A brief (lets not kid ourselves its long) introduction to the continuous and discrete wavelet transforms. Comments on implementations on the computer using MATLAB and other software is also included.

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The differential wave equation can be used to describe electromagnetic waves in a vacuum. In the one dimensional case, this takes the form $\frac{\partial^2\phi}{\partial x^2}-\frac{1}{c^2}\frac{\partial^2\phi}{\partial t^2} = 0$. A general function $f(x,t) = x \pm ct$ will propagate with speed c. To represent the properties of electromagnetic waves, however, the function $\phi(x,t) = \phi _0 sin(kx-\omega t)$ must be used. This gives the Electric and Magnetic field equations to be $E (z,t) = \hat{x} E _0 sin(kz-\omega t)$ and $B (z,t) = \hat{y} B _0 sin(kz-\omega t)$. Using this solution as well as Maxwell's equations the relation $\frac{E_0}{B_0} = c$ can be derived. In addition, the average rate of energy transfer can be found to be $\bar{S} = \frac{E_0 ^2}{2 c \mu _0} \hat{z}$ using the poynting vector of the fields.

Watermarking technique for the image is an efficient method for protecting copyright image, and also a huge topic in cryptography. In this paper, two spread spectrum watermarking scheme, the Convolution Image-based Model (CIM) and the Exponential Convolution Image-based Model (ECIM) are going to be formulated and discussed. The watermarking experiment result will be shown and discussed, focusing on the attack scheme, protectability, and the information encryption of the watermark. We will show that the convolution image-based model for invisible watermark is weak of protectability, but it is able to hide the information (the size of watermark must be less than the original image) and store inside the image.