Peak Models Module

Analytical peak functions and polynomial models using PyTorch.

This module defines several 1D peak profiles (e.g., Gaussian, Lorentzian, Pseudo-Voigt) and polynomial bases commonly used in signal processing, spectroscopy, and diffraction analysis. These functions are differentiable and compatible with PyTorch-based optimization and learning workflows.

Author: Antony Vamvakeros

nDTomo.pytorch.peak_models_torch.chebyshev_polynomial(x, n_terms)[source]

Generate a Chebyshev polynomial of the first kind Tₙ(x), evaluated using recurrence relations.

The Chebyshev polynomials of the first kind are defined by:

T₀(x) = 1 T₁(x) = x Tₙ₊₁(x) = 2x·Tₙ(x) - Tₙ₋₁(x)

Parameters:

  • x (torch.Tensor) – Input tensor of x values.

  • n_terms (int) – Number of terms or the target degree of the polynomial.

Returns:

The Chebyshev polynomial of the first kind Tₙ(x) evaluated at x.

Return type:

torch.Tensor

Notes

This implementation returns the dot product of the coefficient vector and the final recurrence value. It does not return the full basis expansion.

nDTomo.pytorch.peak_models_torch.doniach_sunjic_peak(x, amplitude, x0, sigma, b)[source]

Generate a Doniach-Šunjić peak with asymmetric broadening.

Parameters:

  • x (torch.Tensor) – Input axis values.

  • amplitude (float) – Peak amplitude.

  • x0 (float) – Peak center.

  • sigma (float) – Base width parameter.

  • b (float) – Asymmetry parameter.

Returns:

Doniach-Šunjić peak evaluated at x.

Return type:

torch.Tensor

nDTomo.pytorch.peak_models_torch.exponential_power_peak(x, amplitude, x0, sigma, alpha, beta)[source]

Generate a skewed peak based on an exponential power law.

Parameters:

  • x (torch.Tensor) – Input axis values.

  • amplitude (float) – Peak amplitude.

  • x0 (float) – Peak center.

  • sigma (float) – Width parameter.

  • alpha (float) – Power law decay exponent.

  • beta (float) – Skewness factor.

Returns:

Peak evaluated at x using the exponential power law form.

Return type:

torch.Tensor

nDTomo.pytorch.peak_models_torch.gaussian_peak(x, amplitude, mean, std)[source]

Generate a Gaussian peak.

Parameters:

  • x (torch.Tensor) – Input axis values.

  • amplitude (float or torch.Tensor) – Peak amplitude.

  • mean (float or torch.Tensor) – Peak center.

  • std (float or torch.Tensor) – Standard deviation (controls width).

Returns:

Gaussian peak evaluated at x.

Return type:

torch.Tensor

nDTomo.pytorch.peak_models_torch.lorentzian_peak(x, amplitude, x0, gamma)[source]

Generate a Lorentzian peak.

Parameters:

  • x (torch.Tensor) – Input axis values.

  • amplitude (float or torch.Tensor) – Peak amplitude.

  • x0 (float or torch.Tensor) – Center of the peak.

  • gamma (float or torch.Tensor) – Half-width at half-maximum (HWHM).

Returns:

Lorentzian peak evaluated at x.

Return type:

torch.Tensor

nDTomo.pytorch.peak_models_torch.pearson7_peak(x, amplitude, x0, sigma, m, a, b, c)[source]

Generate a Pearson VII-type peak with flexible tailing.

Parameters:

  • x (torch.Tensor) – Input axis values.

  • amplitude (float) – Peak amplitude.

  • x0 (float) – Peak center.

  • sigma (float) – Width scale factor.

  • m (float) – Tail-shaping parameters.

  • a (float) – Tail-shaping parameters.

  • b (float) – Tail-shaping parameters.

  • c (float) – Tail-shaping parameters.

Returns:

Pearson VII profile evaluated at x.

Return type:

torch.Tensor

nDTomo.pytorch.peak_models_torch.polynomial(x, coefficients)[source]

Evaluate a 1D polynomial at the given x values using provided coefficients.

The polynomial is of the form:

y = c₀ + c₁·x + c₂·x² + … + cₙ·xⁿ

Parameters:

  • x (torch.Tensor) – Input tensor of x values at which to evaluate the polynomial.

  • coefficients (list or torch.Tensor) – Sequence of polynomial coefficients [c₀, c₁, …, cₙ].

Returns:

Evaluated polynomial y(x).

Return type:

torch.Tensor

nDTomo.pytorch.peak_models_torch.pseudo_voigt_peak(x, amplitude, x0, sigma, fraction)[source]

Generate a symmetric Pseudo-Voigt peak composed of a linear combination of Gaussian and Lorentzian components with equal width.

Parameters:

  • x (torch.Tensor) – Input axis values.

  • amplitude (float or torch.Tensor) – Peak amplitude.

  • x0 (float or torch.Tensor) – Peak center position.

  • sigma (float or torch.Tensor) – Common width used for both Gaussian and Lorentzian components.

  • fraction (float) – Mixing ratio of the Gaussian component. Must be in [0, 1]. The Lorentzian component weight is (1 - fraction).

Returns:

Pseudo-Voigt peak evaluated at x.

Return type:

torch.Tensor

nDTomo.pytorch.peak_models_torch.pseudo_voigt_peak_t(x, A, xo, st, fraction, sl, i)[source]

Time-efficient version of a Pseudo-Voigt peak with a linear background.

Parameters:

  • x (torch.Tensor) – Input axis values.

  • A (float) – Amplitude of the peak.

  • xo (float) – Center position.

  • st (float) – Width of the Gaussian/Lorentzian components.

  • fraction (float) – Gaussian fraction; (1 - fraction) is Lorentzian fraction.

  • sl (float) – Slope of the linear background.

  • i (float) – Intercept of the linear background.

Returns:

Composite Pseudo-Voigt + linear background evaluated at x.

Return type:

torch.Tensor

nDTomo.pytorch.peak_models_torch.split_pearson_peak(x, amplitude, x0, sigma1, sigma2, m1, m2, a1, a2, b1, b2, c1, c2)[source]

Generate a two-component asymmetric Pearson peak.

Parameters:

  • x (torch.Tensor) – Input axis values.

  • amplitude (float) – Peak amplitude.

  • x0 (float) – Peak center.

  • sigma1 (float) – Widths for left and right peak components.

  • sigma2 (float) – Widths for left and right peak components.

  • m1 (float) – Tail parameters for left and right components.

  • m2 (float) – Tail parameters for left and right components.

  • a1 (float) – Tail parameters for left and right components.

  • a2 (float) – Tail parameters for left and right components.

  • b1 (float) – Tail parameters for left and right components.

  • b2 (float) – Tail parameters for left and right components.

  • c1 (float) – Tail parameters for left and right components.

  • c2 (float) – Tail parameters for left and right components.

Returns:

Asymmetric Pearson profile evaluated at x.

Return type:

torch.Tensor

nDTomo.pytorch.peak_models_torch.voigt_peak(x, amplitude, x0, sigma, gamma)[source]

Approximate a Voigt peak (convolution of Gaussian and Lorentzian).

Parameters:

  • x (torch.Tensor) – Input axis values.

  • amplitude (float) – Peak amplitude.

  • x0 (float) – Peak center.

  • sigma (float) – Gaussian width.

  • gamma (float) – Lorentzian width (controls tails).

Returns:

Voigt-like peak evaluated at x.

Return type:

torch.Tensor