Meddle Detection Based on Morphological Opening

International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 3 (2018) Spl. © Research India Publications. http://www.ripublication.com

NISHA AB1 AND DR.SASIGOPALAN2

1Department of Mathematics Cochin University of Science and Technology Thrikkakara, Kerala, India. [email protected] 2 School of Engineering, Cochin University of Science and Technology Kerala, India. [email protected]

Abstract—Modish technological methods and II.FUNDAMENTAL OPERATORS IN easily accessible image processing softwares can MATHEMATICAL MORPHOLOGY manipulate images without leaving any trace 1 2 of meddling.So the methods to check the au- Consider two complete lattices[1] LC and LC thenticity of photographs are emerged as a new with a partial order relation ≤ greatest lower field of research.Interposing is normally done in bound∨ and least upper bound ∧. Let F be the 1 2 medical images in order to produce false proof set of all mappings from LC to LC and G be the 2 1 to claim insurance benefits. In this paper a new set of all mappings from LC to LC 1 2 technique using morphological opening based F = {f/f : LC −→ LC } , 2 1 on complete lattices is discussed.This provides G = {f/f : LC −→ LC } The complete lattice 2 a futuristic method to enhance the security of structure of LC can be spread out to F and the 1 digital images. structure of LC can be spread out to G For,let f1, f2 ∈ F Index Terms—opening,closing,morphological 1 f1 ≤ f2 ⇔ f1(x) ≤ f2(X), ∀X ∈ LC where≤ is appendage, idempotence 2 a partial order relation on LC . With respect to this ordering F is a complete lattice with greatest lower bound and least upper bound as follows 1 Let X ∈ LC & A ⊆ F I.INTRODUCTION 1 (∨A)(X) = ∨{fi(X)/fi ∈ A} ∀X ∈ LC 1 (∧A)(X) = ∧{fi(X)/fi ∈ A} ∀X ∈ LC Mathematical morphology provides an alterna- Similar definitions are true in the case of G also The tive approach to image processing based on com- fundamental operators in mathematical morphology plete lattices.Earlier photographs were considered are dilation and erosion .According to Jean Serra[2] to be the most reliable evidences in almost all ,these operators are treated as operators between fields.Now a days digital images are very popular same lattices. and photo editing soft wares made interposing Definition 2.1: Letf ∈ F ,then(i) f is called a 1 2 of images very easy.In this situation we need a dilation from LC to LC if and only if f(∨A) = 1 conscientious way to examine the genuineness of ∨{f(X)/X ∈ A} ∀A ∈ LC (ii) f is called an 1 2 digital images.The detection of digital interposing erosion from LC to LC if and only if f(∧A) = 1 has evolved as a prominent area in the field of ∧{f(X)/X ∈ A} ∀A ∈ LC images. A customary way for interposing a digital Similar definitions are valid in the case of operators 1 2 image is to make use of certain image regions in in G. The set of all dilations from LC to LC form order to hide something or to fabricate a new image a complete lattice . This is true in the case of set with different content.Properties of morphological of all erosions also. 1 2 operators is used here to design a method for Definition 2.2:Dilation from LC to LC followed by 2 1 detecting meddles. erosion from LC to LC is called a closing operator

195 International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 3 (2018) Spl. © Research India Publications. http://www.ripublication.com

1 2 and erosion from LC to LC followed by dilation V. MEDDLINGIN MEDICAL IMAGES 2 1 from LC to LC is called an opening operator Usually images related to medical purposes are stored in digital forms.Such images are very easy III.MORPHOLOGICAL APPENDAGE to be altered for felonious motives.For example,an Let f ∈ F and g ∈ G ,then the pair (f,g) is important scintilla concerned with the diagnosis 1 called a morphological appendage between LC of a particular illness can be erased or added 2 and LC if and only if g(Y ) ≤ X ⇔ Y ≤ from a medial image deliberately for insurance 1 2 1 2 f(X), ∀X ∈ LC ,Y ∈ LC . LC and LC are purposes.Now a days due to advanced technology complete lattices,therefore both are partially ordered the interposed image looks like a natural image sets by definition.Hence morphological appendage as possible and the interposing is undetectable by is a Galois connection[1] between two different conventional inspection. Consider a colour image complete lattices or grey scale image.If the image is colour convert it Proposition 3.1 in to grey scale. This conversion reduces the course If (f, g) is a morphological appendage then f is an of action time .Frequently grey scale images are erosion and g is a dilation considered in three dimensions such as x,y and Proposition 3.2 z-axis. The first two represent the pixel positions If (f,g) is a morphological appendage between two and the third is used to denote the intensity of 1 2 complete lattices LC and LC , then gf is an opening each pixels.Take digital image and apply erosion 1 2 on LC and fg is a closing on LC followed by dilation on the image,As a result we Theorem 1 get an opened image I(O) in the language of There exists closing and opening which forms mathematical morphology. morphological appendage between two complete lattices such that they are idempotent VI.ALGORITHMFOR SPOTTING Proof of this theorem follows from proposition 3.3 INTERPOSES Proposition 3.3 (O) Iff ∈ F, g ∈ G the following conditions are • Segment the opened image I into finite equivalent no of blocks IB1,IB2,IB3 etc. of appropriate (i)(f, g) is a morphological appendage dimension based on the purpose (ii)fg ≥ id1 and gf ≤ id2 where id1 is the identity • Divide each block into pixels and calculate 1 2 the pixel values operator on LC and id is the identity operator on L2 • Apply opening on each blocks to get C (0) (0) (0) Also the above two conditions imply IB1 ,IB2 ,IB3 etc. (O) (iii)fgf = f and gfg = g • Divide each block IBi into pixels of same (iv)fg and gf are idempotent dimension as before and again calculate the Remark pixel values. By proposition 3.1, If (f, g) is a morphological • The difference in pixel values indicate the appendage then f is an erosion and g is a dila- presence of meddle 1 (O) (v) tion.Then by definition 2 ,fg is is a closing on LC |IBi − IBi | = I and gf is an opening on L2 C (v) Proposition 3.4 I = 0, no meddling in block Bi 1 1 (v) The closing operator fg : LC −→ LC and the I 6= 0, presence of meddling in block Bi 2 2 opening operator gf : LC −→ LC are idempotent By reducing the dimension of blocks one can IV. IMPORTANCE OF IDEMPOTENCEIN spot pixel wise meddlees and the union of these meddleded pixels gives the meddled area MATHEMATICAL MORPHOLOGY If an operator is idempotent we can apply it only once,and its further application does not bring VII.CONCLUSION any difference in images.For filtering purposes Now a days some camera making companies the property of idempotence is very useful. Non- sell data validation kit with selected models which idempotent operators are used for filtering processes are very costly.It can stamp digital images wit ,there is no convergence guarantee for its repeated inconspicuous mathematical summary of the im- application .Therefore this property is an important age such that even one microscopic change will requirement for filtering operations. Idempotence of produce a mismatch and flag the photo as an opening operators are used here to design interpose alteration. These techniques are of interest to detection law enforcement officials and defence attorneys

196 International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 3 (2018) Spl. © Research India Publications. http://www.ripublication.com

,because digital evidence an make or break cases. From the medical image centre,provide an opened image with block wise calculated pixel values to the customers.Verification sections of insurance agencies can easily test the authenticity of the digital evidences by applying the proposed method.This process is very fast compared to any other similar techniques . Medical fields the reports of patients are decidedly confidential.Medical images are usually presented as a testament for sickness.Since medical insurance field is dealing with large amount of money,there is high possibility of deceptive claims.

REFERENCES [1] G.Birkhoff, (1984). Lattice theory, American Math- ematical Society, Rhode Island [2] Jean Serra, (1982). Image analisis and mathematical morphology,Academic Press,London [3] G.Gierz,K.H.Hofmann,K.Keimel,J.D.Lawson,M.Mislove and D.S.Scott(1980)A Compendium of continuous lattices,Springer-Verlag,Berlin [4] Pierre Soille (2004).Morphological image analysis:principles and applications,Springer- Verlag,Berlin [5] K.Denecke,M.Erne and S.L Wismath (2004)Galois Connections and Applications ,Springer Science + Business Media B.V [6] C.Ronse and H.J.A.M Heijimanns(1990),The Alge- braic basis of mathematical morphology,Computer vision graphics and image processing Vol.50 ,page no. 245-295 [7] C.Ronse,(1990).Why mathematical morphology needs complete lattices Signal processing,Vol.21,page.no 129-154 [8] C.R Giardina and E R Dougherty,(1980)Morphological Methods in Image and Signal Processing Prentice -Hall, Eaglewood Cliffs,NJ [9] H.J.A.M. Heijmans and C. Ronse(1990), The al- gebraic basis of mathematical morphology; Part I: Dilations and erosions, Comput. Vision Graphics Image Process., Vol. 50, No. 3, pp. 245-295. [10] S.R. Sternberg(1986),Grayscale morphology, Com- put. Vision Graphics Image Process., Vol. 35, No. 3, , pp. 333-355.

197