &{} -1 \\ &{} 1 &{} -1 &{} &{} 1 &{} &{} -1 &{} 1 &{} &{} -1 &{} 1 &{} 1 &{} -1 &{} &{} 1 &{} -1 \\ -1 &{} 1 &{} 1 &{} -1 &{} 1 &{} 1 &{} -1 &{} &{} 1 &{} -1 &{} &{} 1 &{} -1 &{} &{} 1 &{} -1 \\ &{} -1 &{} 1 &{} &{} -1 &{} 1 &{} &{} -1 &{} 1 &{} -1 &{} -1 &{} 1 &{} -1 &{} -1 &{} 1 &{} -1 \\ 1 &{} -1 &{} &{} 1 &{} -1 &{} &{} 1 &{} -1 &{} 1 &{} &{} -1 &{} 1 &{} &{} -1 &{} 1 &{} -1 \\ &{} 1 &{} -1 &{} 1 &{} &{} -1 &{} 1 &{} -1 &{} &{} 1 &{} -1 &{} 1 &{} &{} -1 &{} 1 &{} -1 \\ \end{array}\right] , \end{aligned}$$
$$\begin{aligned} {\mathbf {Q}}_{3}= \left[ \begin{array}{rrrrrrrrrrrrrrrr} -1 &{} 1 &{} &{} -1 &{} 1 &{} -1 &{} &{} 1 &{} -1 &{} 1 &{} &{} -1 &{} 1 &{} -1 &{} &{} 1 \\ 1 &{} -1 &{} &{} 1 &{} -1 &{} 1 &{} -1 &{} &{} 1 &{} -1 &{} 1 &{} -1 &{} &{} 1 &{} -1 &{} 1 \\ -1 &{} 1 &{} &{} &{} 1 &{} -1 &{} 1 &{} -1 &{} &{} 1 &{} -1 &{} 1 &{} -1 &{} 1 &{} &{} -1 \\ 1 &{} -1 &{} 1 &{} &{} -1 &{} 1 &{} -1 &{} 1 &{} -1 &{} 1 &{} &{} &{} 1 &{} -1 &{} 1 &{} -1 \\ -1 &{} 1 &{} -1 &{} &{} &{} &{} 1 &{} -1 &{} 1 &{} -1 &{} 1 &{} -1 &{} 1 &{} &{} &{} 1 \\ 1 &{} -1 &{} 1 &{} -1 &{} 1 &{} &{} &{} &{} -1 &{} 1 &{} -1 &{} 1 &{} -1 &{} 1 &{} -1 &{} 1 \\ -1 &{} 1 &{} -1 &{} 1 &{} -1 &{} 1 &{} -1 &{} 1 &{} &{} &{} &{} &{} &{} -1 &{} 1 &{} -1 \\ 1 &{} -1 &{} 1 &{} -1 &{} 1 &{} -1 &{} 1 &{} -1 &{} 1 &{} -1 &{} 1 &{} -1 &{} 1 &{} -1 &{} 1 &{} -1 \\ &{} &{} &{} &{} &{} &{} &{} &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 \\ &{} &{} &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 \\ &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} &{} &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 \\ &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 &{} &{} &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} &{} &{} -1 \\ &{} 1 &{} 1 &{} 1 &{} 1 &{} &{} &{} -1 &{} -1 &{} -1 &{} -1 &{} &{} 1 &{} 1 &{} 1 &{} 1 \\ &{} 1 &{} 1 &{} 1 &{} &{} -1 &{} -1 &{} -1 &{} &{} 1 &{} 1 &{} 1 &{} &{} -1 &{} -1 &{} -1 \\ &{} &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 &{} &{} &{} &{} 1 &{} 1 \\ &{} -1 &{} -1 &{} -1 &{} &{} &{} 1 &{} 1 &{} 1 &{} &{} -1 &{} -1 &{} -1 &{} -1 &{} &{} 1 \\ \end{array}\right] , \end{aligned}$$
$$\begin{aligned} {\mathbf {Q}}_{4}= \left[\begin{array}{rrrrrrrrrrrrrrrr} -1 &{} 1 &{} -1 &{} &{} 1 &{} -1 &{} 1 &{} &{} -1 &{} 1 &{} -1 &{} &{} 1 &{} -1 &{} 1 &{} \\ -1 &{} &{} 1 &{} -1 &{} 1 &{} -1 &{} &{} 1 &{} -1 &{} 1 &{} &{} &{} 1 &{} -1 &{} 1 &{} \\ 1 &{} -1 &{} 1 &{} -1 &{} &{} 1 &{} -1 &{} 1 &{} -1 &{} &{} &{} -1 &{} 1 &{} -1 &{} 1 &{} \\ 1 &{} &{} &{} 1 &{} -1 &{} 1 &{} -1 &{} 1 &{} &{} &{} 1 &{} -1 &{} 1 &{} -1 &{} 1 &{} \\ -1 &{} 1 &{} -1 &{} 1 &{} -1 &{} 1 &{} &{} &{} 1 &{} -1 &{} 1 &{} -1 &{} 1 &{} -1 &{} 1 &{} \\ -1 &{} 1 &{} &{} &{} &{} -1 &{} 1 &{} -1 &{} 1 &{} -1 &{} 1 &{} -1 &{} 1 &{} -1 &{} &{} \\ 1 &{} -1 &{} 1 &{} -1 &{} 1 &{} -1 &{} 1 &{} -1 &{} 1 &{} -1 &{} 1 &{} -1 &{} 1 &{} &{} &{} \\ 1 &{} -1 &{} 1 &{} -1 &{} 1 &{} -1 &{} 1 &{} -1 &{} &{} &{} &{} &{} &{} &{} &{} \\ -1 &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 \\ 1 &{} 1 &{} 1 &{} &{} &{} &{} &{} &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 \\ -1 &{} &{} &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} &{} &{} &{} -1 &{} -1 &{} -1 \\ -1 &{} -1 &{} -1 &{} -1 &{} &{} &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} &{} -1 &{} -1 &{} -1 \\ 1 &{} &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 &{} &{} 1 &{} 1 &{} 1 &{} 1 &{} &{} &{} -1 &{} -1 \\ -1 &{} &{} 1 &{} 1 &{} 1 &{} &{} -1 &{} -1 &{} -1 &{} &{} 1 &{} 1 &{} 1 &{} &{} -1 &{} -1 \\ 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} &{} &{} &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 \\ 1 &{} 1 &{} 1 &{} &{} -1 &{} -1 &{} -1 &{} -1 &{} &{} 1 &{} 1 &{} 1 &{} 1 &{} &{} -1 &{} -1 \\ \end{array}\right] . \end{aligned}$$

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Radünz, A.P., Portella, L., Oliveira, R.S. et al. Extensions on Low-Complexity DCT Approximations for Larger Blocklengths Based on Minimal Angle Similarity. J Sign Process Syst 95, 495–516 (2023). https://doi.org/10.1007/s11265-023-01848-w

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