Discretized Beltrami energy

In the paper "An Approach Based on Hybrid Genetic Algorithm Applied to Image Denoising Problem", I encounter this fitness function that by minimizing it, a denoised image of a noisy image is obtained.

It is called Discretized Beltrami energy. According to the paper, $$I$$ is the image being evaluated, $$I_0$$ is noisy image, $$\nabla I$$ is total variation (TV) regularizing term, $$\beta$$ and $$\lambda$$ are balancing parameters. $$\Omega$$ is the set of all points of the image.

What is $$\nabla I$$? Determinant of $$\nabla I$$ is a constant, why is there a sum over $$\Omega$$?

In this particular case, $$\nabla I$$ is the gradient because the image $$I$$ is a scalar field. The summation over $$\Omega$$ would make more sense if $$I$$ was a vector field. Since the image pixels are samples over fixed locations in space, the $$\Omega$$ can be dropped.