What is a 3D winding number? This paper mentions the term "3D winding number". Its abstract says:
"We develop a new formulation, mathematically elegant, to detect critical points of 3D scalar images. It is based on a topological number, which is the generalization to three dimensions of the 2D winding number."
I am already familiar with  winding numbers from complex analysis ($\int_{\gamma}\frac{1}{w-z}dw$) and algebraic topology (covering space of $\mathbb{S}^1$). I never heard of 3D winding numbers before. I searched google for 3D winding numbers, but it seems that this paper is the only place where the term occurs.
Could anybody provide me with a "mathematical" reference about 3D winding numbers starting from their definition ?
Thank you
 A: In Section 2 of the paper itself (particularly Subsection 2.2), the authors define what they mean by "3D winding number". :)
At a quick skim, the authors use $\nabla L$, the gradient of the intensity function of a 3-dimensional image, to map voxels to $\mathbf{R}^3$, then use this mapping pull back the closed, non-exact $2$-form 
$$
\omega = \frac{x\, dy\wedge dz + y\, dz \wedge dx + z\, dx \wedge dy}{(x^2 + y^2 + z^2)^{3/2}}
$$
and "integrate" it over a small cube in the space of voxels.
By a standard vector calculus computation/argument, the integral of $\omega$ over a closed, embedded surface $S$ in $\mathbf{R}^3\setminus\{(0,0,0)\}$ is either $0$ (if the origin is outside $S$) or $4\pi$ (if the origin is inside $S$).
Note that (i) $\nabla L$ hits the origin precisely at a critical point of the intensity function $L$, and (ii) the integral of the pullback $(\nabla L)^*\omega$ over a small cube about a critical point is $4\pi$ times the topological degree of $\nabla L$. Presumably that's the origin of the term "3D winding number". (By equation (18) of the paper, the degree of $\nabla L$ at a critical point is the sign of the Hessian of $L$.)
The integral of $(\nabla L)^*\omega$ over an arbitrary small cube in the space of voxels is straightforward to calculate numerically (see the end of Section 2.3), and the result detects critical points of the intensity and distinguishes whether the gradient of the intensity preserves or reverses orientation. (Particularly, the degree distinguishes maxima and minima of $L$.)
