By Kedar Khare

This e-book covers either the math of inverse difficulties and optical structures layout, and contains a overview of the mathematical tools and Fourier optics. the 1st a part of the booklet bargains with the mathematical instruments intimately with minimum assumption approximately earlier wisdom at the a part of the reader. the second one a part of the e-book discusses innovations in optics, quite propagation of optical waves and coherence homes of optical fields that shape the root of the computational types used for snapshot restoration. The 3rd half presents a dialogue of particular imaging platforms that illustrate the ability of the hybrid computational imaging version in improving imaging functionality. a couple of routines are supplied for readers to advance additional knowing of computational imaging. whereas the point of interest of the ebook is basically on optical imaging platforms, the main suggestions are mentioned in a reasonably normal demeanour to be able to offer necessary history for knowing the mechanisms of a various diversity of imaging modalities.

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**Extra resources for Fourier Optics and Computational Imaging**

**Example text**

We will define δ(x) as: ∞ dx δ(x) = 1 −∞ ∞ dx g(x) δ(x − x ) = g(x ). 36) −∞ What is the Fourier transform of δ(x)? We have already seen that: F{ x 1 rect( )} = sinc(2Lfx ). 37) The first zero of the sinc-function is at 2Lfx = 1. When 2L → 0 the first zero effectively occurs at infinity meaning that in the Fourier transform space: F{δ(x)} = 1 for all fx . 38) In other words we may write: ∞ δ(x) = dfx exp(i2πfx x). 39) −∞ Note again that the integral is not valid in the usual sense but is only true in the sense that it represents the idea of an impulse.

Clearly we have to reduce the size 2L over which the average is carried out. We may say that: 1 2L→0 2L ∞ dx I(x) rect( I(0) = lim −∞ x ). 33) 1 x Notice that as the length 2L → 0 the width of the function 2L rect( 2L ) keeps reducing whereas its height keeps increasing such that the area under the curve is unity. This limiting process leads us to an impulse which is also commonly known by the name delta function. We may write: x 1 rect( ). 34) δ(x) = lim 2L→0 2L 2L Although it is commonly referred to as the “delta function” and we will often call it that way, you will appreciate that it is not a function in the usual sense.

The integrand being analytic on and inside the wedge shaped contour, we have the following result: ∞ ∞ 2 dx exp(−πx2 ) = 0 dx exp(−iπ/4) exp(iπx ) − 0 ∞ ⇒ 0 0 1 dx exp(iπx2 ) = exp(iπ/4). 69) The Fourier transform of interest is thus given by: F{exp(iπx2 )} = exp(iπ/4) exp(−iπfx2 ). 70) Note the change of the sign of i in the exponential and the additional phase factor of π/4. 10 Properties of Fourier transform In this section we will discuss several important properties of Fourier transform that will then be used directly throughout the book.