Rotated Gaussian

📐 Definition

A rotated Gaussian is an anisotropic Gaussian whose principal axes are oriented by a rotation matrix. In $\mathbb{R}^d$ , a common unnormalized form is

g(x) = \exp\!\left(-\tfrac{1}{2}(x-\mu)^{\mathsf T}\Sigma^{-1}(x-\mu)\right)

where $\mu\in\mathbb{R}^d$ is the center and $\Sigma$ is a symmetric positive-definite covariance matrix.

Writing $\Sigma$ via an eigen/rotation decomposition,

\Sigma = R \Lambda R^{\mathsf T}, \qquad \Lambda = \operatorname{diag}(\sigma_1^2,\dots,\sigma_d^2)

the level sets are ellipsoids aligned with the columns of $R$ (the principal axes), with widths set by $\sigma_i$ .

Domain and Codomain

Defined for $x\in\mathbb{R}^d$ (often $d=2$ or $3$ ). Outputs are strictly positive real values. When multiplied by the normalization constant, it becomes a probability density or a normalized smoothing kernel.

⚙️ Key Properties

Oriented Coordinates

In rotated coordinates $y = R^{\mathsf T}(x-\mu)$ ,

g(x) = \exp\!\left(-\tfrac{1}{2}\sum_{i=1}^d \frac{y_i^2}{\sigma_i^2}\right)

Normalization (Kernel / Density Form)

The normalized Gaussian density (or convolution kernel) is

G(x) = \frac{1}{(2\pi)^{d/2}\sqrt{\det\Sigma}}\, \exp\!\left(-\tfrac{1}{2}(x-\mu)^{\mathsf T}\Sigma^{-1}(x-\mu)\right)

Fourier Transform

Up to conventions, the Fourier transform of a Gaussian is another Gaussian; for a centered kernel the covariance transforms inversely (precision becomes covariance in frequency space).

🎯 Special Cases and Limits

Isotropic case: $\Sigma=\sigma^2 I$ yields a circular/spherical Gaussian.
Degenerate widths: as some $\sigma_i\to 0^+$ the mass concentrates near lower-dimensional structures (in the unnormalized form).
Broad widths: as some $\sigma_i\to\infty$ the function flattens along those directions; for normalized kernels, amplitude decreases to preserve total mass.

Gaussian convolution kernels specialize to isotropic $\Sigma$ ; rotated Gaussian windows taper signals along oriented axes; Gabor functions add sinusoidal modulation to the same Gaussian envelope.

Usage in Oakfield

Oakfield does not currently expose a first-class “rotated anisotropic Gaussian” spatial kernel (e.g. a 2D/3D covariance-driven oriented blur) in the built-in operator set.

Closest related features in the current codebase:

Gaussian filtering: sieve provides an isotropic 1D Gaussian-like convolution kernel (set by sigma and taps).
Gabor-style oriented structure (1D): stimulus_gabor combines a Gaussian envelope with a carrier (wavenumber), producing directionality in the sense of a chosen spatial frequency.
Phase “rotation” parameters in some complex stimuli refer to complex phase rotation of the written output, not a spatial rotation of an anisotropic Gaussian covariance.

Historical Foundations

📜 Gaussian Forms

Gaussian quadratic forms are central in probability and error theory. The rotated (anisotropic) case follows naturally from linear changes of variables that diagonalize a positive-definite quadratic form.

🌍 Modern Perspective

Oriented Gaussian kernels are widely used in imaging, signal processing, and PDE regularization as a smooth, analytically tractable way to encode directionality.

📚 References

NIST Digital Library of Mathematical Functions (DLMF), §7
Adler & Taylor, Random Fields and Geometry