Commutation theorem: Difference between revisions

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Focus recovery from a defocused image is an ill-posed problem since it loses the component of high frequency. Most of the methods for focus recovery are based on depth estimation theory.<ref>
Most depth recovery methods are simply based on camera focus and defocus. Among those approaches, they usually fall in a depth discontinuity problem.</ref> The [[Linear canonical transform]] (LCT) gives a scalable kernel to fit many famous optical effects. Using LCTs to approximate an optical system for imaging and inverting this system, theoretically permits recovery of a defocused image.
 
==Depth of field and perceptual focus==
[[File:Larger aperture.PNG|Effective DOF interval.|frame|The object is put at the different positions whereas causes to effective focus.]]
 
In photography, [[depth of field]] (DOF) means an effective focal length. It is usually used for stressing an object and deemphasizing the background (and/or the foreground). The important measure  related to DOF is the lens [[aperture]]. Decreasing the diameter of aperture increases focus and lowers resolution and vice versa.
 
==The Huygens-Fresnel principle and DOF==
[[File:Huygens-Fresnel field.PNG|frame|The observation points at two different fields]]
The [[Huygens-Fresnel principle]] describes [[diffraction]] of wave propagation between two fields. It belongs to [[Fourier optics]] rather than  [[geometric optics]].The disturbance of diffraction depends on two circumstance parameters, the size of aperture and the interfield distance.
 
Consider a source field and a destination field, field 1 and field 0, respectively. P<sub>1</sub>(x<sub>1</sub>,y<sub>1</sub>) is the position in the source field, P<sub>0</sub>(x<sub>0</sub>,y<sub>0</sub>) is the position in the destination field. The Huygens-Fresnel principle gives the diffraction formula for two fields U(x<sub>0</sub>,y<sub>0</sub>), U(x<sub>1</sub>,y<sub>1</sub>) as following:
 
:<math>\bold U(x_0,y_0) = \frac{1}{j\lambda}\int\!\int \bold U(x_1,y_1) \frac{e^{jkr_{01}}}{r_{01}}\cos\theta dx_1 dy_1</math>
 
where θ denotes the angle between <math> r_{01}</math> and <math> z</math>. Replace cosθ by <math>\frac{r_{01}}{z}</math> and <math> r_{01}</math> by <br />
<math>[(x_0-x_1)^2+(y_0-y_1)^2+z^2]^{1/2}</math>
 
we get
 
:<math>\bold U(x_0,y_0) = \frac{1}{j\lambda z}\int\!\int \bold U(x_1,y_1) \frac{\exp(jkz[1+(\frac{x_0-x_1}{z})^2+(\frac{y_0-y_1}{z})^2]^{1/2})}{1+(\frac{x_0-x_1}{z})^2+(\frac{y_0-y_1}{z})^2}dx_1 dy_1</math>
 
The further distance ''z'' or the smaller aperture ''(x<sub>1</sub>,y<sub>1</sub>)'' causes a greater diffraction.  A larger DOF can lead to a more effective focused wave distribution. This seems to be a conflict. Here are the notations:
 
<blockquote>
* '''Diffraction'''
** In a real imaging environment, the depths of objects comparing to the aperture are usually not enough to lead to serious diffraction.
** However, a long enough depth of the object can truly blurs the image.
* '''Effective Focus'''
** Small aperture, small blurring radius, few wave information.
** Loses details in comparing to a large aperture.
</blockquote>
 
In conclusion, diffraction explains a micro behavior whereas DOF shows a macro behavior. Both of them are related to aperture size.
 
==Linear canonical transform==
As the meaning of “canonical”, the [[linear canonical transform]] (LCT) is a scalable transform that connects to lots of important kernels such as the [[Fresnel]] transform, [[Fraunhofer]] transform and the [[fractional Fourier transform]]. It can be easily controlled by its four parameters, ''a'', ''b'', ''c'', ''d'' (3 degrees of freedom). The definition:
[[File:Imaging system.PNG|frame|A general imaging system with two free space propagation and one thin lens passing]]
 
:<math>L_M(f(u))=\int L_M(u,u')f(u')du'</math>
 
where
 
:<math>L_M(u,u')=\begin{cases}
                \sqrt\frac{1}{b}e^{-j\pi/4}e^{[j\pi(\frac {d}{b}u^2)-2\frac{1}{b}uu'+\frac{a}{b}u'^2]}, &\mbox{if } b\ne 0 \\
                \sqrt{d}e^{\frac{j}{2}cdu^2}\delta(u'-du) ,&\mbox{if } b=0
                \end{cases}
</math>
 
Consider a general imaging system with object distance ''z<sub>0</sub>'', [[focal length]] of the [[thin lens]] ''f'' and an imaging distance ''z<sub>1</sub>''. The effect of the propagation in freespace acts as nearly a [[chirp]] [[convolution]], that is, the formula of diffraction. Besides, the effect of the propagation in thin lens acts as a chirp multiplication. The parameters are all simplified as [[paraxial approximation]]s while meeting the freespace propagation. It does not consider  aperture size.
 
From the properties of the LCT, it is possible to obtain those 4 parameters for this optical system as:
 
:<math>\begin{bmatrix}
        1-\frac{z_1}{f} \quad &\lambda z_0-\frac{\lambda z_0 z_1}{f}+\lambda z_1 \\
        -\frac{1}{\lambda f} \quad &1-\frac{z_0}{f}
      \end{bmatrix}
</math>
 
Once the values of ''z<sub>1</sub>'', ''z<sub>0</sub>'' and ''f'' are known, the LCT can simulate any optical system.
 
==Notes==
{{reflist}}
 
==References==
* M. Haldun Ozaktas, [[Zeev zalevsky|Zeev Zalevsky]] and M. Alper Kutay, “The fractional Fourier transform with applications in optics and signal processing,” JOHN WILEY & SONS, LTD, New York, 2001.
* M. Sorel and J. Flusser, “Space-variant restoration of images degraded by camera motion blur,” IEEE Transactions on Image Processing, vol. 17, pp.&nbsp;105–116, Feb. 2008.
* Jos. Schneider Optische Werke GmbH, “The way a zoom lens works,” Feb. 2008. [Online]. Available: http://www.schneiderkreuznach.com/knowhow/zoom_e.htm. [Accessed: Mar. 9 2008].
* B. Barshan, M. Alper Kutay and H. M. Ozaktas, “Optimal filtering with linear ca-nonical transformations,” Optics Communications, vol. 135, pp.&nbsp;32–36, Feb. 1997.
 
[[Category:Optics]]
[[Category:Image processing]]

Latest revision as of 13:53, 13 August 2014

by Nas, is very fitting and the film agrees with it. The next step is to visit your Word - Press blog dashboard. I thought about what would happen by placing a text widget in the sidebar beneath my banner ad, and so it went. Transforming your designs to Word - Press blogs is not that easy because of the simplified way in creating your very own themes. Also our developers are well convergent with the latest technologies and bitty-gritty of wordpress website design and promises to deliver you the best solution that you can ever have.

Creating a website from scratch can be such a pain. While direct advertising is limited to few spots in your site and tied to fixed monthly payment by the advertisers, affiliate marketing can give you unlimited income as long as you can convert your traffic to sales. Which is perfect for building a mobile site for business use. You can add new functionalities and edit the existing ones to suit your changing business needs. That's a total of 180$ for each Wordpress theme if you sell 3 links and keep the designer link for your own website, or 240$ if you sell all links.

Minor and medium sized corporations also have a lot to obtain by shelling out in a very good website. Note: at a first glance WP Mobile Pro themes do not appear to be glamorous or fancy. This platform can be customizedaccording to the requirements of the business. The first thing you need to do is to choose the right web hosting plan. Premium vs Customised Word - Press Themes - Premium themes are a lot like customised themes but without the customised price and without the wait.

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Instead, you can easily just include it with our bodies integration field in e - Panel. When you sign up with Wordpress, you gain access to several different templates and plug-in that allow you to customize your blog so that it fits in with your business website design seamlessly. The days of spending a lot of time and money to have a website built are long gone. Working with a Word - Press blog and the appropriate cost-free Word - Press theme, you can get a professional internet site up and published in no time at all. As for performing online business, websites and blogs are the only medium that are available to interact with customers and Word - Press perform this work with the help of cross-blog communication tools, comments and full user registration plug-ins.

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