Extended Depth of Field through Wavefront Coding

by E. Dowski and W. Cathey

The authors worked on extending the depth of field (DoF) of optical systems, the range of distance from which objects can be imaged in full detail. Beyond this range, the image undergoes blurred, mathematically modeled as a convolution between the in-focus image and a point spread function (PSF) associated with the distance of the object. In the more general definition, an object is also considered to be within the DoF if its image can be ideally restored from what is read from the sensor.

Previously the mainstream approach to DoF extension is to block light at the aperture with an apodizer, but the optical power at the sensor is also decreased, resulting in a much longer exposure. The only prior approach with full aperture was Hausler's focus sweep method. However, it is limited in application due to the requirement to continuously change the focus setting during exposure.

The authors proposed to attach a phase mask to the optical system to achieve a PSF that is invariant to misfocus and beneficial to recovery of the full-resolution image, in the sense that its optical transfer function (OTF) has large values within its passband. Thus, the in-focus image could be fully recovered from the sensed image without knowledge about depth of the object.

To compute the profile of the phase mask, the authors firstly computed the OTF as a function of depth and phase profile of the mask. The OTF at a specific distance was proved to be a slice of the ambiguity function across the center, and the slope of the slice corresponds to the amount of misfocus. Therefore, the PSF produced by an optical system is constant to focal distance only when its corresponding ambiguity function is rotationally invariant over the angular region that corresponds to the extended DoF.

Limiting the profile of the phase mask to be monomial, the authors further derived that it has to be in cubic form. The bandwidth of cubic phase masked(cubic-pm) systems was also analyzed as a function of the monomial coefficient to guarantee that the OTF would not have zeros in its passband.

In experiments the authors compared the half-maximum amplitude and Fisher information of the cubic-pm PSFs to standard ones. Although the cubic-pm design greatly outperforms, this comparison may be unfair to the standard one because insensitivity to focus change is not indispensable to DoF extension. Nevertheless, the above experiments did suggest that cubic-pm design avoids the obstacle of PSF identification.

A comparison of restored images was also performed by simulation under noise-free assumption. Again, the cubic-pm optics appeared superior to the standard one. The key to its success lies in a wider support of the OTF. Acknowledging that noise is unavoidable, the authors estimated the signal-to-noise ratio of cubic-pm systems to be more than 20dB. Still, the experiments could have been more persuading had the authors included results in real optical systems to account for not only noise, but also manufactural imprecision of the phase mask.

In summary, this paper presents a novel solution to DoF extension. Although it was published 15 years ago, its influence in Computational Photography (CP) is still significant. On one hand, the need to maximize the amount of light at the sensor is increasingly emphasized in the area and phase masking remains an unique solution to extended DoF under this constraint till now. On the other hand, this paper highlights the importance of ambiguity function, which has recently been found to be the bridge between light field theory in CP and wavefront optics.