A New Family of Spline-Based Biorthogonal Wavelet Transforms and Their Application to Image Compression

Averbuch, Amir; Zheludev, Valery A.

doi:10.1109/tip.2004.827229

Cited by 34 publications

(29 citation statements)

References 33 publications

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“…The analysis wavelets derived from interpolating quadratic spline (left to right): ϕ [3] [k],ψ [3] [k],ψ [2] [k],ψ [1] [k]. Top right the magnitudes of their DTFTs (half band).…”

Section: Fig 131 Top Leftmentioning

confidence: 99%

“…Top right the magnitudes of their DTFTs (half band). Bottom the same for the synthesis wavelets ψ [ j] [k], j = 1, 2, 3, ϕ [3] [k]…”

Section: Fig 131 Top Leftmentioning

confidence: 99%

“…The analysisψ [ j] [k], j = 1, 2, 3,φ [3] [k] and the synthesis ψ [ j] [k], j = 1, 2, 3, ϕ [3] [k] discrete-time wavelets from the first to third decomposition levels, which are derived from the interpolating quadratic spline, are displayed in Fig. 13.1.…”

Section: Fig 131 Top Leftmentioning

confidence: 99%

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Data Compression Using Wavelet and Local Cosine Transforms

Averbuch

Neittaanmäki

Zheludev

2015

Spline and Spline Wavelet Methods With Applications to Signal and Image Processing

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The chapter describes an algorithm that compresses two-dimensional data arrays, which are piece-wise smooth in one direction and have oscillating events in the other direction. Seismic, hyper-spectral and fingerprints data, for example, have such a mixed structure. The transform part of the compression process is an algorithm that combines wavelet and local cosine transform (LCT). The quantization and the entropy coding parts of the compression are taken from the SPIHT codec. To efficiently apply the SPIHT codec to a mixed coefficients array, reordering of the LCT coefficients takes place. On the data arrays, which have the mixed structure, this algorithm outperforms other algorithms that are based on the 2D wavelet transforms combined with the SPIHT coding and on the JPEG 2000 compression standard. The algorithm retains fine oscillating events even at a low bitrate. Its compression capabilities are also demonstrated on multimedia images that have a fine texture. The wavelet part in the mixed transform of the presented algorithm utilizes the library of Butterworth wavelet transforms described in Chap. 12.Currently, wavelet transforms constitute a recognized tool for signal and image processing applications. In particular, they have gained proven success in data compression. A main reason for such a success is a dual time-frequency (spatialfrequency) nature of wavelet transforms. This duality allows separation of smooth areas in a signal (image) from sharp transitions and high-frequency oscillations (edges in images). As a result, a signal (image) can be represented with a sufficient accuracy by a relatively small number of the transform's coefficients. Spatial and Spectral Meaning of Wavelet Transform CoefficientsIn this section, we summarize some known facts that highlight the duality of the wavelet transform coefficients. Multiscale wavelet transform of a signal is implemented via iterated multirate filtering. One step in the transform of a signal x = {x[k]} consists of filtering the signal by an, approximately half-band, low-pass filterh 0 and a high-pass filterh 1 . This filtering is followed by factor-2 downsampling of both filtered signals. Thus,

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“…The analysis wavelets derived from interpolating quadratic spline (left to right): ϕ [3] [k],ψ [3] [k],ψ [2] [k],ψ [1] [k]. Top right the magnitudes of their DTFTs (half band).…”

Section: Fig 131 Top Leftmentioning

confidence: 99%

“…Top right the magnitudes of their DTFTs (half band). Bottom the same for the synthesis wavelets ψ [ j] [k], j = 1, 2, 3, ϕ [3] [k]…”

Section: Fig 131 Top Leftmentioning

confidence: 99%

See 1 more Smart Citation

Data Compression Using Wavelet and Local Cosine Transforms

Averbuch

Neittaanmäki

Zheludev

2015

Spline and Spline Wavelet Methods With Applications to Signal and Image Processing

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“…All overcomplete dictionaries like Gabor, anisotropically refined Gaussian 7 and B-spline, 8 are mathematical formulas approximating the image model. They target all the possible combinations of pixel values.…”

Section: Ica Based Image Modelmentioning

confidence: 99%

A novel efficient image compression system based on independent component analysis

Shahid¹,

Dupont²,

Başkurt³

2009

SPIE Proceedings

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Next generation image compression system should be optimized the way human vision system (HVS) works. HVS has been evolved over millions of years for the images which exist in our environment. This idea is reinforced by the fact that sparse codes extracted from natural images resemble the primary visual cortex of HVS. We have introduced a novel technique in which basis functions trained by Independent Component Analysis (ICA) have been used to transform the image. ICA has been used to extract the independent features (basis functions) which are localized, bandlimited and oriented like HVS and resemble wavelet and Gabor bases. A greedy algorithm named matching pursuit (MP) has been used to transform the image in the ICA domain which is followed by quantization and multistage entropy coding. We have compared our codec with JPEG from the DCT family and JPEG2000 from the wavelets family. For fingerprint images, results are also compared with wavelet scalar quantization (WSQ) codec which has been especially tailored for this type of images. Our codec outperforms JPEG and WSQ and also performs comparable to JPEG2000 with lower complexity than the latter.

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“…Note that the causal Butterworth filters were used by Herley and Vetterli [15] to construct orthogonal non-symmetric wavelets. In our previous papers [1,2] we presented a family of biorthogonal symmetric wavelets related to the Butterworth filters and their application to image compression.…”

Section: Introductionmentioning

confidence: 99%

Interpolatory frames in signal space

Averbuch

Zheludev

Cohen

2006

IEEE Trans. Signal Process.

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We present a new family of frames, which are generated by perfect reconstruction filter banks of linear phased filters. The filter banks are based on discrete interpolatory splines and are related to Butterworth filters. Each filter bank contains one interpolatory symmetric low-pass filter and two high-pass filters, one of which is also interpolatory and symmetric. The second high-pass filter is either symmetric or antisymmetric. These filter banks generate the analysis and synthesis scaling functions and pairs of framelets. We introduce the concept of semi-tight frame. All the analysis waveforms in a tight frame coincide with their synthesis counterparts. In the semi-tight frame we can trade properties of smoothness and number of vanishing moments between the synthesis and the analysis framelets. We construct dual pairs of frames, where all the waveforms are symmetric and all the framelets have the same number of vanishing moments. Although most of the designed filters are IIR, they allow fast implementation via recursive procedures. The waveforms are well localized in time domain despite their infinite support. The frequency response of the designed filters is flat.

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A New Family of Spline-Based Biorthogonal Wavelet Transforms and Their Application to Image Compression

Cited by 34 publications

References 33 publications

Data Compression Using Wavelet and Local Cosine Transforms

Data Compression Using Wavelet and Local Cosine Transforms

A novel efficient image compression system based on independent component analysis

Interpolatory frames in signal space

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