55:148 Digital Image Processing

Chapter 2, Part III
Digitized image and its properties: Digital image properties

Related Reading
Sections from Chapter 2 according to the WWW Syllabus.

Chapter 2.3 Overview:

Metric and topological properties of digital images
Histograms PE 2.C
Visual perception of the image
Image quality
Noise in images PE 2.D

Digital image properties

Metric and topological properties of digital images

• Some intuitively clear properties of continuous images have no straightforward analogy in the domain of digital images

Metric properites of digital images

• Distance is an important example.

• The distance between two pixels in a digital image is a significant quantitative measure.

• The distance between points with co-ordinates (i,j) and (h,k) may be defined in several different ways;
• the Euclidean distance is defined by Eq. 2.42
  



• city block distance ... Eq. 2.43
  



• chessboard distance Eq. 2.44
  

• Pixel adjacency is another important concept in digital images.
• 4-neighborhood
• 8-neighborhood (Fig. 2.6)

• It will become necessary to consider important sets consisting of several adjacent pixels -- regions.

• Region is a contiguous set.

• Contiguity paradoxes of the square grid ... Figures 2.7, 2.8

• One possible solution to contiguity paradoxes is to treat objects using 4-neighborhood and background using 8-neighborhood (or vice versa).

• A hexagonal grid solves many problems of the square grids ... any point in the hexagonal raster has the same distance to all its six neighbors.

• Border R is the set of pixels within the region that have one or more neighbors outside R ... inner borders, outer borders exist.

• Edge is a local property of a pixel and its immediate neighborhood --it is a vector given by a magnitude and direction.

• The edge direction is perpendicular to the gradient direction which points in the direction of image function growth.

• Border and edge ... the border is a global concept related to a region, while edge expresses local properties of an image function.

• Crack edges ... four crack edges are attached to each pixel, which are defined by its relation to its 4-neighbors. The direction of the crack edge is that of increasing brightness, and is a multiple of 90 degrees, while its magnitude is the absolute difference between the brightness of the relevant pair of pixels. (Fig. 2.9)

Topological properties of digital images

• Topological properties of images are invariant to rubber sheet transformations.

• Stretching does not change contiguity of the object parts and does not change the number of holes in regions.

• One such image property is the Euler--Poincare characteristic defined as the difference between the number of regions and the number of holes in them.

• Convex hull is used to describe topological properties of objects.

• The convex hull is the smallest region which contains the object, such that any two points of the region can be connected by a straight line, all points of which belong to the region.

• Fig. 2.10

Histograms

• Brightness histogram provides the frequency of the brightness value z in the image.

• Algorithm 2.1

1. Assign zero values to all elements of the array h.
2. For all pixels (x,y) of the image f, increment h(f(x,y)) by one.

• Histograms may have many local maxima ... histogram smoothing
  

Practical Experiment 2.C - VIP Version

• Start VIP
• You have 3 minutes to complete the following exploratory exercise:
• Open the Histogram project
• Running the project displays histograms of the original image and the equalized image.
• Analyzing the image and histogram differences, can you describe the functionality of the histogram equalization block?

Practical Experiment 2.C - Khoros Version

• Start cantata
• You have 3 minutes to complete the following exploratory exercise:
• Open the Histogram workspace
• Display histograms of the original image and the equalized image using the histogram display capabilities of cantata. You will find the histogram glyph under DataManip -> HistogramOper -> Histogram. Histogram is a single-dimensional array, so you have to use 2D-Plot glyph to display the results, 2D-Plot can be found under Visualization -> PlotDisplay -> 2D-Plot
• Analyzing the image and histogram differences, can you describe the functionality of the histogram equalization block?

Visual perception of the image

Not covered.

Image quality

Not covered.

Noise in images

• Images are often degraded by random noise.

• Noise can occur during image capture, transmission or processing, and may be dependent on or independent of image content.

• Noise is usually described by its probabilistic characteristics.
• White noise - constant power spectrum (its intensity does not decrease with increasing frequency); very crude approximation of image noise
• Gaussian noise is a very good approximation of noise that occurs in many practical cases
• probability density of the random variable is given by the Gaussian curve;
• 1D Gaussian noise - µ is the mean and is the standard deviation of the random variable.
  

• During image transmission, noise which is usually independent of the image signal occurs.

• Noise may be

additive, noise and image signal g are independent

multiplicative, noise is a function of signal magnitude

impulse noise (saturated = salt and pepper noise)

Practical Experiment 2.D - VIP Version

• Since VIP does not have the noise generation capability that Khoros does, examine the following images that represent how noise affects images:

Practical Experiment 2.D - Khoros Version

• Start cantata
• Open the noise workspace, its filename is: ~dip/.public_html/examples/khoros.dir/noise.wksp
• Explore different parameters of the Gaussian noise, view the noise image and the image after additive or multiplicative noise has corrupted the image. What can you conclude? Which noise conditions corrupt the image more severely? Which noise is more serious, additive or multiplicative?
• If you have time, choose some other noise generator from DataManip -> IntroNoise -> (select) and continue in your explorations.