# 物理代写|傅立叶光学代写Fourier optics代考|ECE3225

## 物理代写|傅立叶光学代写Fourier optics代考|Ill-posedness of inverse problems

In the previous section we described the commonly used linear systems model that is commonly used in relation to imaging phenomena. With the computational methods forming an integral part of the imaging system, it is very important to discuss some fundamental issues that arise when solving the inverse problem of obtaining the input function $g_i$ using full or partial measurement of the output $g_o$. We first observe that $g_o$ is typically measured using some detector. If we associate with $g_o$ the output wavefield, then any optical detector will typically detect the light intensity that is proportional to $\left|g_o\right|^2$. For simplicity and for the purpose of discussion here we may assume that $g_o$ is measurable in the laboratory, e.g. by means of an interferometric arrangement. It is important to note that the measurement process is inherently statistical in nature and what is usually available to us is the output function including noise $n(x, y)$ which in its simplest form may be assumed to be additive.
$$\tilde{g}_o(x, y)=\iint d x^{\prime} d y^{\prime} g_i\left(x^{\prime}, y^{\prime}\right) h\left(x-x^{\prime}, y-y^{\prime}\right)+n(x, y) \text {. (5.6) }$$
The noise arises out of a combination of statistical nature of detection process as well as due to the fundamental statistical fluctuations associated with light waves themselves. The noise $n(x, y)$ is never ideally zero in any practical system. Let us further consider the nominal form of impulse response $h(x, y)$ associated with a delta-function input. The transfer function $H\left(f_x, f_y\right)$ for any practical system extends over a finite range of spatial frequencies and the impulse response function $h(x, y)$ has a spread which is inversely related to the spread of the transfer function in spatial-frequency space. The forward operation of going from object to image space is thus typically a blurring operation.

## 物理代写|傅立叶光学代写Fourier optics代考|Inverse filter

A simplistic solution to the problem of obtaining the input $g_i(x, y)$ from a measured output $\tilde{g}_o$ is the inverse filter solution. This method is included here as it clearly illustrates the difficulties due to illposedness as discussed in the previous section. Disregarding the noise in the output and using Eq. (5.5), one may nominally write the inverse solution as:
$$g_i(x, y)=\mathcal{F}^{-1}\left[\frac{G_o\left(f_x, f_y\right)}{H\left(f_x, f_y\right)}\right] .$$
The function $1 / H\left(f_x, f_y\right)$ is referred to here as the inverse filter. Unfortunately we do not have access to $G_o\left(f_x, f_y\right)$ but only its noisy version $\tilde{G}_o\left(f_x, f_y\right)=\mathcal{F}\left[\tilde{g}_o(x, y)\right]$. The solution $\tilde{g}_i(x, y)$ estimated using the noisy output may be written as:
$$\tilde{g}_i(x, y)=\mathcal{F}^{-1}\left[\frac{\tilde{G}_o\left(f_x, f_y\right)}{H\left(f_x, f_y\right)}\right] .$$
In numerical computation one may replace any zero value in $H$ with a small constant to avoid dividing by zero. The illustration in Fig. 5.2 shows the effect of a $20 \times 20$ pixel square averaging filter (with $1 \%$ additive noise) on a picture followed by an attempt at image recovery using the simple inverse filter. The filter $h(x, y)$ and the absolute value of the corresponding transfer function $H\left(f_x, f_y\right)$ are shown in Fig. 5.3. The The recovery in Fig. $5.2$ (c) appears completely meaningless. The reason for this is that we have simply divided the Fourier transform of the blurred image (b) by the Fourier transform $H\left(f_x, f_y\right)$. The function $H\left(f_x, f_y\right)$ has zeros (or very small values) over finite regions or isolated lines/points as seen in Fig. $5.3$ and dividing by these small values greatly enhances the corresponding frequency components. In particular in presence of noise some of these components may produce completely undesirable large oscillations in the recovery as seen in Fig. $5.2$ (c).

## 物理代写|傅立叶光学代写傅里叶光学代考|逆问题的病态性

$$\tilde{g}_o(x, y)=\iint d x^{\prime} d y^{\prime} g_i\left(x^{\prime}, y^{\prime}\right) h\left(x-x^{\prime}, y-y^{\prime}\right)+n(x, y) \text {. (5.6) }$$

## 物理代写|傅立叶光学代写傅里叶光学代考|逆滤波器

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$$g_i(x, y)=\mathcal{F}^{-1}\left[\frac{G_o\left(f_x, f_y\right)}{H\left(f_x, f_y\right)}\right] .$$

$$\tilde{g}_i(x, y)=\mathcal{F}^{-1}\left[\frac{\tilde{G}_o\left(f_x, f_y\right)}{H\left(f_x, f_y\right)}\right] .$$

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