Images, filtering, convolution and edge detection

Images as a functions

Images can be represented as a function:

astronaut image
astronaut image surface

We think of an image as a function, \(f\) or \(I\), from \(\mathbb{R}^2\) to \(\mathbb{R}\):

  • \(f(x, y)\) gives the intensity or value at position \((x,y)\).

A color image is just three functions “stacked” together. We can write this as “vector-valued function”:

\begin{equation*} f(x, y) = \begin{bmatrix} r(x, y) \\ g(x, y) \\ b(x, y) \end{bmatrix} \end{equation*}

In computers images are represented as a set of numbers, not continuous functions:

Image representation in computers

In computer vision we typically operate on digital(discrete) images:

  • Sample the 2D space on regular grid
  • Quantize each sample (round to “nearest integer”)


Noise is just another function that is combined with the original function to get a new one:

\begin{equation*} I’(x, y) = I(x, y) + \eta(x, y) \end{equation*}


  • Impulse (salt): random occurrences of white pixels
  • Pepper: random black pixels
  • Salt and pepper: random occurrences of black and white pixels
  • Gaussian noise: variations in intensity drawn from a Gaussian normal distribution
Example of different noises

To apply a noise it’s enough just to add it to the initial image:

noise = np.random.normal(mean, variance ** 0.5, image.shape)
output = image + noise
Example of applied noise

Effect of \(\sigma\) (standard deviation) on Gaussian noise. Just to remind: \(variance = \sigma^2\).

Gaussian noise depends on sigma