Digital Image Processing Lecture 4 - basu.ac.ir · Digital Image Processing Lecture 4 (Image enhancement in spatial domain) Bu-Ali Sina University Computer Engineering Dep. Fall 2016

Digital Image Processing

Lecture 4(Image enhancement in spatial domain)

Bu-Ali Sina UniversityComputer Engineering Dep.

Fall 2016

OutlineImage Enhancement in Spatial Domain

– Histogram based methods• Histogram Equalization• Histogram Specification• Local Histogram

– Spatial Filtering• Smoothing Filters• Median Filter• Sharpening• High Boost filter• Derivative filter

Image Preprocessing

Enhancement Restoration

SpatialDomain

SpectralDomain

Point Processing• >>imadjust• >>histeq

Spatial filtering• >>filter2

Filtering• >>fft2/ifft2• >>fftshift

• Inverse filtering• Wiener filtering

Image Enhancement (Histogram-based methods)

· The histogram of a digital image with grayvalues is thediscrete function :

· The function represents the fraction of the total number of pixelswith grayvalue .

· Histogram provides a global description of the appearance of the image.

· If we consider the grayvalues in the image as realizations of a random variable R,with some probability density, histogram provides an approximation to thisprobability density. In other words :

Some typical Histograms· The shape of a histogram provides useful information forcontrast enhancement.

Histogram-based methods)Some typical Histograms

Chapter 3: Image Enhancement (Histogram-based methods)Some typical Histograms

Histogram Equalization

· Let us assume for the moment that the input image to be enhanced hascontinuous grayvalues, with r = 0 representing black and r = 1 representing white.· We need to design a grayvalue transformation s = T(r), based on the histogram ofthe input image, which will enhance the image.· As before, we assume that:

T(r) is a monotonically increasing function for(preserves order from black to white)

T(r) maps [0,1] into [0,1] (preserves the range of allowed

grayvalues).

Chapter 3: Image Enhancement (Histogram-based methods)

· Let us denote the inverse transformation by . We assume that theinverse transformation also satisfies the above two conditions.· We consider the grayvalues in the input image and output image as random

variables in the interval [0, 1].· Let and denote the probability density of the grayvalues in theinput and output images.· If and T(r) are known, and satisfies condition 1, we can write(result from probability theory):

· One way to enhance the image is to design a transformation T(.) such thatthe grayvalues in the output is uniformly distributed in [0, 1], i.e.

· In terms of histograms, the output image will have all grayvalues in “equalproportion.”

· This technique is called histogram equalization.



· Consider the transformation :

· Note that this is the cumulative distribution function (CDF) of andsatisfies the previous two conditions.· From the previous equation and using the fundamental theorem of calculus

:

· Therefore, the output histogram is given by :

· The output probability density function is uniform, regardless of the input.


Chapter 3: Image Enhancement (Histogram-based methods)Histogram Equalization


· Thus, using a transformation function equal to the CDF of inputgrayvalues r, we can obtain an image with uniform grayvalues.

· This usually results in an enhanced image, with an increase in the dynamic rangeof pixel values.

· Example: Read example in page 176 of text.· For images with discrete grayvalues, we have

L: Total number of graylevelsnk: Number of pixels with grayvalue rkn: Total number of pixels in the image

· The discrete version of the previous transformation based on CDF is given by:



Example :· Consider an 8-level 64 x 64 image with grayvalues (0, 1, …, 7). The normalized

grayvalues are (0, 1/7, 2/7, …, 1). The normalized histogram is given below:



· Applying the previous transformation, we have (after roundingoff to nearest graylevel):



· Notice that there are only five distinct graylevels --- (1/7, 3/7, 5/7, 6/7, 1) in theoutput image. We will relabel them as(s0, s1, …, s4).· With this transformation, theoutput image will have histogram:

· Note that the histogram of outputimage is only approximately, andNot exactly, uniform. This shouldnot be surprising, since there is noresult that claims uniformity in thediscrete case.



Example :Histogram Equalization


· Histogram equalization may not always produce desirable results, particularly ifthe given histogram is very narrow. It can produce false edges and regions. It canalso increase image “graininess” and “patchiness.”



o Histogram equalization yields an image whose pixels are (in theory) uniformlydistributed among all graylevels.o Sometimes, this may not be desirable. Instead, we may want a transformationthat yields an output image with a prespecified histogram. This technique is calledhistogram specification.o Again, we will assume, for the moment, continuous grayvalues.o Suppose, the input image has probability density . We want to find atransformation z = H(r) , such that the probability density of the new imageobtained by this transformation is , which is not necessarily uniform.o First apply the transformation

o This gives an image with a uniform probability density.o If the desired output image were available, then the following transformationwould generate an image with uniform density:

Histogram Specification


oIt then follows from these two equations that G(z)=T(r) and, therefore, that z mustsatisfy the condition

o For discrete graylevels, we have :



Example :Consider the previous 8-graylevel 64 x 64 image histogram:



· It is desired to transform this image into a new image, using atransformation , with histogram as specifiedbelow:



· The transformation T(r) was obtained earlier (reproduced below):



· Next we compute the transformation G as before :Histogram Specification


· Notice that G is not invertible. But we will do the best possible bysetting :



· Combining the two transformation T and G-1, we get our requiredtransformation H :



· Applying the transformation H to the original image yields animage with histogram as below:



· Again, the actual histogram of the output image does not exactlybut only approximately matches with the specified histogram. Thisis because we are dealing with discrete histograms.



Example :Histogram Specification

Chapter 3: Image Enhancement (Histogram-based methods)Histogram Specification




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Digital Image Processing Lecture 4 - basu.ac.ir · Digital Image Processing Lecture 4 (Image enhancement in spatial domain) Bu-Ali Sina University Computer Engineering Dep. Fall 2016