1. Introduction
As shown in recent work,¹ optimizing digital imaging systems by simultaneously designing both the optical and digital subsystems provides significant advantages over the traditional sequential imaging system design methods. Specifically, simple digital filters can restore the spatial contrast using spatial correlation information about source objects.¹ Such a joint design approach is based on analyzing the entire optical imaging system as a linear system. In this approach, the imaging system is modelled as
(1)
where H represents the system's point spread function, Φ the collection of the optical design parameters (lens thickness, curvatures, glass types, etc), x the ideally captured digital image, and n the noise inherent to photodetection. End-to-end or joint optimization of the optical and digital system is achieved by minimizing the predicted mean-square-error (MSE) as defined by
(2)
where ε represents the statistical expectation operation and R represents the digital filtering subsystem. The statistical expectation operation considers the correlation of the random noise as well as the spatial correlation of the object. Under this framework, both the optical design parameters Φ and the digital filtering subsystem R are varied to find the MSE-optimal imaging system. Basic monochromatic imaging systems designed in this way achieve better contrast at improved signal-to-noise ratios (SNR) while relaxing the optical requirements in terms of aberrations.¹
We extend this concept to include specialized imaging systems in which the objects of interest posses strong spectral correlations. For example, barcode images and many paper documents are typically printed in black and white. The spectral reflectance of these objects are nearly perfectly correlated at every spatial location. In other words, the radiance distribution of the objects is very similar across a range of wavelengths. In the traditional design approach, a system designer would typically utilize a photodetector having a single spectral filter applied to all pixels uniformly since the goals of the imaging system is not include extracting color information. For instance, the imaging system designer might choose a single-filter or monochromatic CMOS or CCD detector array and apply only an infrared (IR) filter to capture a range of wavelengths. In this paper, we explore an alternate approach which applies different color filters to different pixels to segment the spectrum even though the final image is to be grayscale. We call our approach spectrally-coded grayscale imaging.
In this paper, we introduce a new end-to-end design methodology that considers this spectral correlation information during the design of both the optics and the image processing subsystems Our approach relaxes the requirements on the optical aberrations and enhances imaging capabilities, such as extending the depthof- field. First, in Sect. 2 we describe how specialized image processing utilizes the spectral correlation found in grayscale objects to extract information across multiple color channels. Second, in Sect. 3 we describe how to jointly optimize both the optical and the digital subsystems to improve imaging performance and enable new capabilities such as extended depth-of-field imaging. We also illustrate this method for a simple-three lens imaging system. We conclude with some speculations on further directions of this joint design.
