# 物理代写|傅立叶光学代写Fourier optics代考|EE5625

## 物理代写|傅立叶光学代写Fourier optics代考|Usage of $2 D$ Fast Fourier Transform for problems in Optics

Without consideration of any properties of the transformation matrix $\mathbf{F}$, we observe that computing the discrete Fourier transformation would require $N^2$ multiplication operations. A great advance was made by Cooley and Tukey in 1965 when they published an algorithm that utilized the symmetry properties of the $\mathbf{F}$ matrix and reduced the number of multiplication operations to the order of $N \log N$. The corresponding algorithm for computing the DFT is known as the Fast Fourier Transform (FFT). The idea behind FFT algorithm can be traced back in history to a publication by Gauss, but the significance of this received attention after the Cooley and Tukey work. FFT has now become a benchmark against which other algorithms are often compared in the context of their computational complexity. We will not provide detailed discussion behind the FFT theory but only introduce the readers to operational aspects of using FFT algorithms that are now available in widely used computational tools such as MATLAB, SciLab, NumPy, etc.

As we shall see in later chapters of this book, Fourier transforms occur naturally in study of propagation of light waves, optical information processing methods etc. Digital processing methods used in modeling, analysis and synthesis of optical and other imaging systems therefore extensively use the FFT algorithin. In these applications it is important to understand the meaning of results that any standard 2-dimensional FFT tool provides and the correct usage of functions such that the results make sense from Physics point of view.

We will provide an illustration of computation of the 2D Fourier transform of the $2 \mathrm{D}$ rect function. A rect function of size equal to 11 pixels is defined on a $2 \mathrm{D}$ grid of size $255 \times 255$ as shown in Fig. 4.1(a).

## 物理代写|傅立叶光学代写Fourier optics代考|Space-invariant impulse response

The idea of impulse response is particularly useful when the system has an additional property of space-invariance. A system is spaceinvariant if the functional form of the impulse response does not depend on where the delta-function is located in the input plane. The result of shifting the delta impulse in the input plane is simply to translate the response function in the output plane – it will not change its form or shape. If the system is space invariant, we can represent the inpulse response as:
$$h\left(x, y ; x^{\prime}, y^{\prime}\right)=h\left(x-x^{\prime}, y-y^{\prime}\right) .$$
Suppose that a point source of unit strength located at $\left(x^{\prime}, y^{\prime}\right)=$ $(0,0)$ produces an output $h(x, y)$. Space-invariance implies that shifting the input point source to $\left(x^{\prime}, y^{\prime}\right)=(a, b)$ will produce an output $h(x-a, y-b)$. For most common optical imaging systems this space-invariance model is a good approximation – the model is however not exact. For a typical imaging system the space-invariance holds near the optical axis. One may observe distortions in the form of $h$ for point far-off from optical axis. For all practical purposes we will assume that the space-invariance approximation is valid as this will allow us to use the Fourier transform theory – particularly the convolution property – and develop a formalism for understanding and analyzing imaging phenomena. With the space-invariance property the input-output relation for linear systems is a convolution relation:
$$g_o(x, y)=\iint d x^{\prime} d y^{\prime} g_i\left(x^{\prime}, y^{\prime}\right) h\left(x-x^{\prime}, y-y^{\prime}\right) .$$

## 物理代写|傅立叶光学代写傅里叶光学代考|空间不变脉冲响应

$$h\left(x, y ; x^{\prime}, y^{\prime}\right)=h\left(x-x^{\prime}, y-y^{\prime}\right) .$$

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