04/02 Lecture 3

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This week’s recitation

Python and numpy

image sampling

types of images

• binary

  • white and black
  • usually represented by a binary matrix
    • 0 for black and 1 for white

• grayscale

  • have one color channel (but not binary)
  • also represented by matrix
    • each value vary from 0 - 256
    • 0 for darkest and 256 for brightest

• color

  3 color channel

Image Histogram

Histogram captures the distribution of gray levels in the image

How frequently each gray level occurs in the image

Histogram for patches

So we can easily tell there’s three different objects in the second frame because the three spikes in the histogram

Think images as functions instead of matrices

We can think images as a function that takes the real world as input and pixels as output

images could continue on to infinity, but it’s limited because we can’t

Digital images are usually discrete

sample the 2D space on a regular grid

the input to the image function is a pixel location, $[n,m]$

The output to the image function is the pixel intensity

$f[n, m]$ represents the pixel intensity at that value

An image as a function f from $R^2$ to $R^C$

if gray scale, C = 1, if color, C=3

we are gonna learn the different functions that does this job

Filters

• Forming a new image whose pixels values are transformed from original pixel values
• Goals of filters
  • Goal is to extract useful information from images or transform images into another domain where we can modi
• Filter Example 1: moving average
  • $g[n,m] = {1\over 9}\sum _{k=n-1}^{n+1}\sum _{l=m-1}^{m+1}f[k,l] = {1\over 9}\sum _{k=-1}^{1}\sum _{l=-1}^{1}f[n-k,m-l]$

Systems

A bunch of filters one on another

• we define a system as a unit that converts an input function $f[n,m]$ into an response function $g[n,m]$
  • A function in, a function out
• Properties of systems
  • Amplitude properties
    • Additivity $S[f_i[n,m]+f_j[n,m]] = S[f_i[n,m]]+S[f_j[n,m]]$
    • Homogeneity need more construction
    • Superposition
      • kinda like additivity and homogeneity together
    • Stability
      • if $\forall n,m,|f[n,m]| \le k \rightarrow |S[fn,m]]|\le ck$ for some constant c and k
    • Invertable