19-03-2012, 12:02 PM
Image Analysis Filtering and Enhancement
Chapter 1
Image and Image models
Image: Definition
An image may be defined as a 2-D function, f(x, y), where x and y are spatial (plane) coordinates, and the amplitude (value) of f(x, y) at any pair of coordinates (x, y) is called the intensity or gray level of the image at that point. When x, y and the intensity value of f(x, y) are all finite discrete quantities, then image is called digital image. [1]
A Simple Image Formation Model
Figure: 1(a)
Digital image acquisition
An example of the digital image acquisition process 1(a). (1) Energy ("illumination") source. (2) An element of a scene. (3) Imaging system. (4) Projection of the scene onto the image plane. (5) Digitized image.
We denote images by two-dimensional func¬tions of the form f(x, y). The value or amplitude of f at spatial coordinates (x, y) is a positive scalar quantity whose physical meaning is determined by the source of the image. When an image is generated from a physical process, its intensity values are proportional to energy radiated by a physical source (e.g., electromagnetic waves). As a consequence, f(x, y) must be nonzero and finite; that is,
0 < f(x, y) < ∞ (1)
The function f(x, y) may be characterized by two components:
(1) The amount of source illumination incident on the scene being viewed, and
(2) The amount of il¬lumination reflected by the objects in the scene. Appropriately, these are called the illumination and reflectance components and are denoted by i(x, y) and r(x, y), respectively. The two functions combine as a product to form f(x, y):
f(x, y) = i(x, y) r(x, y) (2)
where,
0 < i(x, y) < ∞ (3)
0 < r(x, y) < 1 (4)
Equation (4) indicates that reflectance is bounded by 0 (total absorption) and 1 (total reflectance). The nature of i(x, y) is determined by the illumina¬tion source, and r (x, y) is determined by the characteristics of the imaged ob¬jects. It is noted that these expressions also are applicable to images formed via transmission of the illumination through a medium, such as a chest X-ray[1].
Basic Concepts in Sampling and Quantization
Figure: 1(b)
Generating a digital image
Generating a digital image 1(b) : (1) Continuous image, (2) A scan line from A to B in the continuous image, used to illustrate the concepts of sampling and quantization. (3) Sampling and quantization. (4) Digital scan line.
The basic idea behind sampling and quantization is illustrated in Figure: 1(b). Figure: 1(b-1) shows a continuous image f that we want to convert to digital form. An image may be continuous with respect to the x- and y-coordinates, and also in amplitude. To convert it to digital form, we have to sample the function in both coordinates and in amplitude. Digitizing the coordinate values is called sampling. Digitizing the amplitude values is called quantization.
The one-dimensional function in Figure: 1(b-2) is a plot of amplitude (intensity level) values of the continuous image along the line segment AS in Figure: 1(b-1). The random variations are due to image noise. To sample this function, we take equally spaced samples along line AB, as shown in Figure: 1(b-3). The spatial loca¬tion of each sample is indicated by a vertical tick mark in the bottom part of the figure. The samples are shown as small white squares superimposed on the func¬tion. The set of these discrete locations gives the sampled function. However, the values of the samples still span (vertically) a continuous range of intensity val¬ues. In order to form a digital function, the intensity values also must be con¬verted (quantized) into discrete quantities. The right side of Figure: 1(b-3) shows the intensity scale divided into eight discrete intervals, ranging from black to white. The vertical tick marks indicate the specific value assigned to each of the eight intensity intervals. The continuous intensity levels are quantized by assign¬ing one of the eight values to each sample. The assignment is made depending on the vertical proximity of a sample to a vertical tick mark. The digital samples resulting from both sampling and quantization are shown in Figure: 1(b-4). Start¬ing at the top of the image and carrying out this procedure line by line produces a two-dimensional digital image. It is implied in Figure: 1(b) that, in addition to the number of discrete levels used, the accuracy achieved in quantization is highly dependent on the noise content of the sampled signal [1].
Figure: 1©
Continuous & corresponding Digital image of a scene element
Image models
Basically, there are four types of image models
• RGB image model [1]
• CMY or CMYK image model [1]
• Gray image model [2]
• Binary image model [2]
These models uses different color domain to describe an image. These models are briefly summarized below one after another.
RGB image model
In RGB model, each color appears in its primary spectral components of red, green, and blue. This model is based on Cartesian coordinates system. The color sub space is of interest is the cube shown in figure, in which RGB primary values at three corners; the secondary colors cyan, magenta, and yellow three other corners; black is at origin; and white is at the corner farthest from the origin. In this model the gray scale (points of equal RGB values) extends from black to white along the line joining these two points. The different colors in this model are points on or inside the cube and they are defined by the vectors extending from origin. [1]
Figure: 1(d)
Schematic of RGB color cube
In a single sentence, all the colors (that exist in nature or anthropogenic colors) can be made by varying the composition of Red, Green, and Blue colors.
For example:
Figure: 1(e)
Constituent R, G & B color images & complete RGB image
Figure: 1(f)
RGB image acquired for image processing
Consider an RGB image in which each of the red, green, and blue images (separately) is an 8-bit image, then complete RGB image is having 8 x 3 = 24 bits (8-bits per plane), then the total number of colors in 24-bit RGB image is 224 = 16,777,21. For example the cube shown below is 24-bit RGB image.
Figure: 1(g)
24-bit RGB color cube
CMY or CMYK image model
As we know that cyan, magenta, and yellow are the secondary colors of light. Most devices require CMY data inputs or internally perform RGB to CMY conversion. This conversion is performed using the simple operation
Figure: 1(h)
Conversion formula from RGB to CMY
In case of CMY model, the black color that CMY produces is muddy looking black. So in order to produce the true black, a fourth color, black is added, giving rise to CMYK color model. [1]
For example
Figure: 1(i)
Constituent C,M,Y & K color images & complete CMYK image
Gray image model
It is also known as an intensity, gray scale, or gray level image. Array of class uint8, uint16, int16, single, or double whose pixel values specify intensity values. For single or double arrays, values range from [0, 1]. For uint8, values range from [0,255]. For uint16, values range from [0, 65535]. [2]
Figure: 1(j)
Grayscale image
Binary image model
A binary image is a logical array of 0s and 1s. Pixels with the value 0 are displayed as black; pixels with the value 1 are displayed as white. [2]