2.1.

Cameras Under Test

The two hyperspectral imagers (HSIs) evaluated in this study, referred to as HSI V4 and HSI V6, are pushbroom imagers constructed from COTS components. These were developed for engineering purposes during the Hyperspectral Cubesat for Ocean Observation 1 (HYPSO-1) project. Detailed information about the design and construction of these instruments can be found in Refs. 1 and 2 for the HSI V4 and HSI V6, respectively. Both cameras share the same basic optical design, featuring a transmissive grating as the dispersive element, a high-precision slit, a collimator lens, and a front lens. The cameras are also equipped with the same focal plane arrays for data capture. However, the optical components differ in size: HSI V4 is equipped with smaller S-mount optics, whereas HSI V6 utilizes larger C-mount optics, allowing it to collect more light than HSI V4. All the optics are objective lenses intended for use with machine vision cameras operating in the visible spectral range. The main characteristics of the instruments are summarized in Table 1.

Table 1

Specifications of the HSI V4 and HSI V6.

As the lenses in the cameras are intended for commercial color imaging with broadband illumination, significant deviations from ideality must be expected when they are used in a hyperspectral camera. Notably, the resolution of such lenses is specified in terms of their modulation transfer function (MTF) averaged over the visible spectral range. Behind this average is normally a significant wavelength dependence of image sharpness, as the results below illustrate.

2.2.

Measurement Set-Up

Our measurement setup builds on the work presented in Ref. 9. A preliminary version was reported in Ref. 10. A schematic representation of the setup is shown in Fig. 1. The measurement technique is based on a line source scanned across a pixel while recording data, producing a 1D LSF. By scanning in different directions, it is possible to reconstruct the full 2D shape of the SPSF in each band for the set of pixels covered by the scan.

Fig. 1

Schematic drawing of the SPSF measurement setup

The set-up consists of a collimator (Inframet CDT 11100HR off-axis reflective collimator), a slit ($22\mu \mathrm{m}\times 25\mathrm{mm}$ slit, metal film on glass), and a light source (Inframet LS-DAL). This light source combines two white LEDs and a 2856K halogen bulb in a mixed mode to achieve a consistent spectrum across the visible and near-infrared bands. To minimize external stray light, 3D-printed covers shield the setup’s components.

The slit is mounted on a translation stage that rides on a 360-deg rotating stage with a central optical port. This configuration allows the slit to be scanned across the optical port in any direction. The setup lacks a hardware reference to indicate the scan center position where the slit intersects the rotation stage’s axis. Instead, scans are repeated at an even number of angles around the full circle so that each LSF is recorded twice in opposing directions, enabling determination of the scan center.

Custom-made software controls the scan motion. The camera under test receives hardware trigger input from the motors, thereby syncing the data acquisition with the motor steps. The cameras are controlled by a separate software that captures data with predefined settings (exposure time, pixel clock) and without any preprocessing.

Initial tests were conducted to optimize the scan and camera settings (Table 2). The HSI V4, with its smaller optical components, gathers less light and thus requires a longer exposure time. This limitation reduces the frame rate, consequently extending the overall measurement time. In contrast, the HSI V6, benefiting from larger optics, can operate at shorter integration times, allowing for more rapid data collection.

For each camera 36 LSF, scans were recorded, with a 10-deg step between scans. The LSF scan length is chosen $\sim 15$ times longer than the pixel-to-pixel distance for both cameras. The LSF scans are oversampled to enable noise reduction by binning. The HSI V4 captured 12,000 images per LSF scan, whereas the HSI V6 recorded 10,000 images. Thus, each LSF recording has several hundred points within the nominal pixel width and extends well into the tails of the LSF for the neighborhood of pixels in the center of the scan area. Measurements were made at the center and the edge of the field of view (FOV) for each camera.

Table 2

Parameters and camera settings used during measurements.

2.3.

Data Analysis

The data processing approach follows the methodology described in Ref. 10. Initially, pairs of LSFs recorded in opposing directions are used to determine the scan center. This is achieved by reversing one of the datasets and finding the offset that maximizes the correlation between them. Once the offsets are applied, all LSFs are centered. To mitigate noise, groups of 10 frames are averaged, yielding 1000 and 1200 samples for each LSF after binning. Given that actual LSFs tend to be smooth due to diffraction blur, a Savitzky–Golay filter is applied to the binned LSFs to further reduce high-frequency noise.

The number of frames needed to cover one pixel during a scan is estimated from the number of frames between the LSF peaks of neighboring pixels. The average distance (in frames) between all the detected peaks serves as the conversion factor from frames to pixels in the subsequent plots.

Resolution and coregistration in the along-track and across-track directions are characterized using the LSFs in each direction. Typically, the resolution is quantified by the full width at half maximum (FWHM) of the LSF peak. In accordance with the P4001 draft, LSF peak width is here determined as the FWHM of a Gaussian having the same second moment as the measured LSF. The conventional keystone metric for spatial coregistration between bands in a pixel is expressed as the difference in centroid position between bands.

Finally, the full 2D SPSFs are reconstructed from the measured LSFs through tomographic reconstruction, as described in Refs. 9 and 11. This reconstruction uses the LSFs as 1D projections at various angles to build up a 2D image of the SPSF for each band. Noise can affect the results and is potentially amplified in the reconstruction process. (See Ref. 10 for more details on the exact reconstruction process used here.) The total enclosed energy is calculated as the integral of the SPSF over the image plane. To minimize residual noise, all values below a threshold corresponding to 5% of the total enclosed energy in the SPSF are removed. The resulting SPSFs for each band in a pixel enable evaluation of coregistration error between all band pairs. The 5% truncation threshold was set after an initial estimation of coregistration errors: As shown in Fig. 8, the cameras exhibit coregistration errors significantly larger than 5%; therefore, the distortion of the results due to truncation of the SPSF is significantly smaller than the coregistration errors within the main part of the SPSF. For a camera with a lower level of coregistration error, the truncation threshold would have to be lower, possibly necessitating a longer total measurement time.

3.1.

Spatial Resolution Across and Along Track

Examples of measured LSFs across and along track are shown in Fig. 2 for a set of wavelengths within the nominal spectral range of 400 to 800 nm for both HSI V4 and HSI V6. Measurements were taken from the center of the FOV for both imagers.

Fig. 2

LSFs for bands within the nominal spectral range for (a) HSI V4 across-track, (b) HSI V6 across-track, (c) HSI V4 along-track, and (d) HSI V6 along-track. LSFs are normalized to unit integral.

Figure 3 plots the FWHM along and across track for all bands as a function of wavelength. Sharp spikes in the plots occur because the LSF in those particular bands exhibits significant extra noise compared with neighboring bands. This noise caused the weighted second moment (variance) to be overestimated, resulting in a significantly larger FWHM for those bands.

Fig. 3

Spatial resolution estimated for the along- and across-track direction for (a) HSI V4 and (b) HSI V6. Dashed gray line indicates relative variations in signal strength over the spectral range.

The figure also shows the variation of raw signal level, integrated over the LSF. Signal levels are significantly reduced below 450 nm and above 650 nm due mainly to three factors: the anti-reflective coatings in the optical components are optimized for the visible spectral range, the diffraction efficiency is best for this spectral range, and focus across the spectral range is uneven. (The focus was optimized for shorter wavelengths.) In addition, the quantum efficiency of the image sensor is highest in the 450 to 650 nm range.

Significant variations in sharpness are observed in the across-track direction, clearly resulting from chromatic aberrations in the COTS objective lenses used. Structures in LSFs, such as the triple peak at 550 nm seen for V4 in the across-track direction, are also likely due to chromatic aberration. HSI V4 exhibits the sharpest focus at a wavelength of 450 nm, suggesting that the focus is optimized for shorter wavelengths. This is suboptimal for most applications, where focus should ideally be best at the center wavelength. HSI V6, on the other hand, shows small variations in sharpness between $\sim 400$ and 650 nm, with no features such as triple peaks observed.

For both instruments, the LSFs are significantly wider in the along-track direction. This is due to a design compromise where the slit width is chosen to be larger than the across-track pixel width, to collect more light, accepting some along-track blur as a tradeoff. Overall, the performance of HSI V6 varies less across track and is better in the along-track direction. Both cameras exhibit fair consistency in sharpness across the FOV.

A version of the HSI V6 is currently used on the HYPSO-1 cubesat.¹² The achieved spatial resolution from in-orbit was estimated in Ref. 13, where the FWHM of the across-track LSF was measured to be $\sim 2$ to 3 pixels in the range of 450 to 750 nm. This is consistent with the observations for HSI V6 here.

3.2.

Keystone Distortion

The keystone distortion is the across-track shift of the SPSF with wavelength. This distortion is shown for both cameras in Fig. 4, at the center and edge of the FOV. A nearly linear distortion is seen for the center pixel in both cases, with an amount comparable to the spectral response function peak width. This distortion is clearly due to a residual error in the rotational alignment of the image sensor relative to the diffraction grating because it is the orientation of the grating that determines the direction of the dispersion of light that is subsequently focused onto the image sensor. The edge pixels exhibit additional keystone error, particularly in the shortwave end of the spectrum, which appears to result from chromatic aberration in the lenses.

Fig. 4

Keystone for (a) HSI V4 and (b) HSI V6 as measured and after subtraction of the alignment error.

Previous measurements of keystone distortion for both HSI V4¹⁴ and HSI V6² have shown similar patterns. In addition, the smile and keystone distortion for the version of HSI V6 used on HYPSO-1 were reported in Ref. 15. For HSI V4, keystone distortion was estimated to be $\sim 4$ to 5 pixels in the 400 to 800 nm range. For HSI V6 in both Ref. 2 and HYPSO-1, the shift was found to be $\sim 2$ pixels in the same spectral range. However, in our measurements, we observe a different pattern. This discrepancy is likely due to the use of different camera builds than those in Refs. 15 and 2, with varying calibration and alignment of the components.

3.3.

Reconstructed 2D SPSFs

The full SPSFs reconstructed from the LSFs are shown for selected wavelengths in Figs. 5 and 6 for HSI V4 and HSI V6, respectively. Wavelengths between 450 and 650 nm were chosen due to the low signal-to-noise ratio outside this range.

Fig. 5

Reconstructed SPSFs for HSI V4. The SPSFs for the center pixel (a) and the edge pixel (b). The contour lines indicate the levels at 50%, 60%, 70%, 80%, and 90% of the enclosed energy of the SPSF.

Fig. 6

Reconstructed SPSFs for HSI V6. The SPSFs for the center pixel (a) and the edge pixel (b). The contour lines indicate the levels at 50%, 60%, 70%, 80%, and 90% of the enclosed energy of the SPSF.

For the HSI V4, the wavelength dependence reflects the behavior shown in Fig. 2, with the sharpest focus observed at the shortwave end. The elongation in the along-track direction results from the slit being wider along track than the pixel pitch across track. Significant defocus is seen at the longwave end. This is consistent with Fig. 2, but Fig. 5 shows that the SPSF broadening results from a complex shape of the 2D SPSF.

For HSI V6, focus is set so that the SPSF tends to broaden toward the ends of the spectral range. Generally, the SPSF is better focused than for HSI V4, which is consistent with the observations in Fig. 2. For HSI V6, the SPSF broadens differently than in HSI V4, with tails appearing in the SPSF at longer wavelengths.

In HSI V6, the opposite behavior is observed between the center and edge of the FOV, with smaller SPSFs found in the center. This is expected as better performance is typically observed in the center of the FOV, close to the optical axis.

3.4.

Simplified Approximate SPSF Measurement

Recognizing that the tomographic 2D measurement is complex, the P4001 draft allows a simplified measurement of only two LSFs in orthogonal directions, deriving an approximate SPSF shape as the product of these, as discussed in Ref. 16. Thus, this simplified measurement assumes that the SPSF shape is well represented by such a separable function, which is not necessarily the case, but with the benefit of reduced measurement time and a simpler experimental setup similar to what many labs already have.

As an example, Fig. 7 shows the fully reconstructed SPSF [panel (b)] for the 534 nm band of the HSI V6 edge pixel compared with the simplified SPSF [panel (c)] formed from the LSFs across and along track, shown in panels (a) and (d), respectively. The simplified SPSF in panel (c) is a fair representation of the full reconstructed shape in panel (b). The main features are retained, not least since the scan axes are chosen along the camera axes. Some asymmetry and skew is present in the full SPSF, but lost in the approximation. The full reconstructed shape appears broader in the contour plot, but this is partly because of a higher level of noise and reconstruction artifact in the SPSF tails, leading to contours being drawn at lower signal levels for a given enclosed energy.

Fig. 7

Comparison of (b) fully reconstructed SPSF and (c) simplified SPSF for the HSI V6 edge pixel. Panels (a) and (d) show the LSFs in the across- and along-track directions, respectively.

3.5.

Coregistration Error

Following the P4001 standard, we quantify coregistration as the integrated difference between SPSFs of two different bands in the same pixel:

Fig. 8

Coregistration error matrices for (a) and (c) HSI V4, and (b) and (d) HSI V6. Matrices in panels (c) and (d) are derived from simplified SPSF measurements. The lower and upper halves are the coregistration error matrices for center and edge pixels, respectively. ${\overline{\epsilon }}_{xy}$ is the average coregistration error.

As the matrix (1) is symmetric, we use the two halves in the figure to represent two different pixels: one at the center and one at the edge of the FOV. Unsurprisingly, SPSFs of neighboring bands are more similar and therefore exhibit less coregistration error than bands with larger spectral separation. In terms of the average coregistration error, the edge of the FOV for HSI V4 shows the lowest error at 0.29, followed by the center of the FOV for HSI V4 at 0.32, the center of the FOV for HSI V6 at 0.41, and finally the edge of the FOV for HSI V6 at 0.44. This pattern is consistent with the SPSF plots (Figs. 5 and 6). The simplified measurement gives a slightly larger coregistration error for V6 but overestimates more significantly the error value for the center pixel of V4. Looking back at Fig. 5, the SPSF for this pixel is severely distorted. In terms of use in camera characterization, it can be seen as a good thing that the simplified measurement provides a conservative value.

Notably, although HSI V6 exhibits an overall sharper SPSF, it also has the highest coregistration error. This is evident in Fig. 6, where the sharp SPSF at 650 nm overlaps very little with the SPSF at 450 nm. This misalignment is due to a combination of keystone distortion and SPSF broadening at the blue end of the spectral range. If the observed scene is not uniform around the pixel, this will clearly distort the observed spectrum. For example, consider a pixel region with a vertical edge at $x=0$ in Fig. 5 or 6, with a white area on the left and a black (or shadow) area on the right. The 450-nm band will predominantly see white, whereas the 650-nm band will predominantly see black. Thus, the measured spectrum will be severely distorted.