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Affine differential geometry and smoothness maximization as tools for identifying geometric movement primitives

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Abstract

Neuroscientific studies of drawing-like movements usually analyze neural representation of either geometric (e.g., direction, shape) or temporal (e.g., speed) parameters of trajectories rather than trajectory’s representation as a whole. This work is about identifying geometric building blocks of movements by unifying different empirically supported mathematical descriptions that characterize relationship between geometric and temporal aspects of biological motion. Movement primitives supposedly facilitate the efficiency of movements’ representation in the brain and comply with such criteria for biological movements as kinematic smoothness and geometric constraint. The minimum-jerk model formalizes criterion for trajectories’ maximal smoothness of order 3. I derive a class of differential equations obeyed by movement paths whose nth-order maximally smooth trajectories accumulate path measurement with constant rate. Constant rate of accumulating equi-affine arc complies with the 2/3 power-law model. Candidate primitive shapes identified as equations’ solutions for arcs in different geometries in plane and in space are presented. Connection between geometric invariance, motion smoothness, compositionality and performance of the compromised motor control system is proposed within single invariance-smoothness framework. The derived class of differential equations is a novel tool for discovering candidates for geometric movement primitives.

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Notes

  1. The book Shirokov and Shirokov (1959) provides the most comprehensive treatise on affine differential geometry that I am aware of, see also Guggenheimer (1977). All necessary definitions and formulae are provided further in text.

  2. Coordinates of the minimum-jerk trajectories constrained by via-points are composed of fifth-order polynomials with respect to time, the third-order derivatives of x(t), y(t) are continuous (Flash and Hogan 1985). Minimum-jerk trajectories with a single via-point satisfy isochrony principle stating that different movement portions have nearly the same duration independently of their extent (Viviani and Terzuolo 1982; Bennequin et al. 2009). Movement durations from the start to the via-point and from the via-point to the end point are very similar (Polyakov 2006; Shpigelmacher 2006; Polyakov et al. 2009b).

  3. The model predicts straight point-to-point trajectories when boundary conditions (velocity and higher-order derivatives at start and end points) are parallel to the point-to-point straight line including the case when the derivatives are zero.

  4. Different motor control studies, e.g., Hogan (1984), Flash and Hogan (1985), Viviani and Flash (1995), Todorov and Jordan (1998), Polyakov (2001), Richardson and Flash (2002), Ben-Itzkah and Karniel (2008), Polyakov et al. (2009b), used cost functionals \(J_{\sigma }(\mathbf {r}_L,\, n)\) for the planar (\(L = 2\)) and spatial (\(L = 3\)) curves with orders of smoothness n equal to 2–4.

  5. Constancy of \(\dot{\sigma }\) is preserved under similarity transformations for all planar measurements \(\sigma \) mentioned in Table 2 besides equi-center-affine and center-affine arcs whose speed of accumulation looses constancy under translations.

  6. The last version of Polyakov (2014) contains exposition of this work in a single text.

  7. Say, constancy of \(\dot{\sigma }_\mathrm{eu}\) is required for \(n=3\). Then the following recursive computation in Mathematica implies a specific logarithmic spiral (31) with \(\beta = \pm 1 / \sqrt{5}\):

    $$\begin{aligned}&\pmb {\text {SetAttributes}[\beta ,\text {Constant}]; x[\phi \_]= \mathrm{e}^{\beta \phi } \text {Cos}[\phi ];}\\&\pmb {y[\phi \_]= \mathrm{e}^{\beta \phi } \text {Sin}[\phi ];\mathrm{d}x\mathrm{d}\phi [\phi \_]=\text {Dt}[x[\phi ],\phi ];}\\&\pmb {\mathrm{d}y\mathrm{d}\phi [\phi \_]=\text {Dt}[y[\phi ],\phi ];\text {d}\sigma \mathrm{d}\phi [\phi \_]=\sqrt{\mathrm{d}x\mathrm{d}\phi [\phi ]{}^{\wedge }2 + \mathrm{d}y\mathrm{d}\phi [\phi ]{}^{\wedge }2};}\\&\pmb { \mathrm{d}x\mathrm{d}\sigma [1, \phi \_]{:=} \mathrm{d}x\mathrm{d}\phi [\phi ] / \text {d}\sigma \mathrm{d}\phi [\phi ]; \mathrm{d}y\mathrm{d}\sigma [1, \phi \_]\text {:=} \mathrm{d}y\mathrm{d}\phi [\phi ] / \text {d}\sigma \mathrm{d}\phi [\phi ];}\\&\pmb { \mathrm{d}x\mathrm{d}\sigma [\text {n}\_, \phi \_]{:=}\text {Dt}[ \mathrm{d}x\mathrm{d}\sigma [n-1, \phi ], \phi ] / \text {d}\sigma \mathrm{d}\phi [\phi ]; }\\&\pmb { \mathrm{d}y\mathrm{d}\sigma [\text {n}\_, \phi \_]{:=}\text {Dt}[ \mathrm{d}y\mathrm{d}\sigma [n-1, \phi ], \phi ] / \text {d}\sigma \mathrm{d}\phi [\phi ];}\\&\pmb {\text {FullSimplify}[ \mathrm{d}x\mathrm{d}\sigma [1, \phi ]\text { }\mathrm{d}x\mathrm{d}\sigma [6, \phi ] +\mathrm{d}y\mathrm{d}\sigma [1, \phi ]\mathrm{d}y\mathrm{d}\sigma [6, \phi ] ]}\\&\frac{10 \beta \left( -1+5 \beta ^2\right) }{\left( \mathrm{e}^{2 \beta \phi } \left( 1+\beta ^2\right) \right) ^{5/2}} \end{aligned}$$

  8. Define a curve in L-dimensional space as the system \(F_k(x_1,\, \ldots ,\, x_L) = 0,\; k = 1,\, \ldots ,\, L-1\) representing intersection of \(L-1\) hypersurfaces of dimension \(L-1\). Correspondingly, the system of Euler–Poisson (E–P) equations (Gelfand and Fomin 1961) (called also Euler-Lagrange) with \(L-1\) Lagrange multipliers: (E–P) \(\left[ \sum _{d=1}^{L}{\dddot{x}_d}^2 + \sum _{k=1}^{L-1}\lambda _k(t) F_k \right] = 0\) will lead to the system \((-1)^3 d{x_d}^6/\mathrm{d}t^6 + \sum _{k=1}^{L-1}\lambda _k(t) \partial F_k / \partial x_d = - d{x_d}^6/\mathrm{d}t^6 + \sum _{k=1}^{L-1}\lambda _k(t) \partial F_k / \partial x_d = 0,\, d=1,\ldots ,L\), implying that at every point r(t) of the curve the vector \(\mathrm{d}\varvec{r}^6/\mathrm{d}t^6\) belongs to the hyperplane spanned by \(L-1\) gradients (normals) to corresponding hypersurfaces \(F_k\) and thus vector \(\mathrm{d}\varvec{r}^6/\mathrm{d}t^6\) is orthogonal to the vector \(\dot{\varvec{r}}\) parallel to curve’s tangent; note that the path is parameterized here by \(\sigma (t) = t\). Identical derivation for nth-order smoothness for a trajectory in L-dimensional space leads to Eq. (16).

  9. Equi-affine and similarity arcs of straight segments are zero, center-affine and affine arcs are not defined, and equi-center-affine arc is zero when straight line crosses the origin.

  10. Formula from Shirokov and Shirokov (1959) corresponding to (35) contains misprint corrected here.

  11. Meaning \(x_0 = y_0 = 0\). Equivalently, tangents to a logarithmic spiral centered at the origin have constant angle with the vectors connecting corresponding points on the curve to the origin.

  12. Proven in Appendix D of Polyakov (2006).

  13. Already without requirement for equality of their equi-affine arcs as in case of equi-affine transformation.

  14. Sequences of parabolic-like components revealed in monkey drawing movements (Polyakov et al. 2009a, b) were usually implemented with unchanged direction of motion—either counterclockwise or clockwise.

  15. Planar similarity transformations are uniquely decomposable into Euclidian and scaling transformations in the same manner as affine in (44).

  16. \(\text{ Left-hand } \text{ side } = f(\beta ) \cdot exp(g(\beta ) \cdot \varphi ) = \text{ const }, g(\beta )\ne 0 \, \mathrm{when} \, \beta \ne 0\).

  17. From Shirokov and Shirokov (1959), formula for the spatial equi-affine curvature: \(\chi (\sigma _{\mathrm{ea}3}) = \left| \begin{array}{ccc} x' &{} x''' &{} x^{(4)} \\ y' &{} y''' &{} y^{(4)} \\ z' &{} z''' &{} z^{(4)} \\ \end{array}\right| \); for the equi-affine torsion: \(\tau (\sigma _{\mathrm{ea}3}) = - \left| \begin{array}{ccc} x'' &{} x''' &{} x^{(4)} \\ y'' &{} y''' &{} y^{(4)} \\ z'' &{} z''' &{} z^{(4)} \\ \end{array} \right| \). The differentiation is implemented with respect to \(\sigma _{\mathrm{ea}3}\).

  18. The 1-st order jet of a function f is characterized by three slots: the coordinate x, the value of f at x, \(y = f(x)\) and the value of its derivative \(p = f'(x)\). The latter is the slope of the tangent to the graph of f at the point \(a = (x, f(x) )\) of \(\mathbb {R}\).

  19. This is similar to vectorial concatenation of point-to-point movement elements into a single trajectory (without rest point in the middle) in the double-step paradigm (Flash and Henis 1991).

  20. Such decomposition is not just an exemplar case, but it is a universal property for parabolic segments because it is preserved under arbitrary affine transformations and affine transformations can be used to map the piece of parabola approximating the path into an arbitrary parabolic segment.

  21. Text S1 of Polyakov et al. (2009a) describes regularization procedure used in equi-affine analysis of monkey scribbling movements.

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  1. Department of Mathematics, Ariel University, Ariel, Israel

    Felix Polyakov

  2. Department of Mathematics, Bar Ilan University, Ramat Gan, Israel

    Felix Polyakov

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Polyakov, F. Affine differential geometry and smoothness maximization as tools for identifying geometric movement primitives. Biol Cybern 111, 5–24 (2017). https://doi.org/10.1007/s00422-016-0705-7

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  • DOI: https://doi.org/10.1007/s00422-016-0705-7

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