What is an image

1
What is an image?

• What does it measure?
• How do we represent it?
• How do we get information from it?

2
What is an image? Continuous case.

• We assume a 2-dimensional (i.e., Euclidean)
  surface such that the co-ordinates may be
  designated by two variables (x,y)
• To each point on this plane we assign a
  monochrome brightness value,
• and denote this function by i(x,y).
• We make the following initial
• assumptions about i(x,y)
• i(x,y) is a real-valued function of two real
  variables.
• i(x,y) is bounded as no sensor has unlimited
  capacity for measuring light. Therefore 0 lt
  i(x,y)lt M.
• i(x,y) gt 0 over some bounded region.
• i(x,y) is band-limited. More on this later.
• i(x,y) is called the brightness or intensity.

3
Some additional considerations

• The image may be a function of time, i(x,y,t)
  where t is the (real) temporal value. For the
  present we will deal with static pictures.
• The image may also be a function
  of the wavelength, where is a real
  spectral variable.
• Typically, wavelength is quantised into localised
  spectra such as Red, Green, and Blue.

4
Discretizing the image -- Tessellation and
Quantization

• There are only three ways to tessellate the plane
  with a regular polygons
• Squares The most common method.
• Regular hexagons a useful method due to less
  rotational dependence.
• Consider the ratio of the distance between
  the centre and the corner and the distance
  between the centre and the edge. This ratio is
  lower for the hexagon than the square.
• Equilateral triangles not commonly used.

5
Sampling and quantization

• We shall consider the value at a pixel to be the
  image function sampled at exactly a grid point.
  In reality it is the average brightness over a
  small area.
• This value is equivalent to the brightness at a
  point of an appropriately defocused image.
• Defocusing (blurring) sums brightness over an
  area.
• Assuming point sampling simplifies the analysis.

6
Sampling and quantization

• Three-dimensional space can also be tessellated,
  in voxels, where the values in individual voxels
  may refer to densities.
• For now we shall limit ourselves to equal
  interval quantization of the brightness values at
  each point. For example, 0 through 255.
• We shall denote the tessellated image function
  returning a quantized value by I(X,Y).

7
Hardware and Video Displays

• Today, we have
• Approximately 1920 by 1080 (HDTV) by 24 bits for
  eight bits of each of R, G, and B brightness
  levels.
• North American (NTSC) video is roughly equivalent
  to 512 x 512 x 8 bits of gray. Colour bandwidth
  in NTSC is limited.
• All this capability gives us a vast amount of
  picture data. Humans can, however, take a cue
  from much less data, for example the
  approximately 16 by 16 monochrome image of
  Abraham Lincoln.
• http//www.michaelbach.de/ot/fcs_mosaic/index.html
  
• http//www.cs.ubc.ca/little/right134.jpg
• http//www.cs.ubc.ca/little/right134i.jpg
• http//www.cs.ubc.ca/little/fly-ran-1-h0632.jpg

8
Image Processing

• We always begin by assuming an underlying
  continuous image.
• i(x,y) -gt sampling -gt I(X,Y).
• When is I(X,Y) an exact representation of i(x,y)?
  That is, when can we do the following
• I(X,Y) -gtreconstruction/interpolation-gt
  i'(x,y)
• where we want i'(x,y) i(x,y)?

9

• We need conditions to design a sampling strategy
  and an interpolation scheme so that
• i(x,y) i(x,y)

10

• Suppose now the operation
• i'(x,y) -gt (re)sampling I'(X,Y)
• We shall analyse digital images as if we are
  dealing with I'(X,Y).
• We use i(x,y) and i'(x,y) to judge the adequacy
  of our digital image.
• We use I(X,Y) and I'(X,Y) to analyze the digital
  computations.