Edges and the Image Gradient

The image is a function of position

The gradient measures local change

Finite differences on a pixel grid

Edge strength, orientation and the normal

A photograph shows a dark block against a bright white ground, and one image column crosses the block's left boundary, where intensity rises steeply from the dark block to the bright ground as you move right. At a pixel sitting on this boundary, which direction does the image gradient point?

Four pixels in an image have gradients $(3, 4)$, $(1, 1)$, $(0, 2)$ and $(-2, 1)$. Using edge strength $e(x, y) = \lVert \nabla \ell \rVert$ with threshold $2.5$, which pixels are marked as edges?

A grayscale image satisfies the finite-difference approximation $\frac{\partial \ell}{\partial x} \approx \ell(x, y) - \ell(x-1, y)$. At one pixel the intensity is $\ell(x, y) = 140$ and its left neighbor has $\ell(x-1, y) = 60$. Compute the estimated horizontal derivative at this pixel and write the number.