The majority of imaging systems rely on spherical lens surfaces because they are easy to manufacture and test. During the last century, optical system design involved optimizing the shape and materials of spherical lenses. The goal of this approach was to minimize inherent optical aberrations to achieve the highest-contrast image. Recently, joint analysis of both the optical subsystem and the algorithmic capabilities of digital processing have enabled new classes of digital-optical imaging systems.1 This joint design approach can reduce system cost, improve manufacturing yield, and even transcend historic performance limitations. The key challenge in this framework is to efficiently balance optical and image-processing parameters.
We introduce a concept called spherical coding, in which the optical designer deliberately allows or even enhances the spherical aberration. Then, digital post-processing sharpens the blurry captured images, achieving high contrast. Such a framework provides several advantages. In particular, it improves light collection—providing higher signal-to-noise ratios (SNRs)—and boosts manufacturing yield. In addition, spherical coding offers simplicity: the relationship between optical parameters and spherical aberration is well understood, and the system is supported by modern lens design software.
Figure 1. (left) The curves on the graph compare the transfer functions for a spherically aberrated optical system (dashed), a compensating digital filter (dotted), and the combined digital-optical transfer function (solid). A captured image (center) degraded by spherical aberration can be restored (right) after applying the filter.
Our approach works by exploiting the unique properties of spherical aberration and its effect on the modulation transfer function (MTF). Optical aberrations in general reduce image contrast by creating blurry optical images. The system MTF characterizes the optical blur by quantifying the contrast preserved by the imaging system as a function of spatial frequency. Spherical aberration reduces contrast in a distinctive rotationally symmetric, field-independent fashion that requires only simple, space-invariant digital sharpening filters for compensation. The optical MTF, as traditionally defined, preserves contrast (dashed line in Figure 1) throughout the full range of spatial frequencies, enabling a digital sharpening filter to amplify them (dotted line) and restore digital-optical system contrast (solid line). Figure 1 also shows an example of a captured image blurred by spherical aberration and a restored version after digital processing.
Spherical aberration is often the primary aberration limiting the f-number for moderate field-of-view optical systems, since it grows according to 1/(F#)2. While lens bending can reduce the aberration somewhat, optical designers must either rely on expensive higher-index materials, splitting the optical power (by increasing the number of lenses), or limiting the f-number, and hence the light-gathering capacity of the imaging system.2 Spherical coding offers an alternative to increasing the number of lens elements in an optical system to achieve faster systems (lens splitting). For low-light applications in which thermal read noise dominates the SNR, spherical coding offers a new way to improve such performance.
Errors in optical imaging systems arise from a multitude of process variations from fabrication to assembly. Common process variations create surface power, wedge, decentration, and thickness errors as well as element tilt, decenter, and spacing errors.3 Many of the random process variations create either pupil-dependent or field-dependent focus errors that degrade optical MTF at various spatial frequencies. Intentionally adding spherical aberration to an optical system can make its MTF less sensitive to many of these variations due to the moderate depth-of-focus extension associated with spherical aberration.4
As an example, we compared the as-built MTF curves of a traditional and a spherical coded imaging system. The traditional system is a 10mm-focal-length monochromatic F# 4.5 doublet imaging system with minimal wavefront error (less than one wave of spherical aberration) over a 20 degree field of view. The spherical coded system is similar, except we designed it to have high spherical aberration (five waves). We performed 300 Monte Carlo simulations using the standard tolerances provided by optical manufacturers. While the traditional system has high nominal MTF, the random manufacturing errors cause a large variation in system MTFs, with many systems showing very poor curves that are uncorrectable with digital processing (see the left side of Figure 2). The spherical coded system, however, shows much smaller variation in as-built MTFs, which are closer to the medium-contrast nominal MTF. By leveraging the compensation of digital processing, the spherical coded systems show more than double the manufacturing yield of traditionally designed systems.
Figure 2. (left) Dozens of as-built modulation transfer function (MTF) curves (both on-axis and off-axis field points) for a traditionally designed system under typical random process variations. (right) An overlay of dozens of as-built curves for the spherical coded system shows minimal MTF variation, and hence improved manufacturing yield. lp/mm: Line pairs per millimeter.
These examples underscore the importance of approaching electro-optical imaging systems from a digital-optical perspective. Specifically, rethinking the goal of eliminating all aberrations uniformly can improve system-level performance by leveraging digital processing. The benefits of spherical coding demonstrate that the long-held goal of eliminating all aberrations, including spherical aberration, may be counterproductive. Using controlled amounts of spherical aberration can actually improve system performance and manufacturing yield. Future applications of spherical coding, as well as other joint digital-optical design approaches, require low-cost image-processing techniques and efficient computer-aided design tools.
Related work by the authors was presented in the conference Novel Optical Systems Design and Optimization at the SPIE Optics + Photonics symposium in August 2009 in San Diego.
Dirk Robinson, David Stork
Menlo Park, CA
M. Dirk Robinson received his PhD in electrical engineering from the University of California, Santa Cruz, in 2004. He is currently the manager of the Digital-optics Research group investigating the practical applications of computational imaging.
David G. Stork, who earned his PhD in physics at the University of Maryland, is chief scientist. He has published in the field of optics for over two decades, including Seeing the Light: Optics in Nature, Photography, Color, Vision, and Holography. He serves on the program committee for the SPIE Optics and Photonics: Novel Optical Systems Design and Optimization Conference.