Focus recovery based on the linear canonical transform

For digital image processing, the 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.

[1] The Linear canonical transform (LCT) gives a scalable kernel to fit many well-known optical effects.

In photography, depth of field (DOF) means an effective focal length.

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 describes diffraction of wave propagation between two fields.

The disturbance of diffraction depends on two circumstance parameters, the size of aperture and the interfiled distance.

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

A larger DOF can lead to a more effective focused wave distribution.

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 approximations while meeting the freespace propagation.