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Low-complexity rounded KLT approximation for image compression

Abstract

The Karhunen–Loève transform (KLT) is often used for data decorrelation and dimensionality reduction. Because its computation depends on the matrix of covariances of the input signal, the use of the KLT in real-time applications is severely constrained by the difficulty in developing fast algorithms to implement it. In this context, this paper proposes a new class of low-complexity transforms that are obtained through the application of the round function to the elements of the KLT matrix. The proposed transforms are evaluated considering figures of merit that measure the coding power and distance of the proposed approximations to the exact KLT and are also explored in image compression experiments. Fast algorithms are introduced for the proposed approximate transforms. It was shown that the proposed transforms perform well in image compression and require a low implementation cost.

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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 à Ciência e Tecnologia de Pernambuco (FACEPE), Brazil.

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Correspondence to Anabeth P. Radünz.

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Appendix: 2D transformation and quantization step

Appendix: 2D transformation and quantization step

Let \({\mathbf {A}}\) be a \(8 \times 8\) sub-block from an image. The 2D transformation from \({\mathbf {A}}\) induced by an approximation \(\widehat{{\mathbf {T}}}\) is given by:

$$\begin{aligned} {\mathbf {B}}&= {\left\{ \begin{array}{ll} \widehat{{\mathbf {T}}} \cdot {\mathbf {A}} \cdot {\widehat{{\mathbf {T}}}}^\top , &{} \text {if }{\mathbf {T}}\text { is orthogonal,} \\ \widehat{{\mathbf {T}}} \cdot {\mathbf {A}} \cdot {\widehat{{\mathbf {T}}}}^{-1}, &{} \text {if }{\mathbf {T}}\text { is non-orthogonal,} \end{array}\right. } \nonumber \\&= {\left\{ \begin{array}{ll} ({\mathbf {u}} \cdot {\mathbf {u}}^\top ) \odot ({\mathbf {T}} \cdot {\mathbf {A}} \cdot {\mathbf {T}}^\top ), &{} \text {if }{\mathbf {T}}\text { is orthogonal,} \\ ({\mathbf {u}} \cdot {\mathbf {v}}^\top ) \odot ({\mathbf {T}} \cdot {\mathbf {A}} \cdot {\mathbf {T}}^{-1}), &{} \text {if }{\mathbf {T}}\text { is non-orthogonal,} \end{array}\right. } \nonumber \\&= {\mathbf {R}} \odot \widehat{{\mathbf {B}}}, \end{aligned}$$
(3)

where \({\mathbf {u}} = {\text {diag}}({\mathbf {S}})\) and \({\mathbf {v}}\) is given by the inverse elements from \({\mathbf {u}}\). In the context of JPEG-like compression [48], the quantized coefficient matrix \(\widehat{{\mathbf {B}}}\) is given by:

$$\begin{aligned} \bar{{\mathbf {B}}} = {\text {round}}({\mathbf {B}} \div {\mathbf {Q}}), \end{aligned}$$
(4)

where \({\mathbf {Q}}\) is a quantization matrix and \(\div\) denotes the element-wise matrix division.

By applying Eq. (3) in (4), we obtain

$$\begin{aligned} \bar{{\mathbf {B}}} = {\text {round}}({\mathbf {R}} \odot \widehat{{\mathbf {B}}} \div {\mathbf {Q}}) = {\text {round}}(\widehat{{\mathbf {B}}} \div \tilde{{\mathbf {Q}}}), \end{aligned}$$

where \(\tilde{{\mathbf {Q}}} = {\mathbf {Q}} \div {\mathbf {R}}\). Note that \({\mathbf {R}}\) can be absorbed in the quantization step, thus, the complexity of matrix \({\mathbf {S}}\) can be dismissed in the image compression applications [17, 40, 41].

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Radünz, A.P., Bayer, F.M. & Cintra, R.J. Low-complexity rounded KLT approximation for image compression. J Real-Time Image Proc (2021). https://doi.org/10.1007/s11554-021-01173-0

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Keywords

  • Approximate KLT
  • Image compression
  • Karhunen–Loève transform
  • Low-complexity transforms