Topic 2 Image Data

Image Representation and Analysis, Colors

Devesh Chandra Guest Faculty

1

Agenda

Images in Spatial domain

Mathematics and structures for Images in Spatial domain

Images in Frequency domain

Colors

Color Images

2

Images in spatial domain

Basic notation and mathematical concepts for describing

an image in a regular grid in the spatial domain.

A digital image is defined by integrating and sampling

continuous (analog) data in a spatial domain.

Pixel, Windows

Grid Cells, Grid Points and adjacency

3

Pixels and Windows

Pixels are the atomic elements of an image.

Term pixel is short for picture element.

The images that we see on the screen are composed of

homogenously shaded square cells.

Image values can also be assumed to be labels at grid-

points being the center of grid squares.

4

Grid Cell model and grid point models

5

Image Windows

W((31+171, 241 + 137),

(351 X 275)

6

Histogram

Histogram is a graphical representation showing a visual

impression of the distribution of data

Image Histogram is a type of histogram that acts as a

graphical representation of the lightness/color

distribution in a digital image. It plots the number of

pixels for each value.

7

Normalized Histogram

It is common practice to normalize a histogram by

dividing each of its values by the total number of pixels

in the image, denoted by n.

Normalized histogram gives the estimate of the

probability of occurrence of gray level. Sum of the all

components of a normalized histogram is equal to 1.

8

Why Histograms

Can perform variety of spatial domain processing

techniques ?

Provides useful image statistics and information can be

used for image enhancement.

Horizontal axis corresponds to gray level values.

The vertical axis corresponds to to values of h(rk)=nk or

p(rk)=nk/n if the values are normalized

9

10

Histogram in MATLAB

h = imhist (f, b)

p = imhist (f, b) / numel(f)

Where f, is the input image, h is the histogram, b is number of bins (tick marks) used in forming the histogram (b = 255 is the default).A bin, is simply, a subdivision of the intensity scale. For example, if we are working with uint8 images and we let b = 2, then the intensity scale is subdivided into two ranges: 0 127 and 128 255. the resulting histograms will have two values: h(1) equals to the number of pixels in the image with values in the interval [0,127], and h(2) equal to the number of pixels with values in the interval [128 255]

numel (f): a MATLAB function that gives the number of elements in array f (i.e. the number of pixels in an image.

11

Contrast Image

In image analysis we often classify the windows into

categories such as within a homogenous region (of low

contrast), or showing an edge between two different

regions (of high contrast).

12

Image Contrast

13

Image Contrast

14

Slopes

15

Spatial and Temporal Data Measures

Finding the functions that describe images, such as row by

row in a single image or frame by frame for a given

sequence of images

16

Intensity Profiles ( Value Statistics in an

Intensity profile )

17

Spatial or Temporal Value statistics

Histograms or intensity profiles are example of spatial

value statistics

Consider the image sequences consisting of the frames It

for t =1, 2, 3, ., T, all defined for same carrier

18

Temporal Value Statistics

A plot of two data measures for a sequence of 400 frames

19

Temporal Value Statistics

The same two measure after normalizing mean and variance of

both measures 20

Step-Edges

Discontinuities in images are features that are often useful

for initializing an image analysis procedure

Can be used for simplifying image data and understand

the image

21

illustration for the step

model

Left :Synthetic Image Input

Right: Intensity Profile

- Ideal step edges

- Liner edge

- Smooth Edge

- Noisy Edge

- Thin-line

- Discontinuity in shaded

region

22

What is an edge?

The step-edge model assumes that edges are defined by

changes in local derivatives

23

Synthetic image input with

pixel location (x, y)

Tangential plane in green at

pixel (x, y, I(x,y)), normal n =

[a, b, 1]T

Derivatives and Edges

24

Images in Frequency Domain

The Fourier Transform defines a traditional way for

processing signals

Linear transform

25

The Fourier transform is a representation of an image as

a sum of complex exponentials of varying magnitudes,

frequencies, and phases.

The Fourier transform plays a critical role in a broad

range of image processing applications, including

enhancement, analysis, restoration, and compression.

26

Fourier Series

Fourier series expansion is appropriate for analysis of

periodic functions

Fourier series allows a periodic function to be

represented as an infinite sum of harmonic oscillations at

definite frequencies equal to multiples of the fundamental.

27

Fourier Transform

Fourier transform measures the frequency content of the

signal be it periodic or a-periodic.

Fourier transform allows a-periodic function to be

expressed as an integral sum over a continuous range of

frequencies

With the suitable use of the delta functions the Fourier

transform may be used to cover both periodic and a-

periodic functions. Fourier series can be regarded as the

special case of the Fourier transform.

Fourier transform is frequency dense representation for

non-periodic signals

28

Discrete Fourier Transform

The Discrete Fourier Transform is equivalent to the

continuous Fourier Transform for signals known only at

N instants separated by sample time T

29

2D Discrete Fourier Transform

Formally, the 2D DFT is defined as follows ( Refer Lecture notes):

30

Basis Functions

31

The Complex Plane

(See Lecture Notes)

32

Interpretation of Matrix I (u, v)

(Refer Lecture Notes)

33

Image Analysis in Frequency domain

The complex values of 2D Fourier Transform are defined

in the u-v frequency.

Low frequency and high frequency u , v.

34

Phase Congruency Model for Image Features

35

Phase Congruency Model for Image Features

(Class Notes)

36

Phase Congruency Model

37

Colors

Colors as perceived by human are prone to personal

prejudices and are influenced by emotions

Colors can be important component of the given image

data and it is useful to visualize information by false

image colors

RGB and HSI color models

38

Color Definitions

Electromagnetic Spectrum & Visible Spectrum

39

40

Colors

Red 625 780 nm

Orange 590 625 nm

Yellow 565 590 nm

Green 500 565 nm

Cyan 485 500 nm

Blue 440 485 nm

Violet 380 440 nm

41

Human Vision

Human Eyes

Image Formations

Information Extraction

42

Retina

43

Retina [ Rods & Cones]

Retina contains two photo-receptors, rods and cones.

Rods are 120 million in numbers and are more sensitive

than cones. They are not sensitive to colors

Cones provides the color sensitivity and they are much

more concentrated in macula.

Macula contains the region fovea centralis, a 0.3 mm

diameter rod free area with very thin and densely packed

cones

44

Colored Scanning electron micrograph (SEM) or rods

(blue) and cones (purple) in the retina of eye

45

Retina (Rods and Cones)

46

Response Curves of the Cones

In 1965, experimental evidences were found to confirm

that there are three types of color sensitive cones in the

retina of human being.

The shapes of the curves are obtained by measurement

of absorption by the cones.

The relative height of each type is set equal, due to lack

of data

There is lesser blue cones, yet the sensitivity of blue is

comparable to other colors.

47

Color sensitive cone

48

Observing Colors

When the light strikes a cone, it interacts with visual

pigment which consist of a protein called opsin and a

small molecule called chromophore

Three different kind of opsins respond to short, medium

and long wavelengths of light and lead to three response

curves

For a person to see an object in color at least two kinds

of cones must be triggered, and the perceived color is

based on the relative level of excitation of the different

cones

49

Energy distribution of light source

Monochromatic light is the light that has only one

wavelength. Most of light are not monochromatic in

other words, they radiate a mixture of different

wavelengths.

Spectral power distribution of a given light source

provide information on the total amount of energy (E)

emitted by light source over the electromagnetic

spectrum.

50

Spectral Power distribution of a mercury light source. Y axis indicates the power

per wavelength and the x - axis indicates the wavelength

51

INC Incandescent Lamp ; CF - Fluorescent Lamps ; High pressure Sodium ;

Metal Halide ;

52

Spectral Power distribution of incandescent electric light

53

Energy distribution curve L (lambda).

Response of retina to different

wavelengths i.e. the energy distribution

function

54

Tri-stimulus Values

The weighing functions have been defined by the CIE

within the visible spectrum.

Three curves are scaled such that their integrals are equal

55

The x-y color space of the CIE

Parameters x and y define the 2D CIE color space.

It does not represent brightness, just colors only.

Chromaticity diagram

56

Chromaticity Diagram

57

Chromaticity Diagram

Monochromatic colors :- The convex outer curve in the

diagram contains monochromatic colors (pure spectral

colors)

The bottom line is not mono-chromatic

Less saturated colors in with white at center E = (0.33 ,

0.33)

RGB Primaries

58

HSI- Color Representation

(Refer Textbook or Lecture Notes)

59

References

Slide No

1-9, 12-38,

48, 54, 57

Concise Computer Vision: An Introduction Into Theory and Algorithms,

Reinhard Klette, Springer

10 http://homepages.inf.ed.ac.uk/rbf/HIPR2/hipr_top.htm

11 http://in.mathworks.com/help/images/ref/imhist.html

39 http://www.solarlightaustralia.com.au/2013/02/20/visible-light/

40 http://www.ducksters.com/science/physics/types_of_electromagnetic_wav

es.php

41 http://imgarcade.com/1/visible-light-waves/

http://www.tv411.org/science/tv411-whats-cooking/heat-math-

lesson/activity/1/5

43 -45 The Human Eye and Adaptive Optics, Fuensanta A. Vera-Daz1 and Nathan

Doble, The New England College of Optometry, Boston MA, USA

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61

Slide No

46 http://hyperphysics.phy-astr.gsu.edu/hbase/vision/rodcone.html

51- 52 http://www.math.ubc.ca/~cass/courses/m309-03a/m309-

projects/vaxenga/part2.html

53 http://www.math.ubc.ca/~cass/courses/m309-03a/m309-

projects/vaxenga/part2.html