Gradients

Image Gradient Function

$$\nabla := \begin{bmatrix} \frac{\partial}{\partial x}, \frac{\partial}{\partial y} \end{bmatrix}$$

Direction

The gradient points in the direction of most rapid increase in intensity. For an image $f$, the gradient's direction is given by: $$\theta = \tan ^{-1}(\frac{\partial f}{\partial y} / \frac{\partial f}{\partial x} )$$

Magnitude

The "amount of change" in the gradient is given by its magnitude: $$||\nabla f|| = \sqrt{(\frac{\partial f}{\partial x})^2 + (\frac{\partial f}{\partial y} )^2}$$

Finite Differences

$$\frac{\partial f(x, y)}{\partial x} = f(x + 1, y) - f(x, y)$$

Discrete Gradient

We want an operator that we can apply to a kernel and use as a correlation (or convolution) filter. $$H := \begin{bmatrix} 0 & 0 & 0 \\ -1/2 & 0 & 1/2 \\ 0 & 0 & 0 \end{bmatrix}$$

Sobel Operator

The Sobel Operator is a discrete gradient operator that preserves the "neighborliness" of an image that we discussed earlier when talking about the Gaussian blur filter in Blurring Images. It looks like this: $$S_x := \frac{1}{8} \begin{bmatrix} -1 & 0 & 1 \\ -2 & 0 & 2 \\ -1 & 0 & 1 \end{bmatrix}$$

$$S_y := \frac{1}{8} \begin{bmatrix} 1 & 2 & 1 \\ -0 & 0 & 0 \\ -1 & -2 & -1 \end{bmatrix}$$

$$\nabla I := \begin{bmatrix} g_x, g_y \end{bmatrix} ^T$$

Handling Noise

The images are very noise. To reduce the noise, we apply a smoothing filter.

Derivative Theorem of Convolution

We take advantage of associative of linearity property. $$\frac{\partial}{\partial x}(h * f) = (\frac{\partial}{\partial x}h) * f$$ This saves us one operation so we can calculate derivatives on the kernal

Second derivative of Gaussian