chapter-5 image histogram and morphological...

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CHAPTER-5

IMAGE HISTOGRAM AND MORPHOLOGICAL

OPERATIONS

5.1 Image Histogram

The histogram of a digital image with intensity levels in the range ��

�� is a discrete function� �� , where � is the �� intensity value and �� is

the number of pixels in the image with intensity� �. It is a common practice to

normalize a histogram by dividing each of its components by total number of

pixels in the image, denoted by a matrix�� , where M and N are the row and

column dimension of the image. Thus a normalized histogram is given by the

following equation [36].

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Where �� is an estimate of the probability of the occurrence of intensity

level, � in an image. The sum of all components in a normalized histogram is

equal to 1. The above equation is called probability density function [36]. A

histogram represents the intensity level using X-coordinates going from the darkest

(on the left) to lightest (on the right). Thus, the histogram of an image with 256

levels of grey will be represented by a graph having 256 values on the X-axis and

the number of image pixels on the Y-axis.

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Figure 5.1: A gray level image and its histogram

Figure 5.2: The gray-level histogram for different contrast scenes

Figure 5.2 shows the different characteristics of histogram for different

contrast scene. It is seen that in the dark image the components of the histogram

are concentrated on the low (dark) side of the intensity scale. Similarly, the

components of histogram of the lighter image are biased towards the high side of

the scale. An image with low contrast has a narrow histogram located typically

toward the middle of the intensity scale. For a monochrome image this implies a

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dull, washed-out gray look. The components of histogram of the high contrast

image cover a wide range of the intensity scale. Further, that the distribution of the

pixels is not too far from uniform.

The histogram provides information about the contrast and overall intensity

distribution of an image. Histogram manipulation can be used for image

enhancement, image segmentation, and image registration and image compression.

5.2 Histogram as a tool for image enhancement

There are two primary methods of histogram manipulation for image

enhancement: Histogram Equalization and Histogram Matching.

Histogram Equalization is a technique that generates a gray map which

changes the histogram of an image and redistributing all pixels values [136]. It

allows for areas of lower local contrast to gain a higher contrast. Histogram

equalization automatically determines a transformation function seeking to

produce an output image with a uniform histogram. This method normally

increases the global contrast of images, especially when the functional data of the

image is represented by close contrast values. Through this adjustment, the

intensities can be distributed on the histogram in a better way. Histogram

equalization fulfills this by effectively spreading out the most frequent intensity

values. The probability density function of an image given by [36]

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Based on the probability density function, the cumulative density function

is defined as [36]

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Histogram equalization is a scheme that maps the input image into the

entire dynamic range,��#� $%&� by using the cumulative density function as a

transform function. The transform function '� is defined as [36]

'� # ( $%& � #��

The resulting image has a well distributed histogram with better contrast.

There are two ways of implementing histogram equalization method: non-

adaptive methods, where the transform function remains constant throughout the

image; and adaptive method, where the transform function changes with a

moving window over the pixels.

The second method is the generalization of histogram equalization

technique. Histogram equalization does not allow interactive image enhancement

and generates only one result: an approximation to a uniform histogram. The

method used to generate a processed image that has a specified histogram is called

histogram matching or histogram specification. The procedure can be summarized

as following:

(a) Obtain the histogram of the given image

(b) Apply histogram equalization for the original image.

(c) For the specified histogram, calculate the transform to equalize it. Map the

original histogram to specified histogram by following mapping function:

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))((z 1 rtG −=

Where G-1 is an inverse function.

There are many literatures available suggesting various types of histogram

equalization and matching techniques [136]-[142].

5.3 Histogram based Image Features

Several features of an image can be computed from its histogram. The

image histogram supplies many insights on the character of the image. For

example, a narrowly distributed histogram indicates a low-contrast image. The

following measures have been formulated as quantitative shape descriptions of a

first-order histogram [143]. Some of the typical features are described below:

Mean: The mean value of a set of numbers is simply the average of all of

those values. For a monochrome image, the mean value of the average image is

approximately the middle gray, because the “correct” exposure is assumed to be

one where the dark values balance the bright values, on an average scale. Mean of

an image is defined as [143]

) ��$%&

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Central Moment: An image moment is a certain particular weighted

average of the image pixels intensities, or a function of such moments, normally

chosen to have some interesting property or interpretation. Image moment is useful

to describe objects after segmentation. Simple properties of the image which are

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found via image moments include area, its centroid, and information about its

orientation. Dividing a moment by the 0th order moment, central moment is

obtained. Central moment of an image is defined as [143]

*+� � �)��$%&

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Entropy: The image entropy specifies the uncertainty in the image values.

It measures the averaged amount of information required to encode the image

values. It is a statistical measure of randomness that can be used to characterize the

texture of the input image. An image that is perfectly flat will have an entropy of

zero. Consequently, they can be compressed to a relatively small size. On the other

hand, high entropy images cannot be compressed as much as low entropy images.

It is defined as [143]

, � ��-�./��$%&

�"#

Energy: In Signal Processing, "energy" corresponds to the mean squared

value of the signal. This concept is usually associated with the notion that allows to

think of the total frequency as distributed along "frequencies" and so one can say,

for example, that an image has most of its energy concentrated in low frequencies.

Energy of an image defined as [143]

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5.4 Histogram based Image Segmentation

Histogram-based image segmentation is one of the simplest and most

oftenly used segmentation techniques. It uses the histogram to select the gray

levels for grouping pixels into regions. In a simple image there are two entities: the

background and the object. The background is generally one gray level and

occupies most of the image. Therefore, its gray level is a large peak in the

histogram. The object or subject of the image is another gray level, and its gray

level is another, smaller peak in the histogram [144].

The basic principle of histogram based segmentation is depicted by the

Figures 5.3 below. Figure 5.3 shows an image, its histogram and the segmented

image. The tall peak at gray level 2 indicates the primary gray level for the

background of the image. The secondary peak in the histogram at gray level 8

indicates the primary gray level for the object in the image. The segmented image

consists of pixels of value 8 only, others blanked out. The object is a happy face

emoticon.

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Figure 5.3: Illustration of histogram segmentation

The largest peak represents the background and the next largest peak

represents the object. We choose a threshold point in the valley between the two

peaks and threshold the image. Thresholding takes any pixel whose value is on the

object side of the point and sets it to one; setting all others to zero. The histogram

peaks and the valleys between them are the keys.

The idea of histogram-based segmentation is simple, but there can be

problems in the determination of threshold point. If the point mid-way between the

two peaks (threshold point = 5), is chosen it produces the image of Figure 5.4. This

is not the happy face object desired. Choosing the valley floor values of 4 or 5 as

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the threshold point also produces a poor result. The best threshold point would be

7, but this could be known only by trial and error. This example is comparatively

easy because there are only ten gray levels and the object (happy face) is small.

Figure 5.4: Erroneous segmentation due to wrong threshold

One simple way is to preprocess the image histogram. Histogram

equalization or histogram matching can be used for this purpose. There are three

primary methods for using histogram as a tool for image segmentation. A

technique, which is rarely used, is manual the method of histogram thresholding. In

the manual technique the user inspects an image and its histogram manually. Trial

and error come into play and the result is as good as the user wants it to be Manual

segmentation is good for fine tuning and understanding the operation. Its trial-and-

error nature, however, makes it time consuming and impractical for many

applications [145]. The three methods are discussed below.

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5.4.1 Histogram Peak Technique

This technique finds the two peaks in the histogram corresponding to the

background and object of the image. It sets the threshold halfway between the two

peaks. When the histograms are searched for peaks, peak spacing should be used to

ensure the highest peaks are separated. Another item to watch carefully is to

determine which peak corresponds to the background and which corresponds to the

object [145].

5.4.2 Histogram Valley Technique

The second automatic technique uses the peaks of the histogram, but

concentrates on the valley between them. Instead of setting the mid-point

arbitrarily half way between the two peaks, the valley technique searches between

the two peaks to find the lowest valley [145].

5.4.3 Adaptive Histogram Technique

The final technique uses the peaks of the histogram in first pass and adapts

itself to the objects found in the image in a second pass. In the first pass, the

adaptive technique calculates the histogram for the entire image. It smoothen the

histogram and uses the peak technique to find the high and low threshold values. In

the second pass, the technique segments an image using the high and low values

found during the first pass. Then, it calculates the mean value for all the pixels

segmented into background and object. It uses these means as new peaks and

calculates new high and low threshold values. Now, it segments that area again

using the new values [145].

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5.5 Mathematical Morphology of Image

Mathematical morphology is a well-founded theory of image processing

[146], [147]. Its geometry oriented nature provides an efficient framework for

analyzing object shape characteristics such as size and connectivity, which are not

easily accessed by linear approaches. Morphological operations take into

consideration the geometrical shape of the image objects to be analyzed. The initial

form of mathematical morphology is applied to binary images and usually referred

to as standard mathematical morphology in the literature in order to be

discriminated by its later extensions such as the gray-scale and the soft

mathematical morphology.

Mathematical morphology is theoretically based on set theory. It provides a

wide range of operations to image processing, based on a few simple mathematical

theories. The operators are particularly useful for the analysis of boundary

detection, binary images, image enhancement, noise removal, and image

segmentation. The advantages of morphological operations over linear operations

are straight-forward geometric interpretation, efficiency and simplicity in hardware

implementation. This direct geometric description of set theory makes

morphological approach more suitable for image processing. Mathematical

morphology regards an image as a set and uses another smaller set which is called

as structuring element to probe image. Morphological operations on an image are

defined with regard to a structuring element.

The structuring element consists of a pattern specified as the coordinates of

a number of discrete points relative to some origin. Normally, Cartesian

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coordinates are used and so a convenient way of representing the element is as a

small image on a rectangular grid. The origin is marked by a ring around that point.

Figure 5.5: Some structuring elements

Figure 5.5 shows a number of different structuring elements of various

sizes. In each case the origin is marked by a ring around that point. The origin does

not have to be in the center of the structuring element, but often it is. As suggested

by the figure, structuring elements that fit into a 3×3 grid with its origin at the

center are the most commonly seen type. Morphological operations are based on

basic set theoretic operations. Two set operations translation and reflection of set

are defined before discussing the morphological operations.

5.5.1 Set Operations

Now some definitions of simple set operations are given. The translation

of the set A by the point 1 is defined, in set notation, as:

{ }Aa,acc)A( x ∈−==

For example, if x were at (1, 2) then the first (upper left) pixel in (A)x would

be (3,3) + (1,2) = (4,5); all of the pixels in A shift down by one row and right by

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two columns in this case. This is a translation in the same sense that it seen in

computer graphics, a change in position by specified amount. The reflection of a

set A is defined as:

{ }Aa,acA ∈−==�

Now, the morphological operations are described below.

5.5.2 Dilation

Dilation adds pixels to the object boundary of an image. Let A be a set of

pixels and let B be a structuring element. Dilation is defined as [148]

{ }Bb,Aa,baccBA ∈∈+==⊕

Dilation operation is illustrated by the following example. Let A be the set

of Figure 4.5(a), and let B be the set of {(0, 0) (0, 1)}. The pixels in the set C = A +

B is computed using the last equation which can be rewritten in this case as:

( ) ( )(0,1)A(0,0)A +∪+=⊕ BA

There are four pixels in the set A, and since any pixel translated by (0, 0)

does not change, those four will also be in the resulting set C after computing

C = A + {(0, 1)}:

(3,3) + (0,0) = (3,3) (3,4) + (0,0) = (3,4)

(4,3) + (0,0) = (4,3) (4,4) + (0,0) = (4,3)

The result of A + {(0, 1)} is

(3,3) + (0,1) = (3,4) (3,4) + (0,1) = (3,5)

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(4,3) + (0,1) = (4,4) (4,4) + (0,1) = (4,5)

The set C is the result of the dilation of A using structuring B, and consists

of all of the pixels above (some of which are duplicates). Figure 5.6 illustrates this

operation, showing graphically the effect of the dilation. The pixels marked with

an "X," either white or black, represent the origin of each image. The location of

the origin is important. In this example, if the origin of B were the rightmost of the

two pixels the effect of the dilation would be to add pixels to the left of A, rather

than to the right. The set B in this case would be {(0, −1) (0,0)}

Figure 5.6: Dilation of set A of by set B; (a) Set A (b) the set obtained by adding

(0, 0) to all elements of A; (c) The set obtained by adding (0, 1) to all elements of

A; (d) The union of the two sets is the result of the dilation

5.5.3 Erosion

If dilation can be said to add pixels to an object, or to make it bigger, then

erosion will make an image smaller. In the simplest case, binary erosion will

remove the outer layer of pixels from an object. It is defined as [148]

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( ){ }ABcBA c ⊆=Θ

Figure 5.7: Erosion of A by structuring element B

5.5.4 Opening and Closing

The application of erosion immediately followed by a dilation using the

same structuring element is defined as an opening operation. The name opening is

a descriptive one, describing the observation that the operation tends to "open"

small gaps or spaces between touching objects in an image. This effect is most

easily observed when using the simple structuring element. Figure 5.8 shows

image having a collection of small objects, some of them touching each other.

After an opening using simple the objects are better isolated, and might now

counted or classified [148]

Figure 5.8: Opening: (a) Image with many connected objects, (b) Objects isolated

by opening using simple structuring element, (c) Image with noise (d) The noisy

image after opening.

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When a noisy gray

above the threshold value,

erosion step in an opening will remove isolated pixels as well as boundaries of

objects, and the dilation step will restore most of the boundary pixels without

restoring the noise.

A closing is similar to an op

followed by an erosion using the same structuring element. If an opening creates

small gaps in the image, a closing will fill them, or "close" the gaps.

(a) (b) (c)

Figure: 5.9 (a) Original image

Closing can also be used for smoothing the outline of objects in an image.

Sometimes digitization followed by thresholding can give a jagged appearance to

boundaries; in other cases the objects are naturally rough, and it may be necessary

to determine how rough the outline is. In either case, closing can be used.

However, more than one structuring element may be needed, since the simple

structuring element is only useful for removing or smoothing single pixel

irregularities.

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When a noisy gray-level image is threshold, some of the noise

above the threshold value, and results in isolated pixels in random locations. The



A closing is similar to an opening except that the dilation is performed first,


small gaps in the image, a closing will fill them, or "close" the gaps.

(a) (b) (c)

Original image (b) Opening operation (c) Closing operation




rough the outline is. In either case, closing can be used.



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pixels are

in isolated pixels in random locations. The



ening except that the dilation is performed first,


Closing operation




rough the outline is. In either case, closing can be used.



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5.5.5 Filling

For grayscale images, filling operation brings the intensity values of dark

areas that are surrounded by lighter areas up to the same intensity level as

surrounding pixels. Figure 5.10 depicts the filling operation.

Figure 5.10: Pixels of blue dotted area are changed to match the background

pixels.

5.5.6 Hit or Miss Transform

Hit-or-miss Transform is a powerful tool to detect shapes. It is defined as

the set of matches of structuring element B in image A [148]

234 25 4&�62754/�

Where 4& is the shape to detect and 4/ is the set of element associated with

the corresponding background. The hit-and-miss transform is a general binary

morphological operation that can be used to look for particular patterns of

foreground and background pixels in an image. It is actually the basic operation of

binary morphology since almost all the other binary morphological operators can

be derived from it. The structuring element used in the hit-and-miss is a slight

extension to the type that has been introduced for erosion and dilation, in that it can

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contain both foreground and background pixels, rather than just foreground pixels,

i.e. both ones and zeros. The hit-and-miss operation is performed in much the same

way as other morphological operators, by translating the origin of the structuring

element to all points in the image, and then comparing the structuring element with

the underlying image pixels. If the foreground and background pixels in the

structuring element exactly match foreground and background pixels in the image,

then the pixel underneath the origin of the structuring element is set to the

foreground color. If it doesn't match, then that pixel is set to the background.

5.6 Colour Image Morphology

The extension of mathematical morphology to colour images is not

straightforward. There are two ways representing the morphology of colour image

[149] [150]. The images in consideration are in RGB colour space.

The first approach considers all the colour components of RGB image. In

this component-wise approach, the grayscale morphological operator is applied to

each channel of the color image. The component-wise color dilation of �1� 8� ��91� 8��:1� 8��;1� 8��<�by structuring element

�1� 8� ��91� 8��:1� 8��;1� 8��< in RGB colour space is defined as [149]

�=7��1� 8� ��9=�9�1� 8��:=�:�1� 8��;=�;�1� 8��<

The sign >7 is the component-wise dilation and the sign > on the right

side is gray level dilation. Component-wise color erosion, opening, and closing can

be defined in the same way.

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The second way of computing colour morphology is to consider each pixel

as a vector. Vector-based color morphology makes use of the multivariate ranking

concept. First, reduced ordering is performed. Each multivariate sample is mapped

to a scalar value based on reduced ordering function and then the samples are

ordered according to the mapped scalar value. Given the reduced ordering function

? and the set�,, the value of vector color dilation of � by , at the point 1� 8� is

defined as [149]

�=@��1� 8� A��=@�?B��'BC�DB�'��?E-A'E��

F�BB�A G H�� C�I � C� G ,J

A�?�?A� K ?L�� C�M��A--�� C� G ,N�O�

Similarly, other morphological operations are also defined. Histogram

based features and morphological operations are very useful in image

segmentation. Histogram provides global information of an image as a whole.

Apart from the conventional methods described earlier, there are hosts of

segmentation method based on histogram [151], [152], [153], [154].

Morphological features provide ample opportunities for image

segmentation. The shape information of objects in an image is a very useful aid for

designing a segmentation algorithm. Various segmentation methods exist based on

morphological operations such the Watershed algorithm. Sarkar et al. [155]

proposed a method where region based segmentation is modified with

morphological operations and watershed preprocessing and post-processing to

segment images. A powerful morphological tool for image segmentation is

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watershed transformation [156]. Applying watershed transformation on gradient

image could obtain an initial segmentation, but usually yields an over-segmented

result. Thus, region merging can be performed as the post-processing, and efficient

algorithms have been proposed [157].

Mathematical morphology plays an important role in microscopic medial

images. Morphological cell analysis has been integrated in new methods for

biomedical applications, such as automatic segmentation and analysis of

histological tumour sections [158]. Morphological characteristics analysis of

specific biomedical cells [159], identifying cell morphogenesis in different cell

cycle progression [160] are some examples of morphological operation based cell

image segmentation. Morphological feature quantification for grading cancerous or

precancerous cells is especially widely researched in the literature, such as nuclei

segmentation based on marker-controlled watershed transform, carcinoma feature

extraction and classification, which is important for prognosis and treatment

planning [161], nuclei feature quantification for cancer cell cycle analysis [162],

and using feature extraction including morphological analysis, wavelet analysis,

and texture analysis for automated classification of renal cell [163].

chapter-5 image histogram and morphological...

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