Fractional Wavelet Transform

• Fractional wavelet transform is a generalized family of wavelet analyses that incorporate a tunable fractional parameter through chirp modulation and phase shifts.
• It employs diverse constructions such as FrFT-based kernels, graph-spectral interpolation, and fractional Hilbert transforms to adapt signal representation.
• The transform framework achieves robust inversion, reconstruction, and multiresolution analysis, enabling improved analysis for signals and images.

Fractional wavelet transform denotes a family of wavelet constructions in which a fractional parameter is introduced into the analysis operator, most commonly through the fractional Fourier transform, chirp-modulated dilation–translation kernels, fractional spectral powers, or fractional phase-shift operators. In this broader sense, the topic covers continuous and discrete transforms on $\mathbb R$ and $\mathbb R^n$, distributional extensions, graph-spectral analogues, fractional biorthogonal wavelet bases, and scattering architectures built on fractional wavelet layers. Early work on distance function wavelets explicitly connected fractional DFW transforms with fractal geometry and fractional derivative, while also stating that, in most cases, solid mathematical analysis was missing and the results were in conjecture status [0205063].

1. Conceptual scope and historical emergence

Across the literature, fractional wavelet transform is not a single canonical operator but a class of related generalizations of wavelet analysis. Some constructions modify the wavelet kernel by a fractional Fourier phase factor; some replace ordinary Fourier admissibility by a fractional Fourier admissibility condition; some transfer the idea to graph spectra; and some interpret “fractional” as continuous phase shifting inside a wavelet family. The special parameter value at which the classical transform is recovered is therefore convention-dependent: one-dimensional CFrWT on Hardy and Morrey spaces reduces to the classical continuous wavelet transform at $\theta=1$, several FrFT-based Euclidean formulations reduce at $\alpha=\pi/2$, and SGFRWT reduces to SGWT at $\alpha=1$ (Verma et al., 2021).

Construction	Defining fractional mechanism	Representative source
GNFrWT	Chirp-modulated wavelet kernel with factor $e^{-\frac{i}{2}(t^2-b^2)\cot\theta}$	(Pathak et al., 2014)
CFrWT on $\mathbb R^n$	Fractional Fourier phase built into translated and dilated wavelets	(Verma et al., 2019)
Distributional FRWT	FRWT and synthesis operator on Schwartz and Lizorkin settings	(Maksimović, 22 Sep 2025)
Fractional Hankel wavelet transform	Fractional Hankel translation/dilation and FrHT kernel	(Mahato, 2018)
SGFRWT	Fractional graph Fourier basis $Y=X^\alpha$ and fractional Laplacian $L_\alpha$	(Wu et al., 2019)
Fractional Hilbert-wavelet analysis	One-parameter family of fractional Hilbert shifts acting on wavelet atoms	(0908.3855)

This diversity has two immediate consequences. First, comparison across papers requires attention to parameterization, admissibility constants, and synthesis formulas rather than reliance on the name alone. Second, “fractional” does not always mean the same underlying geometry: in some papers it means a rotation in a time–frequency or space–fractional-frequency plane, in others a graph-spectral interpolation, and in others a continuous phase-shift action on analytic or approximately analytic wavelets.

2. Core Euclidean operator models

A standard FrFT-based template inserts a chirp factor into the wavelet kernel. In the general novel fractional wavelet transform (GNFrWT), the transform of $h(t)$ with respect to a wavelet $\mathbb R^n$0 is

$\mathbb R^n$1

with

$\mathbb R^n$2

This formulation also admits a Parseval-type identity with weight $\mathbb R^n$3 and an inversion formula involving the same synthesis kernel (Pathak et al., 2014).

A closely related FRWT is used in fractional wavelet scattering. There the fractional order $\mathbb R^n$4 is tied to a rotation angle $\mathbb R^n$5, and the kernel is written as

$\mathbb R^n$6

The transform is correspondingly expressed as a chirp-modulated integral, so that the ordinary wavelet transform is recovered when $\mathbb R^n$7. In this interpretation, the fractional wavelet behaves like a band-pass filter in the fractional domain rather than in the ordinary Fourier domain (Liu et al., 2018).

For $\mathbb R^n$8, the continuous fractional wavelet transform is built from the wavelet family

$\mathbb R^n$9

and

$\theta=1$0

The same paper rewrites this as a fractional convolution-type expression and shows that the classical continuous wavelet transform is recovered at $\theta=1$1 (Verma et al., 2019).

Another one-dimensional CFrWT defines fractional wavelets through the fractional Fourier transform $\theta=1$2, with admissibility constant

$\theta=1$3

Its wavelet family uses fractional scaling: the dilation law is $\theta=1$4 rather than $\theta=1$5, the normalization is $\theta=1$6, and all formulas reduce to the classical continuous wavelet transform when $\theta=1$7 (Verma et al., 2021).

3. Admissibility, inversion, convolution, and functional analysis

Once the transform is defined, the classical wavelet infrastructure reappears in fractionalized form. In the GNFrWT, the central auxiliary object is the basic function $\theta=1$8, defined implicitly by a transform identity and reconstructed through the inversion formula. From $\theta=1$9, the paper defines a transform-induced translation operator and a convolution

$\alpha=\pi/2$0

and proves the convolution theorem

$\alpha=\pi/2$1

under stated integrability assumptions (Pathak et al., 2014).

For continuous FrFT-based transforms, the principal structural results are orthogonality, reconstruction, and reproducing-kernel characterizations. In the one-dimensional CFrWT on Hardy and Morrey spaces, the paper establishes an orthogonality relation for two fractional wavelets, a two-wavelet reconstruction formula, a reproducing-kernel criterion for membership in the transform range, convolution and correlation formulas, and a Plancherel-type identity. The normalized operator $\alpha=\pi/2$2 is bounded on $\alpha=\pi/2$3 and on $\alpha=\pi/2$4, with explicit $\alpha=\pi/2$5 scale dependence (Verma et al., 2021).

The $\alpha=\pi/2$6-dimensional CFrWT similarly satisfies an admissibility criterion, an inner-product relation, a reconstruction formula, and a reproducing-kernel Hilbert-space description of its range. The same framework supports Heisenberg-type and local uncertainty inequalities, and for fixed dilation vector $\alpha=\pi/2$7 the operator $\alpha=\pi/2$8 is bounded on the Morrey space $\alpha=\pi/2$9 (Verma et al., 2019). The multidimensional fractional wavelet transform (MFrWT) in $\alpha=1$0 extends this picture further: its range is again an RKHS, and Heisenberg, logarithmic, and local uncertainty principles are obtained by transferring corresponding inequalities from the MFrFT (Kaur et al., 2022).

The functional-analytic setting has also been pushed beyond $\alpha=1$1. For the distributional FRWT, continuity is proved on Schwartz spaces and, by duality, on $\alpha=1$2; the synthesis operator is likewise continuous, and the inversion formula extends to $\alpha=1$3. In the Lizorkin setting $\alpha=1$4, Abelian and Tauberian results relate quasiasymptotic behavior of a distribution to asymptotic behavior of its FRWT (Maksimović, 22 Sep 2025). An analogous continuity program exists for the fractional Hankel wavelet transform, where the transform is shown to be continuous on Gel'fand–Shilov spaces of type $\alpha=1$5 and on ultradifferentiable function spaces (Mahato, 2018).

4. Directional, phase-based, and graph-spectral generalizations

Not every fractional wavelet construction is driven by FrFT chirps. In the dual-tree complex wavelet transform, a distinct approach uses the group of fractional Hilbert transforms

$\alpha=1$6

to reinterpret complex wavelet coefficients as amplitude–phase variables. If

$\alpha=1$7

then the synthesis representation becomes

$\alpha=1$8

For Gabor-like wavelets, the fractional Hilbert transform shifts the oscillatory carrier while leaving the window unchanged, and in two dimensions this extends through directional Hilbert transforms $\alpha=1$9 to direction-selective phase-shifted wavelet superpositions (0908.3855).

On weighted graphs, the fractionalization is spectral rather than geometric. SGFRWT first defines a graph fractional Fourier basis $e^{-\frac{i}{2}(t^2-b^2)\cot\theta}$0 from the unitary Laplacian eigenvector matrix $e^{-\frac{i}{2}(t^2-b^2)\cot\theta}$1, then introduces a fractional graph Laplacian

$e^{-\frac{i}{2}(t^2-b^2)\cot\theta}$2

and finally defines the transform as $e^{-\frac{i}{2}(t^2-b^2)\cot\theta}$3. In spectral form,

$e^{-\frac{i}{2}(t^2-b^2)\cot\theta}$4

At $e^{-\frac{i}{2}(t^2-b^2)\cot\theta}$5, this reduces to the standard SGWT. A fast implementation is derived from truncated Fourier-series approximation, with reported complexity

$e^{-\frac{i}{2}(t^2-b^2)\cot\theta}$6

and the paper emphasizes that the method naturally handles complex-valued fractional operators (Wu et al., 2019).

These variants show that “fractional” can enter wavelet analysis through at least three inequivalent mechanisms: chirp modulation in Euclidean kernels, continuous phase shifting in analytic or approximately analytic wavelet families, and fractional powers of a graph Fourier basis. The shared theme is not a single operator identity but the addition of a tunable parameter that rotates, shifts, or otherwise interpolates the analysis domain.

5. Fractional multiresolution and biorthogonal basis theory

A transform theory becomes substantially more robust once discrete analysis, stable synthesis, and perfect reconstruction are available. This is the role of fractional biorthogonal wavelet theory in $e^{-\frac{i}{2}(t^2-b^2)\cot\theta}$7. There, the necessary and sufficient condition for the translates of a single function to form a fractional Riesz basis for their closed span is the two-sided bound

$e^{-\frac{i}{2}(t^2-b^2)\cot\theta}$8

This is the fractional analogue of the classical periodized Fourier Gramian condition (Ahmad et al., 2020).

The same paper defines fractional multiresolution analyses $e^{-\frac{i}{2}(t^2-b^2)\cot\theta}$9, fractional scaling functions, and dual fractional MRAs. The associated refinement structure is expressed through fractional scaling filters, while biorthogonality of translates is characterized in the FrFT domain by periodized products of primal and dual generators. Perfect reconstruction is enforced by a polyphase-type matrix condition,

$\mathbb R^n$0

which yields decomposition identities paralleling classical filter-bank wavelet theory (Ahmad et al., 2020).

Under mild decay assumptions on the FrFTs of the scaling functions and the corresponding fractional wavelets,

$\mathbb R^n$1

and similarly for the dual system, the fractional wavelet families generate Riesz bases for $\mathbb R^n$2. This basis-theoretic layer provides the Hilbert-space architecture behind discrete fractional wavelet decompositions: stable coefficient maps, dual synthesis, and exact reconstruction are no longer heuristic analogies with classical wavelet theory but explicitly proved structural properties (Ahmad et al., 2020).

6. Applications, limitations, and adjacent interpretations

The principal empirical motivation for fractional wavelet methods is that the additional order parameter can align the analysis with data whose energy is not optimally concentrated in the ordinary wavelet domain. FrScatNet makes this explicit by replacing each wavelet layer in a scattering network with an FRWT layer and computing fractional scattering coefficients through iterative FRWT–modulus cascades. On pathological image patches from the Warwick-QU colon histology dataset and on a texture surfaces dataset, the paper examines the fractional orders

$\mathbb R^n$3

For the benign database and malignant database, the best errors are often achieved at fractional orders different from $\mathbb R^n$4, especially around $\mathbb R^n$5 or $\mathbb R^n$6, whereas for the texture dataset the best performance is reported at $\mathbb R^n$7. In gland segmentation, the paper uses $\mathbb R^n$8, reports performance better than ScatNet, and states that the method achieves comparable results to methods from the 2015 MICCAI Gland Segmentation Challenge (Liu et al., 2018).

SGFRWT supplies a graph-signal counterpart. On a Swiss-roll graph with 500 points, on a $\mathbb R^n$9 Cameraman image converted to a pixel graph, and in MNIST data augmentation for CNN classification, the fractional order changes support and localization of the graph wavelets. The reported MNIST results are: SGWT/SGWT $Y=X^\alpha$0, SGFRWT/SGFRWT $Y=X^\alpha$1, SGWT training versus SGFRWT testing $Y=X^\alpha$2, and classical augmentation training with SGFRWT testing $Y=X^\alpha$3. These numbers show both the usefulness of SGFRWT as a data-augmentation tool and the fact that the fractional parameter can genuinely change the representation class rather than merely perturb the classical SGWT (Wu et al., 2019).

The literature also clarifies several boundaries of the subject. Early distance-function-wavelet work explicitly warned that many of its fractional DFW results were intuitive and conjectural rather than fully analyzed [0205063]. More recent application papers do not support a universal superiority claim for fractional orders different from the classical setting, since the texture surfaces dataset in FrScatNet was best at $Y=X^\alpha$4 (Liu et al., 2018). There are also neighboring areas that are related but distinct. Wavelet analysis of multivariate fractional Brownian motion uses vanishing moments to suppress long-range interdependence, but it does not introduce a new named fractional wavelet transform; rather, the wavelet acts as a generalized differencing operator for a fractional stochastic process (Coeurjolly et al., 2010). Likewise, wavelet Galerkin methods for fractional elliptic differential equations use wavelet Riesz bases adapted to fractional Sobolev spaces, yet that literature is explicitly not about a signal-processing fractional wavelet transform in the classical Fourier/fractional-Fourier sense (Deng et al., 2014).

Taken together, these developments present the fractional wavelet transform as a broad research program rather than a single settled formalism. Its unifying idea is the insertion of a fractional degree of freedom into wavelet analysis so that localization in scale and position can be coupled to fractional phase, fractional spectrum, or fractional-domain geometry. Its divergences—between FrFT kernels, Hilbert-phase models, graph spectra, and basis-theoretic implementations—are therefore not peripheral inconsistencies but constitutive features of the subject.