Digital image processing questions, Exams of Digital Image Processing

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LECTURE NOTES

ON

DIGITAL IMAGE PROCESSING

PREPARED BY

DR. PRASHANTA KUMAR PATRA

COLLEGE OF ENGINEERING AND TECHNOLOGY, BHUBANESWAR

Digital Image Processing

UNIT-I

DIGITAL IMAGE FUNDAMENTALS AND TRANSFORMS

1. ELEMENTS OF VISUAL PERCEPTION

1.1 ELEMENTS OF HUMAN VISUAL SYSTEMS

• The following figure shows the anatomy of the human eye in cross section
• There are two types of receptors in the retina
  • The rods are long slender receptors
  • The cones are generally shorter and thicker in structure
• The rods and cones are not distributed evenly around the retina.
• Rods and cones operate differently
  • Rods are more sensitive to light than cones.
  • At low levels of illumination the rods provide a visual response called scotopic vision
  • Cones respond to higher levels of illumination; their response is called photopic vision

Digital Image Processing

1.2 IMAGE FORMATION IN THE EYE

1.3 CONTRAST SENSITIVITY

• The response of the eye to changes in the intensity of illumination is nonlinear
• Consider a patch of light of intensity i+dI surrounded by a background intensity I as shown in the following figure
• Over a wide range of intensities, it is found that the ratio dI/I, called the Weber fraction, is nearly constant at a value of about 0.02.
• This does not hold at very low or very high intensities
• Furthermore, contrast sensitivity is dependent on the intensity of the surround. Consider the second panel of the previous figure.

1.4 LOGARITHMIC RESPONSE OF CONES AND RODS

• The response of the cones and rods to light is nonlinear. In fact many image processing systems assume that the eye's response is logarithmic instead of linear with respect to intensity.
• To test the hypothesis that the response of the cones and rods are logarithmic, we examine the following two cases:
• If the intensity response of the receptors to intensity is linear, then the derivative of the response with respect to intensity should be a constant. This is not the case as seen in the next figure.
• To show that the response to intensity is logarithmic, we take the logarithm of the intensity response and then take the derivative with respect to intensity. This derivative is nearly a constant proving that intensity response of cones and rods can be modeled as a logarithmic response.
• Another way to see this is the following, note that the differential of the logarithm of intensity is d(log(I)) = dI/I. Figure 2.3-1 shows the plot of dI/I for the intensity response of the human visual system.
• Since this plot is nearly constant in the middle frequencies, we again conclude that the intensity response of cones and rods can be modeled as a logarithmic response.

1.5 SIMULTANEOUS CONTRAST

• The simultaneous contrast phenomenon is illustrated below.
• The small squares in each image are the same intensity.
• Because the different background intensities, the small squares do not appear equally bright.
• Perceiving the two squares on different backgrounds as different, even though they are in fact identical, is called the simultaneous contrast effect.
• Psychophysically, we say this effect is caused by the difference in the backgrounds, but what is the physiological mechanism behind this effect? 1.6 LATERAL INHIBITION
• Record signal from nerve fiber of receptor A.
• Illumination of receptor A alone causes a large response.
• Add illumination to three nearby receptors at B causes the response at A to decrease.
• Increasing the illumination of B further decreases A‘s response.
• Thus, illumination of the neighboring receptors inhibited the firing of receptor A.
• This inhibition is called lateral inhibition because it is transmitted laterally, across the retina, in a structure called the lateral plexus.
• A neural signal is assumed to be generated by a weighted contribution of many spatially adjacent rods and cones.
• Some receptors exert an inhibitory influence on the neural response.
• The weighting values are, in effect, the impulse response of the human visual system beyond the retina.

1.7 MACH BAND EFFECT

• Another effect that can be explained by the lateral inhibition.
• The Mach band effect is illustrated in the figure below.
• The intensity is uniform over the width of each bar.
• However, the visual appearance is that each strip is darker at its right side than its left. 1.8 MACH BAND

• The logarithmic/linear system eye model provides a reasonable prediction of visual response over a wide range of intensities.
• However, at high spatial frequencies and at very low or very high intensities, observed responses depart from responses predicted by the model. 1.11 LIGHT
• Light exhibits some properties that make it appear to consist of particles; at other times, it behaves like a wave.
• Light is electromagnetic energy that radiates from a source of energy (or a source of light) in the form of waves
• Visible light is in the 400 nm – 700 nm range of electromagnetic spectrum 1.11.1 INTENSITY OF LIGHT
• The strength of the radiation from a light source is measured using the unit called the candela, or candle power. The total energy from the light source, including heat and all electromagnetic radiation, is called radiance and is usually expressed in watts.
• Luminance is a measure of the light strength that is actually perceived by the human eye. Radiance is a measure of the total output of the source; luminance measures just the portion that is perceived.

Brightness is a subjective, psychological measure of perceived intensity. Brightness is practically impossible to measure objectively. It is relative. For example, a burning candle in a darkened room will appear bright to the viewer; it will not appear bright in full sunshine.

• The strength of light diminishes in inverse square proportion to its distance from its source. This effect accounts for the need for high intensity projectors for showing multimedia productions on a screen to an audience. Human light perception is sensitive but not linear

2. SAMPLING

Both sounds and images can be considered as signals, in one or two dimensions, respectively. Sound can be described as a fluctuation of the acoustic pressure in time, while images are spatial distributions of values of luminance or color, the latter being described in its RGB or HSB components. Any signal, in order to be processed by numerical computing devices, have to be reduced to a sequence of discrete samples , and each sample must be represented using a finite number of bits. The first operation is called sampling , and the second operation is called quantization of the domain of real numbers. 2.1 1-D: Sounds Sampling is, for one-dimensional signals, the operation that transforms a continuous-time signal (such as, for instance, the air pressure fluctuation at the entrance of the ear canal) into a discrete-time signal, that is a sequence of numbers. The discrete- time signal gives the values of the continuous-time signal read at intervals of T seconds. The reciprocal of the sampling interval is called sampling rate F s= 1/T

. In this module we do not explain the theory of sampling, but we rather describe its manifestations. For a a more extensive yet accessible treatment, we point to the Introduction to Sound Processing. For our purposes, the process of sampling a 1-D signal can be reduced to three facts and a theorem.  Fact 1: The Fourier Transform of a discrete-time signal is a function (called spectrum ) of the continuous variable ω , and it is periodic with period 2 π. Given a value of ω , the Fourier transform gives back a complex number that can be interpreted as magnitude and phase (translation in time) of the sinusoidal ^ component at that frequency. 

Let us assume we have a continuous distribution, on a plane, of values of luminance or, more simply stated, an image. In order to process it using a computer we have to reduce it to a sequence of numbers by means of sampling. There are several ways to sample an image, or read its values of luminance at discrete points. The simplest way is to use a regular grid, with spatial steps X e Y. Similarly to what we did for sounds, we define the spatial sampling rates F X= 1/X F Y= 1/Y As in the one-dimensional case, also for two-dimensional signals, or images, sampling can be described by three facts and a theorem.  Fact 1: The Fourier Transform of a discrete-space signal is a function (called spectrum ) of two continuous variables ω X and ω Y, and it is periodic in two dimensions with periods 2 π. Given a couple of values ω X and ω Y, the Fourier transform gives back a complex number that can be interpreted as magnitude and phase (translation in space) of the sinusoidal component at such spatial  frequencies.  Fact 2: Sampling the continuous-space signal s ( x , y ) with the regular grid of steps X , Y , gives a discrete-space signal s ( m , n ) = s ( mX , nY ) , which is a function of the discrete variables m and n .  Fact 3: Sampling a continuous-space signal with spatial frequencies F X and F Y gives a discrete-space signal whose spectrum is the periodic replication along the grid of steps F X and F Y of the original signal spectrum. The Fourier variables ω X and ω Y correspond to the frequencies (in cycles per meter) represented by the variables f X= ω X/ 2 πX    And fy= ω Y / 2 πY 

. The Figure 2 shows an example of spectrum of a two-dimensional sampled signal. There, the continuous-space signal had all and only the frequency components included in the central hexagon. The hexagonal shape of the spectral support (region of non-null spectral energy) is merely illustrative. The replicas of the original spectrum are often called spectral images.

Spectrum of a sampled image Figure 2 Given the above facts, we can have an intuitive understanding of the Sampling Theorem.

3. QUANTIZATION

With the adjective "digital" we indicate those systems that work on signals that are represented by numbers, with the (finite) precision that computing systems allow. Up to now we have considered discrete-time and discrete-space signals as if they were collections of infinite-precision numbers, or real numbers. Unfortunately, computers only allow to represent finite subsets of rational numbers. This means that our signals are subject to quantization. For our purposes, the most interesting quantization is the linear one, which is usually occurring in the process of conversion of an analog signal into the digital domain. If the memory word dedicated to storing a number is made of b bits, then the range of such number is discretized into 2 b quantization levels. Any value that is found between two quantization levels can be approximated by truncation or rounding to the closest value. The Figure 3 shows an example of quantization with representation on 3 bits in two's complement. Sampling and quantization of an analog signal

The number of distinct colors that can be represented by a pixel depends on the number of bits per pixel (bpp). A 1 bpp image uses 1-bit for each pixel, so each pixel can be either on or off. Each additional bit doubles the number of colors available, so a 2 bpp image can have 4 colors, and a 3 bpp image can have 8 colors:  1 bpp, 2 1 = 2 colors (monochrome)  2 bpp, 2^2 = 4 colors  3 bpp, 2^3 = 8 colors  8 bpp, 2^8 = 256 colors   16 bpp, 2 16 = 65,536 colors ("Highcolor" )  24 bpp, 2 24 ≈ 16.8 million colors ("Truecolor") For color depths of 15 or more bits per pixel, the depth is normally the sum of the bits allocated to each of the red, green, and blue components. Highcolor, usually meaning 16 bpp, normally has five bits for red and blue, and six bits for green, as the human eye is more sensitive to errors in green than in the other two primary colors. For applications involving transparency, the 16 bits may be divided into five bits each of red, green, and available: this means that each 24-bit pixel has an extra 8 bits to describe its blue, with one bit left for transparency. A 24-bit depth allows 8 bits per component. On some systems, 32-bit depth is opacity (for purposes of combining with another image). Selected standard display resolutions include: Name Megapixels Width x Height CGA 0.064 320× EGA 0.224 640× VGA 0.3 640× SVGA 0.5 800× XGA 0.8 1024× SXGA 1.3 1280× UXGA 1.9 1600× WUXGA 2.3 1920×

5. BASIC GEOMETRIC TRANSFORMATIONS

Transform theory plays a fundamental role in image processing, as working with the transform of an image instead of the image itself may give us more insight into the properties of the image. Two dimensional transforms are applied to image enhancement, restoration, encoding and description.

5.1. UNITARY TRANSFORMS

5.1.1 One dimensional signals For a one dimensional sequence { f ( x ), 0  x  N 1} represented as a vector f   f (0) f (1) f ( N 1) T^ of size (^) N , a transformation may be written as N  1 g  T  f  g ( u )   T ( u , x ) f ( x ), 0  u  N^ ^1 x  0 where g ( u ) is the transform (or transformation) of f ( x ) , and T ( u , x ) is the so called forward transformation kernel. Similarly, the inverse transform is the relation N  1 f ( x )   I ( x , u ) g ( u ), 0  x  N  1 u  0 or written in a matrix form f  I  g  T ^1  g where I ( x , u ) is the so called inverse transformation kernel. If I  T ^1  T  T