02 topic imagedata
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image dataTRANSCRIPT
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Topic 2 Image Data
Image Representation and Analysis, Colors
Devesh Chandra Guest Faculty
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Agenda
Images in Spatial domain
Mathematics and structures for Images in Spatial domain
Images in Frequency domain
Colors
Color Images
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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
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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.
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Grid Cell model and grid point models
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Image Windows
W((31+171, 241 + 137),
(351 X 275)
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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.
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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.
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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
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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.
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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).
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Image Contrast
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Image Contrast
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Slopes
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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
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Intensity Profiles ( Value Statistics in an
Intensity profile )
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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
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Temporal Value Statistics
A plot of two data measures for a sequence of 400 frames
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Temporal Value Statistics
The same two measure after normalizing mean and variance of
both measures 20
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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
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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
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What is an edge?
The step-edge model assumes that edges are defined by
changes in local derivatives
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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
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Images in Frequency Domain
The Fourier Transform defines a traditional way for
processing signals
Linear transform
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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.
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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.
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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
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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
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2D Discrete Fourier Transform
Formally, the 2D DFT is defined as follows ( Refer Lecture notes):
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Basis Functions
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The Complex Plane
(See Lecture Notes)
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Interpretation of Matrix I (u, v)
(Refer Lecture Notes)
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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.
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Phase Congruency Model for Image Features
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Phase Congruency Model for Image Features
(Class Notes)
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Phase Congruency Model
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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
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Color Definitions
Electromagnetic Spectrum & Visible Spectrum
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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
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Human Vision
Human Eyes
Image Formations
Information Extraction
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Retina
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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
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Colored Scanning electron micrograph (SEM) or rods
(blue) and cones (purple) in the retina of eye
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Retina (Rods and Cones)
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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.
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Color sensitive cone
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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
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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.
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Spectral Power distribution of a mercury light source. Y axis indicates the power
per wavelength and the x - axis indicates the wavelength
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INC Incandescent Lamp ; CF - Fluorescent Lamps ; High pressure Sodium ;
Metal Halide ;
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Spectral Power distribution of incandescent electric light
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Energy distribution curve L (lambda).
Response of retina to different
wavelengths i.e. the energy distribution
function
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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
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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
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Chromaticity Diagram
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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
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HSI- Color Representation
(Refer Textbook or Lecture Notes)
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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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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