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Digital image fundamentals

Dr. Huidae Cho
Institute for Environmental and Spatial Analysis...University of North Georgia

1   Principal energy source for digital images

Principal energy source for images: Electromagnetic (EM) energy spectrum.

EM waves: Stream of massless particles (protons), each traveling in a wave-like pattern at the speed of light

Each massless particle has a bundle of energy called photon

fig0210-electromagnetic-spectrum.png

Figures from Gonzalez and Woods (2002)

2   Waves

Wavelength: $\lambda=\frac{c}{\nu}$

  • $c$: Speed of light ($2.998\times 10^8 \mathrm{m/s}$)
  • $\nu$: Frequency in Hz
  • 1 Hz is 1 cycle of a sinusoidal wave per second
  • Units of wavelength: m, microns ($\mathrm{\mu m}, 10^{-6} \mathrm{m}$), nanometers (nm, $10^{-9} \mathrm{m}$)

Energy: $E=h\nu$

  • $h$: Planck’s constant ($4.135,667,66\times 10^{-15} \mathrm{eV}$)
  • Units of energy: electron-volt (eV)

3   Alternative energy sources

Acoustic

  • Geological applications: Mineral and oil exploration using low-frequency waves (e.g., 100 Hz)
  • Low-frequency waves travel a longer distance because they lose less energy to the medium (less active $\rightarrow$ less loss)

Ultrasound

  • Medical applications: Baby
  • High-frequency waves (highly active $\rightarrow$ high loss)

Electronic

  • Electron microscopy

4   Imaging sensors

  • Single imaging sensor (point)
  • Line sensor (linear)
  • Array sensor (rectangular)

sensor

5   Digital image

$f(x,y)=i(x,y)\times r(x,y)$

  • $i(x,y) \in [0,\infty)$: Illumination, characteristics of the energy source
  • $r(x,y) \in [0,1]$: Reflectance, characteristics of the object sensed

6   Image sampling and quantization

fig0216-generating-a-digital-image.png

Figures from Gonzalez and Woods (2002)

7   Representation of digital images

$M$ rows, $N$ columns

A digital image $=$ An $M$-by-$N$ matrix

By Python convention (because we use Python)

  • Rows: $0,1,\cdots,M-1$
  • Columns: $0,1,\cdots,N-1$

$f(x,y)=\begin{bmatrix} f(0,0)& \cdots& f(0,N-1)\\ \vdots& \ddots& \vdots\\ f(M-1,0)& \cdots& f(M-1,N-1) \end{bmatrix}$

$\mathbf{A}=\begin{bmatrix} a_{0,0}& \cdots& a_{0,N-1}\\ \vdots& \ddots& \vdots\\ a_{M-1,0}& \cdots& a_{M-1,N-1} \end{bmatrix}$

8   Bits

bitsLSB: Least significant bit

MSB: Most significant bit

1 byte $=$ 8 bits

Unsigned integer $\in [2^0,2^8]=[0,256]$

What is this number in decimal notation?

9   Gray levels

Number of bits per pixek: $k$

Gray levels: $L=2^k$

$f(x,y)=\begin{cases} 0& \text{black}\\ L-1& \text{white} \end{cases}$

$k$$L$Description
12Binary image in black and white
66464 gray levels, limit of human visual system
8256Typical gray level resolution (1 byte per pixel)

10   Storage bits and bytes

Storage bits: $b=M\times N\times k$

Storage bytes: $B=\left\lceil\frac{b}{8}\right\rceil$

Problems

  1. $M=N=32, L=2$
  2. $M=N=32, L=128$

11   Spatial and gray-level resolution

$M$ rows, $N$ columns, $L$ gray levels

Spatial resolution: $M\times N$ pixels, sometimes in pixels per inch (ppi)

Gray resolution: $L$ levels

Convention: $M\times N, L$-level image

For example,

  • $512\times 512, 8$-level image
  • $1024\times 1024, 256$-level image

12   Homework 1

  1. The frequency of commercial alternating current in the United States is 60 Hz. Find its wavelength.
  2. How many storage bits and bytes do you need to store a digital image of size 128-by-128 pixels and 256 gray levels?

Show your work for full credits!