(a) Reference [561] (as cited in [560]) used experimental...

(a) Reference [561] (as cited in [560]) used experimental data measured using a human eye with 3.0 mm pupil diameter to obtain an analytic fit to the line spread function: $I(i)=0.47 \exp \left(-3.3 i^2\right)+0.53 \exp (-0.93|i|)$. The distance on the retina from the fovea is $i$, in arc min. Show that Fig. 11.30 accurately plots this function. (b) Plot on this same set of axes this same function after each point has been broadened by 1 arc min , and compare them. (One relatively crude, way to do this is divide the $x$-axis into bins that are 0.1 arc min wide, so you have a set of rectangles that have heights $I(i)$ and widths 0.1 arc min centered at $i=0.0, \pm 0.1, \pm 0.2, \pm 0.3, \ldots$ arc min. Replace these by rectangles with widths of 1.0 arc min . Sum the contributions at each $i$ and then normalize all points by the value at $i=0$. This can be made less crude by using a smaller width or a broadening function that is smoother than a rectangle.)

(a) Reference [561] (as cited in [560]) used experimental data measured using a human eye with 3.0 mm pupil diameter to obtain an analytic fit to the line spread function: $I(i)=0.47 \exp \left(-3.3 i^2\right)+0.53 \exp (-0.93|i|)$. The distance on the retina from the fovea is $i$, in arc min. Show that Fig. 11.30 accurately plots this function.
(b) Plot on this same set of axes this same function after each point has been broadened by 1 arc min , and compare them. (One relatively crude, way to do this is divide the $x$-axis into bins that are 0.1 arc min wide, so you have a set of rectangles that have heights $I(i)$ and widths 0.1 arc min centered at $i=0.0, \pm 0.1, \pm 0.2, \pm 0.3, \ldots$ arc min. Replace these by rectangles with widths of 1.0 arc min . Sum the contributions at each $i$ and then normalize all points by the value at $i=0$. This can be made less crude by using a smaller width or a broadening function that is smoother than a rectangle.)

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Key Concepts

Line Spread Function

The line spread function (LSF) represents the response of an imaging system to a line object. It characterizes how a point (or line) of light is spread out on the detector, thus providing crucial information about the spatial resolution and quality of the optical system.

Analytic Fitting

Analytic fitting involves finding a mathematical function that matches experimental data. This technique is used to approximate the behavior of physical systems, allowing for easier manipulation, analysis, and comparison with theoretical predictions.

Convolution and Broadening

Convolution is the mathematical process of combining two functions to produce a third that expresses how the shape of one is modified by the other. In the context of image processing, applying a broadening (or smoothing) function—such as a rectangular or other window function—effectively 'blurs' the original data, simulating effects like optical blur or experimental error.

Data Binning and Normalization

Data binning involves dividing continuous data into discrete intervals or bins, which simplifies analysis and visualization. Normalization scales the data so that comparisons are made on a consistent basis, ensuring that key features such as peak values can be directly compared across different conditions or after transformations like broadening.

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