Velocity-Adapted Affine Gaussian Derivatives

• The paper introduces velocity-adapted affine Gaussian derivatives, combining affine Gaussian kernels with explicit velocity shifts to yield closed-form direction and speed tuning curves.
• It defines scale-normalized spatio-temporal derivative operators that ensure invariance under affine and Galilean transformations for robust feature detection.
• The approach benefits both computational neuroscience and computer vision by providing filters that adapt to changes in scale, orientation, and motion with practical applications in feature matching and dynamic scene analysis.

Velocity-adapted affine Gaussian derivatives are a mathematically precise class of spatio-temporal receptive fields designed to model the direction and speed selectivity of simple and complex cells in the visual cortex, and to provide a theoretically grounded family of Galilean-covariant filters for computer vision. Defined as scale- and orientation-normalized derivatives of affine Gaussian kernels that move at specific velocities, these fields enable rigorous modeling and prediction of both the physiological and computational responses to drifting textures and motions. Their analytic tractability provides closed-form expressions for direction and speed tuning, and their parameterization supports matching across changes in viewing direction, motion, and observer.

1. Affine Gaussian Kernels and Their Spatio-Temporal Extension

The foundational spatial kernel is the affine Gaussian

$$G(x;\Sigma) = \frac{1}{2\pi\sqrt{\det\Sigma}} \exp\left(-\frac{1}{2} x^T \Sigma^{-1} x\right)$$

where $x \in \mathbb{R}^2$, and $\Sigma \succ 0$ is a positive-definite $2 \times 2$ spatial covariance matrix encoding the elliptical anisotropy and orientation of the filter. When $\rho=0$, the matrix reduces to an axis-aligned ellipse of widths $\sigma_1, \sigma_2$.

The velocity-adapted spatio-temporal kernel extends this by introducing an explicit image velocity $v \in \mathbb{R}^2$ and a temporal smoothing kernel $h(t; \sigma_t)$. The spatial kernel is dynamically shifted according to $x - v t$, representing a Galilean motion: $$T(x,t;\Sigma,\sigma_t, v) = G(x-vt;\Sigma) \cdot h(t; \sigma_t)$$ with $h(t; \sigma_t)$ typically chosen as a 1D (possibly causal) Gaussian in time or, for strictly causal systems, as the limit of cascaded first-order filters.

2. Velocity-Adapted Derivative Operators

The framework defines derivatives adapted both to spatial structure and velocity:

• The $m$th-order spatial directional derivative along angle $\varphi$:

$$\partial_\varphi = \cos\varphi\,\partial_{x_1} + \sin\varphi\,\partial_{x_2}$$

• The velocity-adapted temporal derivative (“material derivative”):

$$\partial_{\bar t} = \partial_t + v_1\partial_{x_1} + v_2\partial_{x_2}$$

These are combined to generate fully scale-normalized, velocity-adapted affine Gaussian derivative fields: $$T_{m,n}(x,t) = \sigma_\varphi^m \sigma_t^n \left(\partial_\varphi^m \partial_{\bar t}^n\right) \left[G(x-vt;\Sigma) h(t;\sigma_t)\right]$$ where the scale-normalization guarantees invariance and comparability across filter size and orientation.

3. Analytical Direction and Speed Selectivity

A key theoretical advantage is the existence of closed-form response and tuning curves. When probed by sine gratings of form

$$f(x, t) = \sin\left(\omega (x_1\cos\theta + x_2\sin\theta) - u t + \beta \right)$$

the response amplitude at optimal frequency is: $$A_m(\theta,u) = \alpha_m \left(\frac{\cos\theta}{\sqrt{\kappa^2 \sin^2\theta - 2(u/v)\cos\theta + 2\cos^2\theta + (u/v)^2}}\right)^m$$ where $\kappa = \sigma_2 / \sigma_1$ encodes spatial elongation, $m$ is the spatial derivative order, and $v$ is the preferred velocity. For $\theta=0$ (preferred direction) and $u=rv$, the speed tuning curve simplifies to

$R_m(r) = (r^2 - 2r + 2)^{-m/2}$

independent of $\kappa$.

The orientation-selectivity curve at optimal velocity ($u = v\cos\theta$) matches the purely spatial result: $$\begin{aligned} A_1(\theta;\kappa) &= \frac{|\cos\theta|}{\sqrt{\cos^2\theta + \kappa^2\sin^2\theta}} \ A_2(\theta;\kappa) &= \frac{\cos^2\theta}{\cos^2\theta + \kappa^2\sin^2\theta} \end{aligned}$$ and the bandwidth shrinks strictly with increasing $\kappa$ and $m$.

4. Covariance under Affine and Galilean Transformations

These fields are constructed to be covariant under affine spatial transformations and Galilean motion:

• If spatial coordinates are transformed as $x' = A x + b + u_0 t$, $t' = t$, the parameters must change as

$$\Sigma' = A \Sigma A^T \,,\quad v' = A v + u_0\,,\quad \sigma_t' = \sigma_t$$

for the output to remain structurally equivalent. This ensures normalized derivatives transform into themselves up to permutation, providing a mechanism for robust feature detection under changes in viewpoint and observer motion.

A canonical summary table:

Transformation	Spatial Covariance	Velocity	Temporal Variance
Affine	$\Sigma' = A \Sigma A^T$	$v' = A v$	$\tau' = \tau$
Galilean (motion)	no change	$v' = v + u_0$	no change

This guarantees matching across locations, scales, orientations, and image velocities in both biological and artificial contexts (Lindeberg, 2023).

5. Neurophysiological Relevance

Velocity-adapted affine Gaussian derivatives provide a parsimonious model for several classes of V1 neurons:

• Strict velocity tuning, with direction and speed selectivity arising analytically as a function of parameter regime (direction tuning narrows as $\kappa$ or $m$ increase).
• Classes of neurons observed empirically (velocity-low–pass, broad–band, and velocity-tuned) correspond to different composite or mixtures of these fields.
• Observed peak-speed shifts with eccentricity can be modeled by a family of fields with different velocities, consistent with V1 data and the existence of direction-selective columns for multiple velocities (Lindeberg, 11 Nov 2025).
• The Galilean covariance property supports the hypothesis that the cortex implements a bank of such filters, each tuned to a distinct velocity and orientation.

6. Applications in Computer Vision and Pattern Analysis

Velocity-adapted affine Gaussian derivatives furnish the theoretical foundation for affine- and motion-covariant feature detectors:

• Interest point detection (e.g., affine-Harris, affine-Hessian, velocity-adapted saliency points) invariant under viewpoint and motion changes.
• Spatio-temporal descriptors for dynamic scene analysis, tracking, and action recognition where explicit velocity and affine normalization is required.
• Matching of local surface patches and dynamic events across views is enabled by parameter adjustment as dictated by the affine+Galilean covariance laws (Lindeberg, 2023).
• These filters are also suitable for simulation and fitting to neural data in computational neuroscience.

A plausible implication is that any vision application requiring invariance (or covariantly matched response) to motion and local affine changes can exploit these kernels as optimal filters.

7. Key Parameters and Tuning Regimes

The behavior of velocity-adapted affine Gaussian derivatives is governed by the set

$$\left\{m,\,n,\,\sigma_\varphi,\,\sigma_t,\,\varphi,\,v,\,\kappa\right\}$$

where:

• $m,n$: spatial and temporal derivative orders
• $\sigma_\varphi, \sigma_t$: spatial and temporal scales
• $\varphi$: preferred orientation
• $v$: preferred velocity
• $\kappa = \sigma_2/\sigma_1$: spatial elongation (anisotropy)

Key regimes and their consequences:

• Direction tuning sharpens with increasing $\kappa$ and $m$.
• Speed tuning (at $\theta = 0$) is controlled exclusively by $m$.
• Affine-covariant normalization factors ensure comparability irrespective of raw kernel size or shape.
• The joint spatio-temporal covariance matrix

$$\Sigma_{3D} = \begin{pmatrix} \Sigma + \sigma_t^2 v v^T & \sigma_t^2 v \ \sigma_t^2 v^T & \sigma_t^2 \end{pmatrix}$$

represents the true receptive field in joint space-time, with determinant $\sigma_t^2\det\Sigma$.

These theoretical frameworks allow direct prediction and control of filter selectivity for both physiological modeling and engineered vision systems. All claims and formal results are supported by the referenced literature (Lindeberg, 11 Nov 2025, Lindeberg, 2023, Lindeberg, 2023).