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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.

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
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