Watermarking of Digital Image Based on Complex Number Theory

In the paper it is demonstrated that a valid path to a contradiction in complex number theory exists. In the path use is made of Euler’s identity and simple trigonometry. Each step can be easily verified.

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Complex Number Theory without Imaginary Number (i)

OALib ◽

10.4236/oalib.1100856 ◽

2014 ◽

Vol 01 (07) ◽

pp. 1-13

Author(s):

Deepak Bhalchandra Gode

Keyword(s):

Number Theory ◽

Complex Number ◽

Imaginary Number

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Formalization of the Complex Number Theory in HOL4,

Applied Mathematics & Information Sciences ◽

10.12785/amis/070135 ◽

2013 ◽

Vol 7 (1) ◽

pp. 279-286 ◽

Cited By ~ 1

Author(s):

Zhiping Shi ◽

Liming Li ◽

Yong Guan ◽

Xiaoyu Song ◽

Minhua Wu ◽

...

Keyword(s):

Number Theory ◽

Complex Number

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Characteristic Functions

10.1093/oso/9780192844507.003.0011 ◽

2021 ◽

pp. 213-234

Author(s):

James Davidson

Keyword(s):

Number Theory ◽

Characteristic Function ◽

Complex Number ◽

Random Variables ◽

Infinite Divisibility ◽

Characteristic Functions ◽

Conditional Distributions ◽

Inversion Theorem ◽

Sums Of Random Variables

This chapter begins with a look at convolutions and the distribution of sums of random variables. It briefly surveys complex number theory before defining the characteristic function and studying its properties, with a range of examples. The concept of infinite divisibility is introduced. The important inversion theorem is treated and finally consideration is given to characteristic functions in conditional distributions.

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Complex number theory

Dynamics of Offshore Structures ◽

10.1016/b978-0-408-01074-0.50015-x ◽

1989 ◽

pp. 378-379

Keyword(s):

Number Theory ◽

Complex Number

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An Overview of Digital Image Processing and Recognition (Invited Paper)

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100055837 ◽

1982 ◽

Vol 40 ◽

pp. 700-702

Author(s):

R. C. Gonzalez

Keyword(s):

Image Processing ◽

New York ◽

Digital Image Processing ◽

Digital Image ◽

Practical Implementation ◽

Submarine Cable ◽

Vigorous Growth ◽

Digitized Pictures ◽

Processing Techniques ◽

And Storage

Interest in digital image processing techniques dates back to the early 1920's, when digitized pictures of world news events were first transmitted by submarine cable between New York and London. Applications of digital image processing concepts, however, did not become widespread until the middle 1960's, when third-generation digital computers began to offer the speed and storage capabilities required for practical implementation of image processing algorithms. Since then, this area has experienced vigorous growth, having been a subject of interdisciplinary research in fields ranging from engineering and computer science to biology, chemistry, and medicine.

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A method to measure pores and fissures in geologic materials under S.E.M. by digital image processing

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100108209 ◽

1978 ◽

Vol 36 (1) ◽

pp. 212-213

Author(s):

L. Montoto ◽

M. Montoto ◽

A. Bel-Lan

Keyword(s):

Image Processing ◽

Optical Microscopy ◽

Digital Image Processing ◽

Digital Image ◽

Digital Processing ◽

Free Surfaces ◽

Geologic Materials ◽

Automatic Information ◽

Detector Output ◽

Rock Specimens

INTRODUCTION.- The physical properties of rock masses are greatly influenced by their internal discontinuities, like pores and fissures. So, these need to be measured as a basis for interpretation. To avoid the basic difficulties of measurement under optical microscopy and analogic image systems, the authors use S.E.M. and multiband digital image processing. In S.E.M., analog signal processing has been used to further image enhancement (1), but automatic information extraction can be achieved by simple digital processing of S.E.M. images (2). The use of multiband image would overcome difficulties such as artifacts introduced by the relative positions of sample and detector or the typicals encountered in optical microscopy.DIGITAL IMAGE PROCESSING.- The studied rock specimens were in the form of flat deformation-free surfaces observed under a Phillips SEM model 500. The SEM detector output signal was recorded in picture form in b&w negatives and digitized using a Perkin Elmer 1010 MP flat microdensitometer.

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Digital image processing methods applied to the study of annealing time on semiconductor-metal eutectic composites

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100106302 ◽

1988 ◽

Vol 46 ◽

pp. 848-849

Author(s):

J. Hefter

Keyword(s):

Image Processing ◽

Digital Image Processing ◽

Digital Image ◽

Electronic Device ◽

Processing System ◽

Average Diameter ◽

Eutectic Solidification ◽

Metal Silicide ◽

The Matrix ◽

Microscopic Studies

Semiconductor-metal composites, formed by the eutectic solidification of silicon and a metal silicide have been under investigation for some time for a number of electronic device applications. This composite system is comprised of a silicon matrix containing extended metal-silicide rod-shaped structures aligned in parallel throughout the material. The average diameter of such a rod in a typical system is about 1 μm. Thus, characterization of the rod morphology by electron microscope methods is necessitated.The types of morphometric information that may be obtained from such microscopic studies coupled with image processing are (i) the area fraction of rods in the matrix, (ii) the average rod diameter, (iii) an average circularity (roundness), and (iv) the number density (Nd;rods/cm2). To acquire electron images of these materials, a digital image processing system (Tracor Northern 5500/5600) attached to a JEOL JXA-840 analytical SEM has been used.

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