View source for Focus recovery based on the linear canonical transform

{{technical|date=October 2009}}
For [[digital image processing]], the [[Deblurring|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 well-known 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|alt=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 interfiled 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>\mathbf U(x_0,y_0) = \frac{1}{j\lambda}\int\!\int \mathbf 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 cos&nbsp;θ 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>\mathbf U(x_0,y_0) = \frac{1}{j\lambda z}\int\!\int \mathbf 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 many important kernels such as the [[Fresnel]] transform, [[Joseph von Fraunhofer|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==
* {{cite book
  |author1=Haldun M. Ozaktas |author2=[[Zeev Zalevsky]] |author3=M. Alper Kutay
  |title     = The fractional Fourier transform with applications in optics and signal processing
  |publisher = [[John Wiley & Sons]]
  |location  = New York
  |year      = 2001
  |isbn      = 978-0-471-96346-2
  }}
* 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.
* {{cite web
  |publisher   = Jos. Schneider Optische Werke GmbH
  |url         = http://www.schneiderkreuznach.com/knowhow/zoom_e.htm
  |title       = The way a zoom lens works
  |date        = February 2008
  |archiveurl  = https://web.archive.org/web/20120508065349/http://www.schneiderkreuznach.com/knowhow/zoom_e.htm
  |archivedate = 2012-05-08
  }}
* B. Barshan, M. Alper Kutay and [[Haldun Ozaktas|H. M. Ozaktas]], "Optimal filtering with linear ca-nonical transformations", ''Optics Communications'', vol. 135, pp.&nbsp;32–36, Feb. 1997.

[[Category:Image processing]]