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A global method for invertible integer DCT and integer wavelet algorithms. (English) Zbl 1053.65103
The author presents a new global approach to derive integers transforms from given linear transforms. More specifically, for a given linear transform $$\hat{F}: \mathbb{R} ^{n}\longrightarrow \mathbb{R} ^{n}$$ given by $$\hat{F}\left( x\right) =H_{n}X$$, where $$H_{n}\in \mathbb{R} ^{n}$$ is an invertible matrix, one can find an invertible integer transform $$F: \mathbb{Z} ^{n}\longrightarrow \mathbb{Z} ^{n}$$ approximating $$\hat{F}$$ by $$F\left( x\right) =\text{rd}\left( H_{n}X\right)$$ if $$H_{n}$$ satisfies this condition. Since $$H_{n}$$ does not satisfy this condition, one can blow up the matrix $$H_{n}$$ with a suitable expansion factor $$a_{n}>1$$ such that $$a_{n}H_{n}\left( \left( -1/2,1/2\right] ^{n}\right)$$ completely covers the unit cube $$\left[ -1/2,1/2\right) ^{n}$$. An invertible mapping $$F: \mathbb{Z} ^{n}\longrightarrow \mathbb{Z} ^{n}$$ can now simply be defined by $$F\left( x\right) =\text{rd}\left( a_{n}H_{n}X\right)$$, and is very close to the exact (scaled) transform $$a_{n}H_{n}X$$ since the error $$a_{n}H_{n}X-F\left( x\right)$$ is at most 1/2 in each component. This idea is applied in order to derive a new integer discrete cosine transform (DCT)-II algorithm of radix-$$2$$ length and new integer wavelet algorithms.

##### MSC:
 65T50 Numerical methods for discrete and fast Fourier transforms 65G50 Roundoff error 94A08 Image processing (compression, reconstruction, etc.) in information and communication theory 65T60 Numerical methods for wavelets
binDCT
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