1.

Introduction

In remote sensing, push-broom cameras can generate two-dimensional (2D) images of the Earth by combining successive one-dimensional (1D) images captured by a linear array detector¹ (Fig. 1). In such configurations, the camera’s field of view is large in the satellite’s across-track (ACT) direction and small in the along-track (ALT) direction. The linear array detector is parallel to the ACT direction. During the time $dt$ between two acquisitions, the instrument rotates in the ALT direction by the equivalent of the pixel’s instantaneous field of view in the ACT direction. The integration time $t_{\mathrm{int}}$ is ideally equal to $dt$, or slightly smaller, giving minimal information loss.

Fig. 1

(a) Acquisition of successive 1D images in a push broom configuration. (b) 2D reconstructed image for a checkerboard extended scene.

Off-axis three-mirror anastigmats are ideal optical configurations for such instruments, as they provide a large field of view in a single direction.^2,3 They are used, for example, in ProbaV⁴ and Sentinel-2 MSI.⁵ Off-axis configurations with four mirrors add an extra degree of freedom and are used, for example, in Landsat 8’s Operational Land Imager.⁶ Currently in development, the cloud imager (CLIM) instrument from the CO2 monitoring (CO2M) mission^7,8 uses a similar optical design to ProbaV but with a reduced field of view of $f_x=\pm 12.8\mathrm{deg}$. With three linear array detectors, each with a spectral filter on top, it enables the reconstruction of a 2D image of the Earth at three different wavelength channels (670, 753, and 1370 nm).

Stray light (SL) poses a significant challenge for Earth observation applications,^9–12 including those employing a push-broom camera. As SL affects image quality and radiometric accuracy, it must be controlled through appropriate opto-mechanical design and material selection.^13,14 In a three-mirror anastigmat (TMA) such as CLIM, baffles, apertures, and light traps are strategically placed to mitigate SL¹⁵ [Fig. 2(a)]. Spectral rejection is achieved by utilizing a narrow band spectral filter atop a glass slab, complemented by a broadband filter on the bottom surface [Fig. 2(b)]. Spectral crosstalk is prevented using black masks. However, ghost reflections are present due to partial reflections at glass interfaces and on the detector surface. Figure 2(c) depicts the SL pattern on the linear array detector at 670 nm, predicted by ray tracing when the central pixel $x_0$ is illuminated by a point-like source. Here, SL is normalized to a nominal signal of 1. The SL diminishes gradually away from $x_0$ due to scattering effects, whereas ghost reflections are concentrated around $x_0$. Moreover, it shows that the reflection on the detector increases the number of ghosts compared with the case of a perfectly absorbing detector. A worst-case assumption of 20% reflectivity at the detector is used, though the actual reflectivity likely falls between that of a fully absorbing detector and 20%.

Fig. 2

(a) Opto-mechanical design of the CLIM instrument. (b) Spectral rejection architecture. (c) SL kernel predicted by ray tracing at 670 nm. (d) and (e) Estimated SL associated with the checkerboard reference extended scene, computed with the SL kernel assumed rotationally symmetrical at 670 nm.

When the SL requirement is very strict, control by design can be insufficient, necessitating additional SL correction by post-processing.¹¹ In CLIM, the requirement is defined based on the checkerboard scene illustrated in Fig. 1, with transitions between areas of bright and dark radiances. Extended scenes with a transition between bright and dark areas is a common way of specifying SL in Earth observation, as it reproduces large extended areas of lands, oceans, or clouds.^11,12 For mission success, SL on any pixel should not exceed 2% of the measured radiance, except for those within 20 pixels from a transition. At the 1370-nm channel, however, this distance is 10 pixels as its linear array detector uses pixels twice bigger than in the visible spectral range. Based on ray tracing simulations, it is expected that a correction algorithm reducing the SL by at least one order of magnitude is necessary.

In the past, SL correction by post-processing was mostly limited to deconvolution.^16–18 Recently, the demand for higher-performing instruments has driven the development of advanced correction methods.^19–21 In Metop-3MI, a remarkable correction factor of two orders of magnitude has been reported, using a matrix method where SL kernels are modulated by the signal coming from various positions within the field of view.^11,22,23 With a square field of view of up to $\pm 57\mathrm{deg}$, SL kernels in that instrument have been measured with a dynamic range of ${10}^{-8}$, and their dependence within the field of view was interpolated using a local symmetry assumption.¹¹ In this paper, we describe a correction approach for the specific case of push-broom cameras with a linear detector array, with the CLIM instrument as a case study. We describe the calibration method and a direct approach for interpolation of SL kernels. The algorithm’s performance is demonstrated using ray-traced data at 670 nm, assuming, without loss of generality, a rotationally symmetrical dependence of its SL properties.

2.

SL Contribution to the Image for a Given Scene

On a given line $y$ of the recombined 2D image, the measured signal $I_{\mathrm{mes}}(x)$ is the sum of a nominal contribution, $I_{\mathrm{nom}}$, originating from the image-forming beam and a SL contribution, $I_{\mathrm{SL}}$ [Eq. (1)]. The SL can be decomposed into spatial components, in-field SL ($I_{\mathrm{SL}\mathrm{IF}}$) and out-of-field SL ($I_{\mathrm{SL}\mathrm{OOF}}$), and a spectral component, $I_{\mathrm{SL}\mathrm{spectral}}$ [Eq. (2)]. The latter represents SL originating from wavelengths different from the one being considered in the channel. In CLIM, $I_{\mathrm{SL}\mathrm{spectral}}=0$ thanks to the optimized spectral filter architecture. Spatial components arise from SL at the wavelength of the considered channel, caused by illumination at various field angles. In CLIM, SL occurs when the illumination angle in the ALT direction is within about ± 10 deg of the angle corresponding to the direct illumination of the linear array detector. In the ACT direction, SL from illumination within the instrument field of view ($|f_x|\le 12.8\mathrm{deg}$) is considered in-field ($I_{\mathrm{SL}\mathrm{IF}}$), whereas for illumination outside that range, it is considered out-of-field ($I_{\mathrm{SL}\mathrm{OOF}}$)

We define the SL kernel, ${\mathrm{SPST}}_{x_f}(x,y_f)$, as the SL on pixel $x$ when the instrument observes a point-like source illuminating pixel $x$ when the instrument observes a point-like source illuminating pixel $(x_f,y_f)$ on the recombined image, such that the linear detector array is directly illuminated when $y_f=0$ and giving a nominal signal of 1. Here, SPST stands for Spatial Point Source Transmittance. With that definition, the SL contribution to the measured scene is expressed by Eq. (3), representing a sum over the field of the SL kernel modulated by the nominal signal at the corresponding field. The factor $dt/t_{\mathrm{int}}$ accounts for energy conservation when the integration time is not equal to the time interval between two successive lines

3.

SL Correction Principle

The SL correction of an image is performed one line at a time. First, an estimation of the SL component is obtained with Eq. (4), which is analogous to Eq. (3) except that the modulation of the kernel is done with the measured signal. Then, subtracting this result from the measured signal gives an estimate of the corrected signal [Eq. (5)]. Performing this operation for each line provides the estimated 2D corrected image

4.

SL Kernel Measurements and Processing

4.1.

Calibration Grid

SL kernels are measured over a calibration grid $(x_f,y_f)$ using point-like source illumination at the corresponding field angles. The matrix ${\mathrm{SPST}}_{x_f}(x,y_f)$ is measured by illuminating the instrument at a fixed value of $x_f$ and varying $y_f$. For each value of $y_f$, a line of the matrix ${\mathrm{SPST}}_{x_f}(x,y_f)$ is obtained along the $x$ direction. This process is repeated for various values of $x_f$, effectively providing the various kernels ${\mathrm{SPST}}_{x_f}$.

A calibration grid with $x_f=[1:N]$ and $y_f=[-\mathrm{\Delta }y:\mathrm{\Delta }y]$ in steps of 1 would result in an unrealistically long measurement campaign. Therefore, a restricted calibration grid is considered, and interpolation will be performed to provide the kernels over the full-resolution grid. For CLIM, SL has a smooth behavior along $y_f$ but shows rapid variations in the near-nominal region due to localized ghosts. For adequate sampling, a grid with one-pixel steps is selected in the range $y_f=[-35:35]$. Beyond this range, larger steps can be used: $y_f=\pm [50,250,500,750,1000,1250,1484]$. In the $x_f$ direction, SL variation is mostly a shift, as the instrument is quasi-telecentric. Therefore, a grid with 200 pixels steps is selected. Finally, the full calibration grid is shown in Fig. 3.

Fig. 3

Calibration grid for SL calibration.

4.3.

SPST Map Normalization

SL kernels must be normalized to the nominal signal. If the image-forming beam produces a sub-pixel nominal image at the detector, the measurements are simply normalized to the signal at pixel coordinate $(x_f;y_f=0)$. If the nominal image is spread over more than one pixel, due to aberrations or the test collimator’s angular extent, the nominal signal is obtained by summing the signal across those pixels. In CLIM, we consider the nominal signal as extended over $(x_f\pm 1;0\pm 1)$. Finally, the signal on nominal pixels is removed from the measurement to retain only the SL. Figure 4(a) illustrates the results of these measurements and processing, simulated by ray tracing and confirming the adequate sampling of the SL variations.

Fig. 4

(a) Calibrated SPST map. (b) and (c) SPST interpolation along the $y_f$ coordinate (ALT). (c) Theoretical SPST in full resolution and (d) interpolation error. Color maps are in log₁₀ scale, with normalization to the nominal signal, corresponding to the direct image forming a beam.

4.5.

Interpolation Along x_f (ACT)

At this stage, the matrices ${\mathrm{SPST}}_{x_f}(x,y_f)$ are known in full resolution for each calibrated value of $x_f$. Interpolation is then performed to derive the kernels for all values of $x_f$ between 1 and $N$, with steps of one pixel. A local shift invariance assumption is made, which is reasonable for a quasi-telecentric instrument such as CLIM.

For a given value $x_f=x_{f\mathrm{interp}}$, the matrix is deduced by searching for the closest neighboring calibrated fields, $x_{f1}$. A shift along $x$ of the associated matrix is performed to obtain the interpolated kernel: ${\mathrm{SPST}}_{x_{f\mathrm{interp}}}(x,y_f)={\mathrm{SPST}}_{x_{f1}}(x-(x_{f\mathrm{interp}}-x_{f1}),y_f)$. This process results in missing signals on one of the edges along $x$, which is filled by applying the equivalent transformation to the kernel associated with the second closest neighbor, $x_{f2}$, located on the other side. This process is illustrated in Fig. 5(a). Finally, this process provides $N$ full-resolution SL kernels, ${\mathrm{SPST}}_{x_f}$, each with dimension $N\times (2·\frac{\mathrm{\Delta }y}{2}+1)$.

Fig. 5

SPST interpolation along the $x_f$ coordinate: we search for the first (a) and second (b) neighboring calibration fields and apply shifts to reconstruct the interpolated SPST (c). Color maps are in a log₁₀ scale.

5.

SL Correction Performance

SL correction performance is assessed by applying the algorithm on the image of the reference extended scene, shown in Fig. 1. This is done with ray tracing data, considering the assumption of an absorbing detector, or the worst-case scenario of a reflective detector. In both cases, the theoretical SL is computed with Eq. (3), and the correction method follows all the steps described in the previous sections.

Figure 6(a) shows the estimated SL associated with the reference scene, obtained using the correction algorithm with enough iterations to reach convergence. Figure 6(b) displays the residual SL error, obtained by subtracting the estimated SL from the measured image. Figure 6(c) presents the profiles along $x$ for the nominal signal, initial SL, and residual SL. Specifically, the residual SL is shown for a single iteration and at convergence, which is reached with two or more iterations. The SL reduction factor is on average 1/25 with a single iteration, whereas a factor of $\sim 1/100$ is achieved at convergence. Overlaying the performance requirement on the graph shows that a single iteration suffices to reach a satisfactory residual SL level. Interestingly, there is a larger error at the transition zone, which arises from the field binning effect. With an appropriate selection of the field binning factor, this error is correctly restricted to the transition area not included in the performance requirement.

Fig. 6

Estimated SL (a), residual SL after correction (b), and profiles (c) in the assumption of an absorbing detector. (d)–(f) Analogous results for a reflective detector.

Figures 6(d)–6(f) show the analogous results for the case of a reflective detector. In this scenario, the initial SL level is larger due to additional ghosts. Consequently, two iterations are necessary to bring the residual SL below the performance requirement. Moreover, the convergence is reached with at least three iterations. In practice, the real properties of the detector are somewhere in between these two scenarios.

6.

ACT Out-of-Field SL

SL coming from outside the field of view in the ACT direction, $I_{\mathrm{SL}\mathrm{OOF}}$, has been neglected because its level is much lower than $I_{\mathrm{SL}\mathrm{IF}}$. Moreover, correcting for this contributor is challenging as the nominal signal from such angles is unknown. For Metop-3MI, out-of-field SL kernels are measured, and the in-field image is mirrored to estimate the out-of-field input radiance.²⁷ This method works well for scenes with large uniform features; however, it cannot predict localized objects such as isolated clouds.

Following a similar approach, an algorithm correction of both $I_{\mathrm{SL}\mathrm{IF}}$ and $I_{\mathrm{SL}\mathrm{OOF}}$ could be built for CLIM. ${\mathrm{SPST}}_{x_f}$ is calibrated for out-of-field ACT angles, considering $x_f$ from $1-n_{\mathrm{OOF}}$ to $N+n_{\mathrm{OOF}}$ with $n_{\mathrm{OOF}}$ the limit for non-zero out-of-field SL. Here, we link the out-of-field pixel coordinate with the field angle by extrapolating the distortion curve. Because there is no nominal illumination for OOF SPST maps, the normalization is done to the monitoring then to the nominal signal of the closest in-field SPST map. Next, the image $I_{\mathrm{mes}}^{*}$ going from pixel $1-n_{\mathrm{OOF}}$ to $N+n_{\mathrm{OOF}}$ is obtained by an ACT mirroring of $I_{\mathrm{mes}}$. Finally, the estimated SL is obtained with Eq. (6) instead of Eq. (4)

7.

Conclusions

In this paper, we presented an SL correction approach for space telescopes equipped with linear detector arrays in a push-broom configuration, using the CLIM instrument as a case study. Our approach successfully reduces the SL level by two orders of magnitude, effectively fulfilling the users’ requirements. The methodology involves estimating the SL pattern associated with an input scene by modulating SL kernels corresponding to point-like source illumination from various field angles.

The SL kernels consist of two main components: ghost patterns near the nominal area and a smooth scattering effect in the long-range area. Due to their large dynamic range, measuring these kernels requires multi-level acquisitions, capturing the signal at various input powers or integration times. Initially calibrated over a specific grid, the kernels are then derived for any field angle illumination through interpolation. In the along-track direction, the method employs simple linear interpolation, making no initial assumptions about the SL properties. The primary requirement is a sufficiently dense calibration grid, particularly near the nominal area, to ensure that rapidly varying ghost features are well sampled. For the across-track directions, we apply a local shift assumption in our interpolation strategy, resulting in a comprehensive database of SL kernels. Ultimately, spatial and field binning is performed, significantly reducing both data volume and computation time.

We evaluated the algorithm’s efficacy on a checkerboard scene, which featured transitions between areas of bright and dark radiance. Two scenarios were considered: in the first, we assumed an absorbing detector, whereas in the second, we accounted for a larger SL with the assumption of a reflective detector. In the absorbing detector scenario, one iteration of the correction algorithm sufficed to meet the user’s performance requirements, and convergence was achieved in two iterations. For the reflective detector, two iterations were required to meet the performance requirements, with convergence achieved after three iterations.

Finally, we extended our algorithm to consider out-of-field SL in the across-track direction. For that, we use a mirroring technique to deduce the scene in the out-of-field area, as employed for the case of the Metop-3MI instrument. This combined approach provides an effective SL correction algorithm able to improve the SL of linear detector array instruments in push-broom configuration, considering all kinds of SL present in the instrument.