Commutation theorem

2013-07-28T22:00:17Z

80.121.86.210:

{{context|date=October 2009}}

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. P1(x1,y1) is the position in the source field, P0(x0,y0) is the position in the destination field. The Huygens-Fresnel principle gives the diffraction formula for two fields U(x0,y0), U(x1,y1) 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 
<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 ''(x1,y1)'' 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 ''z0'', [[focal length]] of the [[thin lens]] ''f'' and an imaging distance ''z1''. 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 ''z1'', ''z0'' 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. 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. 32–36, Feb. 1997.

[[Category:Optics]]
[[Category:Image processing]]

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Commutation theorem