Focus recovery based on the linear canonical transform

Focus recovery from defocused image is an ill-posed problem since it loses the component of high freqency. Most of the methods for focus revocery are based on depth estimation theory [Most of depth recovery methods are simply based on camera focus and defocus. Among those approaches, they usually fall in a depth discontinuity problem.] . 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 inverse this system we can theoretically recover a defocused image.

Depth of field (DOF) relating to perceptual focus

In photography, depth of field (DOF) means an effective focused interval. It is usually used for stressing an object and relatively ignoring the background (or the foreground). The important instrument related to DOF is the lens aperture, while we decrease the diameter of aperture the hole scene will get more perceptually focused whereas is low resolution due to its less optical information than a larger aperture.

Relation between the Huygens-Fresnel principle and DOF

The Huygens-Fresnel principle describes diffraction of wave propagation between two different fields. It belongs to Fourier optics comparing to geometric optics.The disturbance of diffraction depends on two circumstance parameters, the size of aperture and the distance between two fields.

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:

:

where θ denotes the angle between $r\_\{01\}$ and $z$. Replace cosθ by $frac\{r\_\{01\{z\}$ and $r\_\{01\}$ by
$[(x\_0-x\_1)^2+(y\_0-y\_1)^2+z^2]\; ^\{1/2\}$

we get

:

The farer distance "z" or the smaller aperture "(x₁,y₁)" causes to a serious disturbance. That is, a larger DOF also causes to a serious diffraction. Whereas a larger DOF can lead to a more effective focused wave distribution. This seems to be a conflict. Here are the notations:

*Diffraction
**In a real imaging environment, the depths of objects comparing to the aperture are usually not far enough to lead to a serious diffusion (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.

In conclusion, we can give a summary that a diffraction explains a micro behavior whereas the DOF shows a macro behavior. Both of them are related to aperture size.

Optical system referred to the linear canonical transform (LCT)

As the meaning of “canonical”, the linear canonical transform (LCT) is a scalable transform that build a connection to lots of important kernels such as the Fresnel transform, Fraunhofer transform and the fractional Fourier transform and so on. It can be easily controlled by its 4 parameters, "a", "b", "c", "d" (3 degrees of freedom). The definition:

:$L\_M(f(u))=int\; L\_M(u,u\text{'})f(u\text{'})du\text{'}$

where

:

Consider a general imaging system. There exists an 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. Most importantly, the chirp convolution is an approach among the paraxial approximation. The parameters are all simplied as paraxial approximation while meeting the freespace propagation. It actually lacks of the consideration on aperture size!

From the properties of the LCT, we can get those 4 parameters for this optical system as:

:

Once the values of "z₁", "z₀" and "f" are known, the LCT can simulate any optical systems.

Combine the LCT with the Huygens-Fresnel principle

Notes

References

*M. Haldun Ozaktas, 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.