Two-channel nonseparable wavelets statistically matched to 2-D images

Gupta, Anubha; Joshi, Sachindra

doi:10.1016/j.sigpro.2010.07.001

Cited by 8 publications

(6 citation statements)

References 23 publications

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“…ϕ⟩ which implies sgn ′ = 2δ. Using the FT of derivative of a function (24) and delta function, we can write F{sgn ′ } = F{2δ} =⇒ (j2πf ) sgn(f ) = 2. Thus, FT of sgn can be written as…”

Section: Examplementioning

confidence: 99%

“…Using the duality of FT, we can write 1 πt ⇌ −j sgn(f ) and using the differentiation property of FT from (24), we can write (−1)…”

Section: Examplementioning

confidence: 99%

“…Many studies present recent advancements and applications of the Fourier representation [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. In fact, wavelet transform that emerged as a generalization of the Fourier theory to capture joint time-scale relationship also finds umpteen applications in signal processing [22][23][24][25][26][27][28].…”

mentioning

confidence: 99%

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On the Convergence of Fourier Representations and Schwartz Distributions

Singh¹,

Singhal²,

Fatimah³

et al. 2023

Preprint

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Fourier theory is one of the most important tools used ubiquitously for understanding the spectral content of a signal, extracting and interpreting information from signals, and transmitting, processing, and analyzing the signals and systems. Undergraduate engineering students are exposed to these concepts, usually in their second year, to build their foundation in the areas of signal processing and communication engineering. However, the popular signal processing literature \cite{book1,book2,book3,book4} does not offer a clear explanation regarding the convergence or the existence of Fourier representations for certain well-known signals. Because of this subtle gap, it becomes hard for young students to assimilate the Fourier theory with clarity, and they are forced to be familiar with some of these concepts without understanding them. To bring clarity to the existence and the convergence of Fourier representation, including Fourier series and transform, lecture notes were published recently in IEEE Signal Processing Magazine's September 2022 issue \cite{PSDCs}. This lecture note is in the continuation with technical details added from yet another mathematical topic of distribution theory that connects delta Dirac functions with the Fourier theory. The distribution theory by Schwartz in 1945 is one of the great revolutions in mathematical function analysis. It is considered as a completion of differential calculus, similar to how the revolutionary measure theory or Lebesgue integration theory proposed in 1903, is considered as a completion of integral calculus. Both these theories unlocked new paradigms of mathematical development. Although distribution theory is a powerful tool for understanding Fourier theory, it is ignored in engineering textbooks. In this lecture note, we utilize the concepts of this theory to show how some signals that fail to exhibit FT in the conventional sense can have FT in the distributional sense.

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Section: Examplementioning

confidence: 99%

“…Using the duality of FT, we can write 1 πt ⇌ −j sgn(f ) and using the differentiation property of FT from (24), we can write (−1)…”

Section: Examplementioning

confidence: 99%

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On the Convergence of Fourier Representations and Schwartz Distributions

Singh¹,

Singhal²,

Fatimah³

et al. 2023

Preprint

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“…We also compare the proposed lifting factoring method with that of other two-dimensional two-channel non-separable wavelet transform. The famous two-dimensional two-channel nonseparable filter bank [3,16,25] On comparing the performance analyses shown in Figs. 9b, c and 10b, c, we can draw the conclusion that the proposed factoring is better than the original two-dimensional two-channel nonseparable filter banks presented in (2) or (3) and the famous biorthogonal symmetric non-separable wavelet transform in sparsity.…”

Section: Sparsity Analysis Of the Proposed Lifting Factoringmentioning

confidence: 99%

“…We also compare the proposed lifting factoring method with that of other two‐dimensional two‐channel non‐separable wavelet transform. The famous two‐dimensional two‐channel non‐separable filter bank [3, 16, 25] constructed by J. Kovacevic et al is selected for evaluation. The low‐pass and high‐pass filters are listed in (26).…”

Section: Performance Evaluationmentioning

confidence: 99%

Factoring two‐dimensional two‐channel non‐separable stripe filter banks into lifting steps

Liu

2018

IET image process

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Since the division with remainder cannot be implemented in multivariable polynomials, the two-dimensional nonseparable wavelet transform cannot be lifted by using a similar way as that of univariate wavelet transforms. To solve this problem, a general lifting factoring method of two-dimensional two-channel non-separable stripe filter banks is presented. The constructing form of the polyphase matrices of the stripe filter banks is deduced and the general factoring of the polyphase matrices is given. Compared with the separable lifting wavelet transform, the proposed lifting factoring method can extract better texture information. The lifting form is more succinct than that of the tensor product lifting wavelet transform. The computation amount of the proposed factoring method for image decomposition is a quarter of the two-dimensional two-channel nonseparable stripe filter bank and the original two-dimensional two-channel non-separable wavelet system is quickened. Moreover, the proposed lifting factorising method is faster than the traditional two-dimensional two-channel non-separable wavelet transform based on the Fourier transformation framework in which the size of each filter is greater than N + 1. The proposed lifting factorising method has better sparsity than that of the original wavelet transform and the famous two-dimensional twochannel biorthogonal symmetric non-separable wavelet transform.

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