Illumination Normalization Paper: Help understand equation
It is indeed a gradient but in particular a relative gradient or normalized gradient as the paper authors put it, they also give its definition:
$$
\nabla_rF(x,y) = \dfrac{\nabla F(x,y)}{|F(x,y)|+c}
$$, where $c$ is a constant to avoid division by zero (since $F(x,y)$ represents pixel intensities it is reasonable for a zero intensity value to occur, encoding a black pixel).
Because an image is represented in a discretized 2D spatial domain the gradient can be computed through finite differences around a neighborhood for each pixel's coordinates. A pixel in a rectilinear grid has at most 8 and at least 3 (for corner pixels) immediate neighbors, thus computing its relative differential involves taking 8 discrete differences, not including the difference with the center pixel itself.
Now the authors claim that this relative gradient is illumination insensitive because the image, in the context of lambertian illumination model, can be analyzed into two components: a reflectance and an illuminance component, with the latter varying slowly in a small neighborhood around a pixel and the first depending heavily on the characteristics of the face's surfaces.
$$
F(x,y) = R(x,y) \times I(x,y)
$$, where $R$ is the reflectance component and $I$ the illuminance component.
Combining the above, this leaves us with:
$$
WLD(F'(x,y)) = \arctan\left(\alpha\cdot\sum_{i \in A}\sum_{j \in A}\dfrac{F'(x,y)-F'(x-i\Delta x, y-j\Delta y)}{F'(x,y)+c}\right) \\
= \arctan\left(\alpha\cdot\sum_{i \in A}\sum_{j \in A}\dfrac{R(x,y)\times I(x,y)-R(x-i\Delta x, y-j\Delta y)\times I(x-i\Delta x, y-j\Delta y)}{R(x,y)\times I(x,y)+c}\right)
$$
Slow illuminance variations mean that we can locally approximate $I(x-i\Delta x, y-j\Delta y) \approx I(x,y) $.
Thus, by factoring out $I(x,y)$, ignoring $c$ and then adding a new constant $c'$:
$$
\arctan\left(\alpha\cdot\sum_{i \in A}\sum_{j \in A}\dfrac{R(x,y)\times I(x,y)-R(x-i\Delta x, y-j\Delta y)\times I(x-i\Delta x, y-j\Delta y)}{R(x,y)\times I(x,y)}\right)\\
\arctan\left(\alpha\cdot\sum_{i \in A}\sum_{j \in A}\dfrac{R(x,y)\times I(x,y)-R(x-i\Delta x, y-j\Delta y)\times I(x, y)}{R(x,y)\times I(x,y)}\right)\\
\arctan\left(\alpha\cdot\sum_{i \in A}\sum_{j \in A}\dfrac{R(x,y)-R(x-i\Delta x, y-j\Delta y)}{R(x,y)+c'}\right)
$$
which approximates the facial pixel intensity variations while being relatively invariant to intensity variations regarding the illuminance component of the image.
The final step of taking the arctangent of the product of the double sum and the scaling parameter $\alpha$, is used as a saturation step to avoid large values of the descriptors and to measure the direction of the variations.