3. Grayscale Triplet Imaging System
In the previous sections, we explained how prior information about spectral correlation allows the use of spectrally encoding depth information to extended the imaging system's depth-of-field. In this section, we analyze a triplet lens design based on this spectral-coding design principle. Specifically, we compare the performances of a triplet imaging systems designed using the traditional, single color filter architecture with one designed using the digital-optical design framework and spectral coding.
3.1. Triplet Specifications
The triplet specifications correspond roughly to a 40 degree field-of-view (FOV) VGA web-camera specifications and a 1/5" sensor as shown in Table 1. The triplet system comprises two glass spherical elements and a third plastic aspheric element which corrects field errors. The plastic element is defined by even-ordered aspheric surfaces up to the 8th order rotationally-symmetric polynomial. The 20 optical design variables include the spherical lens curvatures, the aspheric terms, the lens and air thicknesses, and the glass or plastic types.
| sensor size | 1/5" |
| resolution | VGA |
| pixel pitch | 4.5 µm |
| spectral range | 0.47...0.63 |
| focal length | 4.75 mm |
| FOV | 40° |
| Glass Types | Glass, Glass, Plastic |
| Max. Chief Ray Angle | 16° |
| Track Length | ≤ 7 mm |
| Max. F# | 3.0 |
| Max. Distortion | 3.0% | Table 1. Triplet system specifications |
|---|
3.2. Traditional Triplet
First, we used a traditional methods to optimize the triplet system focussing three test wavelengths (0.48, 0.54, 0.62 nm) onto a single focal plane at a working distance of 750 mm. We achieved this by balancing glass types in order to minimize chromatic aberration. The merit function used to optimize the optical system was based on the RMS optical path difference (OPD) wavefront error. The upper left side of Fig. 1 shows this aberration minimizing design. After global optimization, the design form followed a traditional positive-negative-positive triplet form in crown-flint-crown glass. After optimization using the Schott catalog the glass types are N-FK51A and N-SF10. The plastic is a high index, low dispersion COC type plastic E48R.
We find that the design provides acceptable performance at F# 3.0. Below this f-number, the lens begins to suffer from a loss in contrast over the range of wavelengths. The curves below the graph in Fig. 1 show the field curvature for the three different RGB test wavelengths. The curve shows that the optical system does a reasonable job of focussing all three color channels onto a single focal plane. To achieve this, however, the system suffers from a bit of astigmatism.
3.3. Spectral Coding Triplet
In the second design approach, we assume that the sensor uses a standard set of RGB color filters to segment the spectrum. We optimized the optical design using a merit function based on the average MSE over a range of seven depth locations according to Eq. 7. We use a simple spatial covariance model where the covariance between neighboring pixels is given by 0.9k where k is the spatial separation in pixels.5 We assume the systems's SNR is 40 dB. We achieve joint optimization of both the optical and digital processing subsystems by using the user-defined operand capability of Zemax, a commercially-available lens design software tool. We created a user defined operand to compute the predicted MSE according to Eq. 7. In this fashion, we can leverage the optimization capabilities of the Zemax lens design software.1
The depths were chosen to uniformly sample the depth-of-focus range for a nominal object distance of 750 mm. The depth locations used during optimization were infinity, 2000, 1000, 750, 380, 255, 190, and 150 mm. Again, we performed global optimization over the optical design parameters using the traditional triplet as the starting design.
The resulting design is shown in the upper right side of Fig. 1. The triplet form again follows a positive-negative-positive design form. The glass types, however, are both high index flints N-SF6 and LASF32. The plastic is also a low Abbe number polycarbonate plastic which is much less expensive than the high Abbe number COC plastic used in the traditional design. The design achieves increased light gathering capacity (1.5X) over the traditional design. The curves in the bottom right of Fig. 1 show the field curvature for the three color wavelengths. The field curvature plots show the strong separation between the three wavelengths focal plane due to strong axial color aberrations in the spectrally-coded system.
Figure 1. The lens on the left represents the traditional optical design approach which focusses all three wavelengths at a single focal plane. The bottom left curve shows the field curvature plots for the three test wavelengths (RGB). The traditional optical system brings the three color planes into focus at nearly the same focal plane. The system does, however, suffer from a bit of astigmatism. The spectral-coding design on the right achieves increased light gathering capacity (1.5X). The field curvature plots show the strong separation between the three wavelengths focal plane due to strong axial color aberrations.
3.4. Depth-Of-Field Comparison
We compare the effective depth-of-field performance of our traditional, single channel, triplet and our spectrally-coded triplet. In grayscale imaging the final image quality depends on the spectral sensitivity of the detectors. The top graphs of Fig. 2 show the spectral sensitivities for the single channel detector (left) and the spectrally-coded system (right). Both system cover same spectral range and reflect the typical sensitivities of commercially-available sensors combined with IR cutoff filters.
The curve in the bottom left of Fig. 2 shows the through-focus polychromatic MTF for the traditional system using a set of nine wavelengths weighted by spectral sensitivity of the single channel sensor. The through-focus polychromatic MTF shows the MTF at 50 lp/mm spatial frequency for the on-axis field point. As we would expect, the MTF falls off as we move away from the focal plane due to defocus. The system shows a maximum depth-of-focus of about 120 µm. While this design provides reasonable quality at the proper focal distance, the limited depth-of-field shows that the imaging system will provide defocussed images at a working distance of about 250 mm which would require a focal shift of about 150 µm. Unfortunately, a fixed-focus lens system designed under this constraint will work only within a particular depth range around the chosen object distance. Furthermore, the depth-of-field decreases with decreased F# (and hence light sensitivity) creating an undesirable tradeoff.
The curves on the bottom right show the polychromatic MTF curves for the spectrally-coded system. The MTF again reflects a polychromatic average over different sets of nine spectrally-weighted wavelengths. As expected, the different color channels focus at different depth planes. The depth-of-focus for the spectrally coded system is extended to about 240 µm. Over the focal range, however, at least one of the color channels provides strong contrast. Furthermore, for every depth plane, at least one of the wavelengths has significantly poor contrast suggesting the ability to infer object depth using color channel image sharpness. Also, the spectrally-coded triplet has an increased light gathering capacity of F# 2.4.
Figure 2. The top left curve shows the spectral sensitivity of the single-channel sensor used in the traditional imaging system. The bottom left curve shows the polychromatic MTF of the traditional triplet system using nine spectrally-weighted wavelengths for the on-axis field point. The system shows reasonable performance within about ± 60 µm from the nominal focus position. The top right curve shows the spectral sensitivity for the three color channel spectrally-coded triplet system. The three curves on the bottom right show the polychromatic MTF of the using different spectrally-weighted wavelengths according the color channel sensitivities for the on-axis field point. The three curves reveal the three different color focal planes. The thick line shows the effective MTF combining the best MTF among the three color channels. The system shows that at least one of the color channels is in focus within ± 120 µm of the nominal focal distance.
Image Simulation Results
We simulated images produced by these two systems using our imaging system simulation tool.¹ Our image simulation tool is similar to that described in6 with the extension of adding multi-spectral weighting according to the pixel spectral sensitivity. We use a traditional Air Forces resolution target as our simulated object, a binary target having either uniformly broadband radiance or none simulating a perfectly correlated object. When simulating the captured images, we use three spectral samples per color channel to simulate the spectral integration of the detector.
Figure 3 compares portions of the target at two different object depths. We show a cropped portion of the image so as to reveal the resolution properties of the image. The leftmost image column shows the target at 1.5 meters (top) and 130 millimeters (bottom) for the traditional single channel imaging system. The image shows good contrast at 1.5 meters, but very poor contrast at 130 millimeters due to depth-of-field limitations. The images in the second and third columns show the same images for the Red channel (middle) and Blue channel (right) at the two object depths. At 1.5 meters, the red image shows almost equivalent contrast to the single channel imaging system while the blue image shows very low contrast. Alternatively, the red image shows very low contrast when the object is at 130 millimeters, while the blue image shows sharp contrast. These images visualize the contrast predicted by the through focus MTF shown in Fig. 2.
Figure 3. The left column shows the images of a resolution target produced by the traditional single channel imaging system for an object located at 1500 mm (top) and 130 mm (bottom). The system shows good contrast at 1500 mm but very low contrast at the short working distance of 130 mm due to limited depth-of-field. The second and third columns shows the red and blue channel images respectively. The red image shows good contrast at 1500 mm while the blue image shows good contrast at 130 mm.
To evaluate the ability to discern object depth using the spectrally-coded imaging system, we simulated imaging the resolution target located at 2 m, 1 m, 750 mm, 380 mm, 250 m, 190 mm, and 130 mm. We then applied a simple Laplacian sharpness filter to each of the color channel images and compared the relative magnitude of the filtered images for a small patch near the center of the image. To compute the relative magnitude, we first integrate the energy in the filtered images for a 50 × 50 pixel patch at the center of the image for three color channels. Then, we normalize the three color channel values so that the sum of the energies equals one. This approximates the percentages of the total high frequency image energy present in the three different color channel images. Figure 4 compares the relative sharpness for the three color channel images as a function of object depth. The images in the left and right columns show the magnitude of the Laplacian filtered images for the object located at 130 mm and 1.5 m respectively. The curves demonstrate the clear relationship between object distance and relative sharpness. The simplest application of the relative sharpness is to find the sharpest image over the collection of image planes to use as the captured image.
Figure 4. The curve shows the relative sharpness for at the center of each color channel image versus the distance of the object from the camera. The sharpness was computed as the average magnitude of the color channel filtered by a Laplacian filter. The column of images at the left and right visualize the magnitude of the Laplacian filtered images at 130 mm and 1.5 m respectively. The curves demonstrate the clear relationship between object distance and the computed sharpness metric.
