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The invariant $a^v$, is also found in [9] and is denoted accel. The reasoning leading to Theorem 4 can be repeated for the divergence based tangent gauge (26).

Theorem 5. A complete set of scalar invariants for scalar functions on $\Gamma_3$ at points where the gradient and flow line curvature are non-vanishing are acceleration in the tangent and gradient direction,

$$a^u=\frac{f_{ss}{f}_{vv}}{f_v{f}_{uv}}-\frac{f_{ssv}}{f_{uv}},a^v=-\frac{f_{ss}}{f_v},(31)a^u\; =\; \backslash frac\{f\_\{ss\}f\_\{vv\}\}\{f\_v\; f\_\{uv\}\}\; -\; \backslash frac\{f\_\{ssv\}\}\{f\_\{uv\}\},\; \backslash quad\; a^v\; =\; -\backslash frac\{f\_\{ss\}\}\{f\_v\},\; \backslash qquad\; (31)$$

divergence in the tangent and gradient direction, skew in the gradient direction while moving in the tangent direction,

$${\delta }^u=\frac{f_{su}{f}_{vv}}{f_v{f}_{uv}}-\frac{f_{suv}}{f_{uv}},{\delta }^v=-\frac{f_{su}}{f_v},{\sigma }^u=-\frac{f_{svv}}{f_{uv}},(32)\backslash delta^u\; =\; \backslash frac\{f\_\{su\}f\_\{vv\}\}\{f\_v\; f\_\{uv\}\}\; -\; \backslash frac\{f\_\{suv\}\}\{f\_\{uv\}\},\; \backslash quad\; \backslash delta^v\; =\; -\backslash frac\{f\_\{su\}\}\{f\_v\},\; \backslash quad\; \backslash sigma^u\; =\; -\backslash frac\{f\_\{svv\}\}\{f\_\{uv\}\},\; \backslash qquad\; (32)$$

and isophote and flow line curvature, (see Theorem 1).

6.4 Hessian Gauge

On points where the isophote surface is tangent to the spatial surface, the tangent gauge is not defined. As long as the Hessian is non-degenerate, which generically is the case, we can define an adapted $\Gamma_3$-frame, ${\partial_r, \partial_p, \partial_q}$ that diagonalize the Hessian, i.e. $f_{pq} = f_{rp} = f_{rq} = 0$. Using the fact that the spatio-temporal vector in the adapted frame must be on the form,

$${\mathrm{\partial }}_r={\mathrm{\partial }}_t+\beta {\mathrm{\partial }}_x+\gamma {\mathrm{\partial }}_y\mathrm{.}(33)\backslash partial\_r\; =\; \backslash partial\_t\; +\; \backslash beta\backslash partial\_x\; +\; \backslash gamma\backslash partial\_y.\; \backslash qquad\; (33)$$

Starting by diagonalizing the Hessian in the spatio-temporal direction we get the constraints $f_{rx} = f_{ry} = 0$, and by using (33) and solving for $\beta$ and $\gamma$, we get

$$\beta =\frac{f_{ty}{f}_{xy}-f_{tx}{f}_{yy}}{f_{xx}{f}_{yy}-f_{xy}^2},\gamma =\frac{f_{tx}{f}_{xy}-f_{ty}{f}_{xx}}{f_{xx}{f}_{yy}-f_{xy}^2}\mathrm{.}(34)\backslash beta\; =\; \backslash frac\{f\_\{ty\}f\_\{xy\}\; -\; f\_\{tx\}f\_\{yy\}\}\{f\_\{xx\}f\_\{yy\}\; -\; f\_\{xy\}^2\},\; \backslash quad\; \backslash gamma\; =\; \backslash frac\{f\_\{tx\}f\_\{xy\}\; -\; f\_\{ty\}f\_\{xx\}\}\{f\_\{xx\}f\_\{yy\}\; -\; f\_\{xy\}^2\}.\; \backslash qquad\; (34)$$

This gives the first part of the attitude transformation, a spatio-temporal shear $A$. If we project $\partial_r$ on the spatial plane we get the same vector field as when the optical flow constraint equation is used on the gradient of the image [17]. As the next step the frame must be rotated in the spatial plane s.t. the spatial Hessian is diagonalized. Here we can use the results for the Hessian gauge for $E_2$ reviewed in Section 4.2. Combining these steps we get,

where $\tan 2\phi = f_{xy}/(f_{yy}-f_{xx})$. We proceed using (3) and the same reasoning as for the tangent based frames in the preceding section and arrives to the following theorem.