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W. Blaschke's Theory Application in Digital Image Processing

O. Samarina*, V. Slavsky

Abstract: Texture classification is one of the basic images processing tasks. In this paper we present a geometrical approach to the images analysis and processing. We introduce topological invariants of the RGB-image based on W. Blaschke's web geometry. The approach to images processing presented in the given work, can be used at the solving of images classification problems, their recognition, and also at construction of statistical methods of group of images processing.

Keywords: invariants, multichannel image, three-webs

Received December 21, 2012; Revised June 05, 2013; Accepted June 06, 2013

Citation: O. Samarina*, V. Slavsky, W. Blaschke's Theory Application in Digital Image Processing. Journal of Mathematical Sciences and Applications, 1 (2). 17-23. January 2013. doi:10.12691/jmsa-1-2-1

1. Introduction

This paper will be discussed methods of texture classification, based on topological images invariants. In the central topics of this paper will be researched such invariants as connection form, element of surface, curvature and RGB-images canonical form. The algorithm of numerical calculation for invariants will be also described by the authors. Additionally there will be introduced the degree of regularity. This characteristic will be analysed for three types of images: picture of Earth surface from space, photo and simulated image.

2. Basic Constructions

Let’s consider a three-channel RGB-image. Suppose that RGB-image is a set of three non-negative functions in a two-dimensional domain. Families of lines for this functions can be written as follows:

Call these three families of lines an image's topographical grid (or three-webs) [1]. The three-webs function is any function nonidentically equal to a constant such that in domain:

W. Blaschke's suggested to consider “topological” differential geometry.

He studied differential-geometrical (the local!) properties of various objects invariant to topological transformations. Thus use of the classical differential geometry forces to be limited transformations set by functions, differentiated sufficient times or analytical.

3. Pfaffian Form

Let be the families of curves:

Multiplying by we get the Pfaffian form .

Curves are defined by differential equation Multiplies are any functions distinct from zero.

Pfaffian forms can be normalized in such a way as the normalization condition (for some)

[picture] | (1)

hold. Pfaffian form is possible to represent as

where

Now we find. For this purpose define,:

Then normalization condition is equivalent to system of equalities

where is scalar product of vectors. Make use vector product and put

From here we will receive vector components:

Clearly

From (1) for form follows

Equation is called element of surface

However depends from normalization condition

External differentials of Pfaffian form for web is differ only by scalar multiplier from element of surface:. It follows from (1) that . Values are called Christoffel symbols for three-web. To find expression for let's enter some designations

In this designations

Construct the Pfaffian form

.

This form is called connection form for web. Normalized can be transformed so [1]:

[picture] | (2)

This way, connection form changes only on full differential. Hence, in force (2) integral

[picture] | (3)

is renormalization invariant. In the same way the norm element of surface is invariant.

The function is called the web curvature. It follows from last equation that

This means that the web curvature is a relative invariant of weight two (e.g. [3]). The integrated Formula (3) reminds the Gauss formula

It follows that

where functions и are

For three-web curvature we have:

4. Connection Form, Element of Surface and Curvature

We may assume that at the point satisfy a condition

Decompose the web function in a series :

Here are private derivatives of the first and second order with respect to.

In the paper [1] connection form, norm element of surface and curvature are defined as three-web invariants:

where :

Curvature for three-channel image can be defined in another way. Let's assume (using local diffeomorphism) that first two functions are coordinate functions , and the third function is three times differentiable function. Using Taylor decomposition third order with centre in the arbitrary point of area we take:

From the discrete grid (numeral images are defined on the discrete grid of points) it's easy to calculate coefficients of Taylor decomposition [2]. For this functions the RGB-images curvature take the form

Thus, the connection form, element of surface and curvature can be used as RGB-images invariants to wide group of transformations at the decision of various tasks of digital image processing.

5. RGB-Images Canonical Form

In [1] it is proved that web-function by some transformations can take the canonical form in a vicinity of any point:

where constants satisfy to condition

The first type of transformations has the form [1]:

That is parameter replacement in each family by means of unequivocal functions (this transformations correspond to channels calibration). The second and third transformations have the following form

where These transformations allow to normalize definitely channels , and web-function in a vicinity of the chosen point .

Let's consider algorithm of reduction to canonical representation for RGB-images web-function. Assume that condition the point is satisfied. Decompose web-function in sedate a number on :

We will apply to functiontransformation (II)

where

Now take such that in decomposition remove elements,,

Thus we delete from decomposition all mixed derivative functions of the second order. Further at calculation we will consider that members , and are absent also function looks like.

Let's enter following replacements

Choose a representation of such what will depend on. For this purpose consider equality

[picture] | (4)

We get system of equations

Solving this system of equation concerning and we receive following expressions:

Further we received and in expression (4).

Now take and , such that ..., in equality (4) on the left and on the right have coincided. Adding the condition to system of 12 equations and ,, to 10 factors, we will receive system of 13 equations on 13 unknown variables:

As a result we will receive required expressions for ..., and ,,.

Using this expressions we can take the canonical representation for :

Canonical form of three-web function is invariant characteristic of the numerical RGB-image (e.g. [1,2]). It can be used at the decision of various tasks in images analysis, such as pattern recognition, analysis of biomedical images and other tasks.

6. Numerical Calculation of RGB-images Invariants

Let's define the algorithm for a numerical finding of three-webs function for three-channel discrete digital image. Decompose to the third order in :

At all points satisfies to condition

Thus values of three-web function in rectangular grid in the size of 5×5 is equal to zero.

Let's delete central and angular points from rectangular grid (black points in Figure 1). Then receive twenty points for definition of three-web function. Set the system from 20 equations on 19 unknown variables:

Let be a row-vector of decompositions coefficients to the third order, and be a zero vector.

Figure 1. Rectangular grid

For a finding of non-trivial numerical decision of the system we will add to the considered list of the equations a following condition for functions:

Define the system of 21 equations in 19 unknown variables. Solving this system by the least squares method we have

where be a pseudo-inverse matrix.

The factors of three-web function decomposition may be used for calculate without effort three-channel images connection form, norm element of surface and curvature.

7. Numerical Calculation of the Parameter a

In this section we introduce approach to numerical calculation of the parameter which is present in canonical representation of RGB - images web-function:

For experiment we will consider three thematic types of images: a picture of Earth surface, photo and simulated image. For each kind of images we will spend a separate series of tests.

To investigate dependence of on a point choice, we will calculate for nine points of the image located on a grid in the size 3×3 (e.g. Figure 2).

For estimation of the received resulting importance we will notice that for continuously differentiated function on a plane from its regularity in the vicinity centre (i.e. the gradient is distinct from zero) follows that the contour lines passing through the vicinity centre breaks its two coherent components. By analogy we will make definition of a regularity discrete function defined on pixels.

We will construct for each pixel of the image a matrix in the size 3×3. The central element of a matrix we will accept for zero. Boundary values will be accepted for “+”, if it is more than central element and boundary values will be accepted for “−” if it is less than central element.

If as a result the set of boundary pixels is divided into two coherent pieces we will name the image regular on the given pixel. If a component of connectivity is two or one we will name the image extended regular. If a component is more than two the image is irregular.

We will define degree of regularity for digital image as probability of that the image is extended regular in casually chosen pixel. The given property of the digital image is important for correct morphological images processing.

Figure 2. Rectangular grid 3×3

Let's estimate probability of coherent areas formation for any stochastic function.

We will display set of boundary pixels in the form of circular area for visual representation (e.g. Figure 3). Number of all possible arrangements of signs equally .

By the direct calculations gets the Table 1.

Table 1. Number of connectivity components

Where is number of arrangements to which corresponds i=1,2,4,6,8 component of connectivity of set of boundary pixels. Hence, for the random digital image regularity degree is equal:

Let's estimate degree of images regularity for various types of the images.

Figure 3. Set of boundary pixels

7.1. Picture of Earth’s Surface

This image is an eight-channel picture of earth’s surface received from Sputnik Landsat 7 (e.g. Figure 4). We will choose for processing three first channel with ranges of the spectral permission 0.45-0.52, 0.52-0.6 and 0.63-0.69, corresponding to blue, green and red colour zones.

Figure 4. Picture of earth’s surface

For this type of images 10 000 tests have been spent, in 61% positive results have been received. That is the set of pixels was divided into one or two coherent areas. Degree of regularity for this image is.

7.2. Photo

The given type of the image is presented of a format jpg (e.g. Figure 5). Pixels values contain in a range [0,255] for three colour channels.

Figure 5. Photo

Degree of a regularity of the given image has made.

7.3. Simulated Image

Simulated image has been generated with use of mathematical package Matlab (e.g. Figure 6).

Degree of regularity for the image is equal.

Using Laplace integration theorem it is easy to convince that the received values can't be random.

Figure 6. Simulated image

8. Conclusion

Automatic processing of the visual information is one of the major directions in the field of artificial intellect and now the greatest attention is being paid to it all over the world. The important part in the theoretical base of processing systems, analysis and identification of images is performed by the images invariants construction device.

Images invariants are the effective characteristics which can be used in the various applied problems. It can be used at biomedical images analysis, geological researches, problems of images classification and their recognition (e.g. [2]).

Acknowledgement

The research was supported by the State Maintenance Program for Young Russian Scientists and the Leading Scientific Schools of the Russian Federation (Grant NSh-921.2012.1) , FCP "Scientific and pedagogical persons of innovative Russia" 2009-2013(No 02.740.11.0457) and FCP (Grant № 8206, 2012-1.1-12-000-1003-014).

Statement of Competing Interests

The authors have no competing interests.

References

[1]   | Blaschke, W., Einfuhrung in die Geometrie der Waben, Russian transl.: GITTL, Moskva, 1959.
[2]   | Samarina, O. V., Group images invariants, Germany: Lambert Academic Publishing, 2010.
[3]   | Samarina, O. V., W. Blaschke's curvature as RGB-images invariant. [Online]. Available: http://conference.kemsu.ru/ conf/GA2011.