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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
Software:
binDCT
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References:
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