Efficient method to measure the spectral distortions using periodically distributed slit in hyperspectral imager

Jinsuk Hong; Youngsoo Kim; Byungin Choi; Seongryung Hwang; Dohwan Jeong; Jun Ho Lee; Yeonsoo Kim; Hyunsook Kim

doi:10.1364/OE.25.020340

1. Introduction

Airborne multispectral imagery is one of the well-known remote sensing techniques that have been used since early 1960s [1]. The hyperspectral imager can be classified as the instruments that provide images of more than 100 continuous spectral bands with the spectral resolution smaller than 1/100 of the full spectral range [2,3]. The hyperspectral instruments are widely used for target detection, material mapping and identification and surface properties mapping [4,5].

These spectrometers generally show geometric distortions such as the keystone along the spectral axis and the smile along the spatial axis as the spectral range extends [6]. The airborne systems usually have spectral range up to 3000 nm useful for identification and detection of the material composition, such as HYDICE and Hyperion which have the spectral range of up to 2500 nm [7–9]. The geometric distortions of these hyperspectral imagers have been measured by a statistical method in systems such as the Hyperspectral Infrared Imager (HyspIRI), Ultra-Compact Imaging Spectrometer (UCIS) and CompAQS [10–12].

A brief concept of the conventional measurement method of the keystone and smile is illustrated in the Fig. 1 [13]. To measure the keystone in the conventional statistical method, it is required to create a concentrated ray source with multiple wavelengths on the slit as shown in the Fig. 1 [13]. When it comes to measure the smile with the conventional method, the spectrometer requires to measure in multiple field points by scanning along the slit [13]. Furthermore, the method requires repeated measurement of a certain field position in a certain wavelength to calculate the smile and the keystone in a statistical method [13]. As a result, the conventional method requires separate measurement setups, a precision scanning along the slit and tediously repeated measurement [13]. This method is proven effective and accurate enough despite of the time consumption due to the number of the sampling points and precision optical setup [13]. The optical setup for the measurement often requires precision optical component such as the etalon and is hard to establish the setup for the measurement of other characteristics at the same time. It is also highly important to maintain the integrity of the etalon during the measurement process as the gap of the two flat surfaces facing each other directly affects the full width half maximum (FWHM) of the output beam [14]. Also, the optical distance between plates of the etalon needs to be changed according to the measurement wavelength [13]. Moreover, scanning along the spectral and spatial axis requires the precision tilt of the etalon and shift of the source with the etalon [13].

Fig. 1 A brief concept of the conventional measurement method of the keystone and smile.

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To complement this statistical measurement method, a periodically sampled measurement (PSM) was applied to measure and correct the keystone and smile. The method adopts a newly designed tool called the field identifier (FI) and an integrating sphere small enough to fill the slit with the calibration lamp. The FI is fairly easy to manufacture by etching method, easy to remove and reattach to the setup during the alignment procedure. The integrating sphere light source is for the keystone and smile measurement and doesn’t need to be exactly aligned. It is also easy to maintain the integrity and repeatability of the measurement as the method doesn’t require a precision movement for the scanning mechanism. Furthermore, it is easy to identify each field of the optical system since the FI separates the continuous slit line image into multiple field points. The magnification can be calculated directly just by measuring the distance of the points of each end.

The PSM accuracy is calculated to be more than 99% when sampled more than 10 points and central coordinate difference from the scanned measurement is to be less than 1% so that the method is highly effective, accurate, easy and fast.

2. Optical Design of the VNIR and the SWIR channels

The basic optical layout of the spectrometer is based on the Offner relay optics with the convex diffraction grating on the 2nd mirror surface since many compact spectrometers successfully used it to maximize the efficiency of overall spectral range and minimize optical aberrations [15–17]

The goals for the VNIR and SWIR channel are described in the Table 1. The keystone and smile of both channels are to be smaller than 0.15 pixel. WFE should be smaller than 0.39λ@650 nm in VNIR channel and 0.25λ@1300 nm in SWIR channel.

Table 1. The Optical requirements for the VNIR and SWIR channels.

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The optical layout of the spectrometer channels is as shown in the Fig. 2. Two spectral channels, the VNIR and the SWIR channels, are sharing the slit and the beam splitter (BS). The BS has a crossover frequency of 925 nm from VNIR to SWIR spectral region.

Fig. 2 Optical Layout of VNIR and SWIR channels of Spectrometer.

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The specifications of the spectrometer channels are as shown in the Table 2. Since the detectors used in each channel have different pixel pitch, the VNIR and the SWIR channel have different magnifications of 1 and 1.04 respectively. The primary mirror (VM1) and the tertiary mirror (VM3) surfaces of the VNIR channel are designed to be one aspheric surface which is concentric on the optical axis. They are combined into one large aspheric mirror VM1-M3 to simplify assembly and alignment procedure. The SWIR channel has a spherical primary mirror (SM1) and an aspheric tertiary mirror (SM3). The aspheric surfaces are implemented to minimize the keystone and smile. The aspheric surface of SM3 and its tilt angle are for the magnification 1.04 of the SWIR channel.

Table 2. Specification of VNIR and SWIR Channels of the Spectrometer.

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The diffraction grating of the VNIR channel (VDG) and the diffraction grating of the SWIR channel (SDG) have spherical reflective surfaces. The blaze angle of VDG is 2.31° which gives the maximum efficiency of the 1st order diffraction more than 80% in 550 nm. That of SDG is to be 3.2° so that the maximum efficiency of the 1st order diffraction to be more than 80% in 1200 nm in the same manner. Both channels have the same effective f number of 2.55. The spatial length is different from each other since the magnification of the SWIR channel is 1.04 while the VNIR channel 1. After the sensitivity analysis, the VDG, SDG and SM3 in the each spectrometer channel are assigned as the optical compensators to optimize the keystone, the smile and the WFE performance.

3. Periodically Sampled measurement of keystone and smile

To assemble and aligning the spectrometer, it is necessary to measure the keystone, the smile and the WFE characteristics repeatedly. To measure these characteristics more efficiently, a PSM method is implemented and the FI is manufactured for the measurement. To verify effectiveness of the FI, the accuracy of the PSM method is calculated. Also, the centroid coordinates differences induced by the measurement setting are calculated.

3.1 The accuracy of the periodically sampled measurement

The smile can be approximated by a quadratic function with one inflection point [18]. When the smile is approximated as $f (x)$ , it can be categorized into two cases according to the position of the inflection point as shown in Fig. 3.The horizontal axis represents the spatial axis (along slit) and the vertical axis represents the spectral axis. $x_{i n i t i a l}$ and $x_{f i n a l}$ represents the end of the slit and $x_{p e a k}$ is the inflection point. If $x_{p e a k}$ is out of the region from $x_{i n i t i a l}$ to $x_{f i n a l}$ , the smile S can be described as the Eq. (1), and if it exists within the a $x_{i n i t i a l}$ nd $x_{f i n a l}$ , the Eq. (2).

S = | f (x_{f i n a l}) - f (x_{i n i t i a l}) |

S = f (x_{p e a k}) - M i n (f (x_{i n i t i a l}), f (x_{f i n a l}))

If the region from

x_{i n i t i a l}

to

x_{f i n a l}

is divided into evenly spaced n sub-regions, n + 1 points of

x_{n}

along spatial axis can be obtained as shown in the Fig. 4.When the smile calculated by the values of

f (x_{n})

is called the sampled smile and noted as

S_{s}

and the true smile

S_{t}

, they can be expressed as shown in Eqs. (3) and (4).

S_{s} = M a x (f (x_{0}), f (x_{1}), \dots, f (x_{n})) - M i n (f (x_{0}), f (x_{1}), \dots, f (x_{n}))

S_{t} = f (x_{p e a k}) - M i n (f (x_{0}), f (x_{1}), \dots, f (x_{n}))

_{$A_{s}$}, the accuracy of the

S_{s}

, can be calculated as shown in the Eq. (5).

A_{s} = \frac{S_{s}}{S_{t}} \times 100

The accuracy decreases as

S_{t}

increases and

S_{s}

decreases as shown in the Eq. (5). Unless

x_{p e a k}

coincide with one of

x_{n}

,

S_{t}

is always larger than

S_{s}

. When

Δ S

is defined as the difference between

S_{t}

and

S_{s}

as shown in the Eq. (6),

A_{s}

becomes minimized when

Δ S

is maximized.

Δ S = S_{t} - S_{s}

When the nearby

x_{n}

s of

x_{p e a k}

is to be

x_{k}

and

x_{k + 1}

,

Δ S

is maximized when

x_{p e a k} = (x_{k} + x_{k + 1}) / 2

and

x_{p e a k} = (x_{i n i t i a l} + x_{f i n a l}) / 2

at the same time as shown in the Fig. 5(a). When

x_{p e a k}

is not in the center of the region of

x_{k} - x_{k + 1}

and

x_{i n i t i a l} - x_{f i n a l}

,

S_{s}

curve (dashed line) tends to move closer to

S_{t}

curve (solid line) as shown in the Fig. 5(b) and

Δ S

becomes smaller and

A_{s}

becomes larger than the case shown in the Fig. 5(a).As noted above,

A_{s}

is minimized when

S_{t} = f (x_{p e a k}) - f (x_{0})

and

S_{s} = f (x_{k}) - f (x_{0})

,

A_{s - m i n}

, the minimum of

A_{s}

, can be calculated as the Eq. (7).

A_{s - m i n} = \frac{f (x_{k}) - f (x_{0})}{f (x_{p e a k}) - f (x_{0})} \times 100

Since

f (x)

can be approximated as a form of a quadratic polynomial with one inflection point when rotational term is well removed by clocking and the coordinate is set so as the inflection point to be (0,0) as shown in the Fig. 6, the coefficients of the 1st and 0th term can be neglected and f(x) can be simplified as the Eq. (8) below.

f (x) = a x^{2}

When Eq. (8) is assigned to Eq. (7), the 2nd order coefficient “a” can be canceled and Eq. (7) becomes the following Eq. (9).

A_{s - m i n} = \frac{f (x_{k}) - f (x_{0})}{f (x_{p e a k}) - f (x_{0})} \times 100 = \frac{a x_{k}^{2} - a x_{0}^{2}}{a x_{p e a k}^{2} - a x_{0}^{2}} \times 100 = \frac{x_{k}^{2} - x_{0}^{2}}{x_{p e a k}^{2} - x_{0}^{2}} \times 100

As

x_{0}

is constant and

x_{p e a k}

is 0 since

f (x)

is symmetric about

x_{p e a k}

, the accuracy becomes the function of

x_{k}

. When the region from

x_{0}

and

x_{n}

is evenly divided into n sub-regions,

x_{k}

can be a function of n as shown in the Eq. (10).

x_{k} = g (n) = \frac{x_{0}}{n}

Substituting Eq. (10) into Eq. (9),

A_{s - m i n}

can be expressed as Eq. (11).

A_{s - m i n} = f (n) = \frac{{(x_{0} / n)}^{2} - x_{0}^{2}}{- x_{0}^{2}} \times 100 = (1 - \frac{1}{n^{2}}) \times 100.

Since n is the number of the sampling points,

A_{s - m i n}

according to n can be expressed by the following graph as shown in the Table 3 and the Fig. 7.

Fig. 3 Smile. (a) when $x_{p e a k}$ is out of the region $x_{i n i t i a l}$ and $x_{f i n a l}$ . (b) $x_{p e a k}$ is within the region of $x_{i n i t i a l}$ and $x_{f i n a l}$ .

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Fig. 4 Scheme of the Periodically Sampled smile measurement.

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Fig. 5 (a) when $x_{p e a k}$ is in the middle of the region $x_{k} - x_{k + 1}$ and $x_{i n i t i a l} - x_{f i n a l}$ at the same time. (b) when $x_{p e a k}$ is not in the middle of the region $x_{k}$ and $x_{k + 1}$ .

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Fig. 6 The coordinate reset so as the inflection point is to be the origin.

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Table 3. Calculated Accuracy according to the number of sampling points.

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Fig. 7 Calculated Accuracy according to the number of sampling points.

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As shown in the Table 3, the minimum accuracy reaches 99% when n is 10 when other measurement errors are not taken account. Since $S_{s}$ and $S_{t}$ are same in the sub-region from $x_{0}$ to $x_{k}$ and from $x_{k + 1}$ to $x_{n}$ as shown in the Fig. 5, we need only $x_{0}$ , $x_{k}$ , $x_{k + 1}$ and $x_{f i n a l}$ out of all the sampling points to calculate $S_{s}$ . The desired n is an odd number so that the center field should be included in the measurement.

3.2 Assembly and Alignment Procedure with the FI

To apply the PSM measurement to the alignment procedure, an evenly distributed perpendicular slit, the FI, was designed and manufactured as shown in the Fig. 8(a). The FI has 21 slits holes perpendicular to the actual slit to make an odd number as described in 3.1. Although only 10 measurement points are needed to reach the accuracy 99%, the total slit number is increased because of the manufacturability concern. The width of each of the tool’s slit which is related to the spatial resolution is to be 20 μm, which is smaller than actual slit width, since the purpose of the tool is to measure the keystone and smile efficiently. The overlaid slit and the FI are as shown in the Fig. 8(b)

Fig. 8 The FI. (a) The comparison of the actual slit attached in the spectrometer and the FI. (c) The overlaid slit and the FI in the measurement setup.

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Another error which can occur from implementation of the FI to the PSM method is the air gap between the slit and the FI. The centroid difference seems to have no apparent relationship with the gap and shows a noise-like fluctuation of ± 0.3%. Though it is hard to measure the actual air gap between the tool and the slit, it is to be well below 100 μm considering the manufacturer’s report.

Each spectrometer channel frame is as large as 190 mm wide and 140 mm deep and several instruments are concentrated in the region nearby the optics, the Shack-Hartmann Sensor was selected as the WFE measurement equipment which is relatively small and highly effective. It saves space and effort while maintaining the required measurement accuracy. Overall alignment setup is described in the Figs. 9(a) and 8(b).

Fig. 9 Alignment and Measurement setup. (a) Setup for keystone and smile measurement with the folding mirror (FM). (b) Setup for WFE measurement without the FM.

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After the installation of the reference laser, optical components and Shack-Hartmann Sensor, the alignment iteration begins which includes the fine adjustment of the compensators along with the characteristics measurements. As the adjustment of the compensator affects not only the WFE but also the keystone and smile, the characteristics of WFE, keystone and smile must be measured in all iterations when the compensators are moved.

The integrating sphere source with the Krypton lamp and the monochromic source of 632.8 nm are placed in front of the slit. They’re easily interchangeable according to the measurement purpose. Also, the monochromatic source and the Shack-Hartmann Sensor are on the separate height-changeable stages so that the WFE measurement can be performed in the desired field rapidly. The detector and the Shack-Hartmann sensor are placed in the desired position at the same time.

When the FM is attached, the setup is automatically changed for the keystone and smile measurement as shown in the Fig. 9(a). The FM should be detached from the setup for the WFE measurement so that the ray path is directed to the Shack-Hartmann Sensor as shown in the Fig. 9(b). The slit and the FI are attached at the end of the slit adaptor. The FI may be removed during the WFE measurement as the error of reattaching the tool to the setup is small enough to neglect. With this manner, these characteristics can be minimized at the same time.

3.3 Keystone and smile measurement

When the keystone and the smile are measured, the FI is assembled in front of the slit and lit by the integrating sphere with the spectral calibration lamp of Kr as described in the Fig. 9. As shown in the actual measurement signal in Fig. 10(a), the FI separates each spectral line into point signal corresponding to each field point. Total 21 field points are identified in the measured image as shown in Fig. 10(b), which agrees with the purpose of the tool.

Fig. 10 VNIR detector image of the krypton lamp. (a) Image without the FI. (b) Image with the FI.

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The Gaussian curve fitting algorithm is used to calculate the central coordinate of each point as shown in the Fig. 11. The cross sections of each sampling point along spatial (X) and spectral axis (Y) are taken at the maximum irradiance location approximated by two-dimensional Gaussian fitting.Once the central coordinate $(x_{m, n}, y_{m, n})$ is calculated, the smile in the desired wavelength and the keystone in the desired field can be calculated at the same time. The smile of n^th raw can be calculated by subtracting minimum of $(y_{1, n}, y_{2, n}, y_{3, n} \dots y_{m, n})$ from maximum of $(y_{1, n}, y_{2, n}, y_{3, n} \dots y_{m, n})$ as shown in the Eq. (12). The keystone of m^th column can be calculated by subtracting minimum of $(x_{m, 1}, x_{m, 2}, x_{m, 3} \dots x_{m, n})$ from maximum of $(x_{m, 1}, x_{m, 2}, x_{m, 3} \dots x_{m, n})$ as shown in the Eq. (13).

S_{n} = \max (y_{1, n}, y_{2, n}, y_{3, n} \dots y_{m, n}) - min (y_{1, n}, y_{2, n}, y_{3, n} \dots y_{m, n})

K_{m} = \max (x_{m, 1}, x_{m, 2}, x_{m, 3} \dots x_{m, n}) - min (x_{m, 1}, x_{m, 2}, x_{m, 3} \dots x_{m, n})

A reference coordinate axes are defined to quantify the movement of the compensators and their effects as shown in the Fig. 12.“u”, “v” and “w” are rotational direction along “X”, “Y” and “Z” respectively. The axis of the assembly and alignment setup for each channel coincide with the reference coordinate axes.

Fig. 11 Signal Reference coordinates for keystone and smile calculation.

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Fig. 12 (a) Reference Axes of each compensator's orientation and movement. (b) VNIR Channel Orientation coincided with the reference axis. (c) SWIR channel Orientation coincided with the reference axis.

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3.4 Performance of the VNIR and SWIR channel

The final performance of each channel is shown in the Table 4, the Table 5 and the Fig. 13. The worst WFE measurement results are 0.34λ@650 nm in VNIR channel and 0.22λ@1300 nm in SWIR channel as shown in the Table 4.

Table 4. Worst WFE of each spectrometer channel.

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Table 5. The final Performance of the VNIR and SWIR channel of the Spectrometer after fixation.

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Fig. 13 The performance of the VNIR and SWIR channel after alignment and fixation. (a) Keysotne of the VNIR channnel (b) Keystone of the SWIR channel. (c) Smile of the VNIR Channel (d) Smile of the SWIR channel.

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The keystone and smile results are as shown in the Table 5 and the Fig. 13. The measured keystone in the VNIR channel is 0.08 pixel and 0.10 pixel in the SWIR channel. The measured smile in the VNIR channel is 0.13 pixel and 0.04 pixel in the SWIR channel. Comparing the requirement of the keystone, smile and the WFE, the measured results are all satisfying the requirement.

The response tendencies of the keystone and the smile according to the movement of the compensators along each axis were surveyed in each channel. The detailed the keystone and the smile measurement curves are as shown in the Fig. 13 after the keystone and the smile are optimized according to the surveyed tendencies. Figures 13(a) and 13(b) are the curves of the keystone in VNIR and SWIR channel, Figs. 13(c) and 12(d) are the smile curves respectively. Only 9 well-etched sampling points were taken while the FI has 21 perpendicular slit holes and still the result has 99% accuracy since the selected sampling fields has 0.2 to 0.3 field from the nearby selected points. The error bars in each graph in Fig. 13 signify the corresponding 3σ of measurement error. To estimate the measurement error of the method, central coordinates of a certain point in different images were acquired and plotted. The calculated 3σ along spatial axis is 0.019 pixel and applied to the keystone measurement. The 3σ along spectral axis is 0.012 pixel and applied to the smile measurement in the same manner. Some of the sampled points look off the smile curve because of the contamination on the slit. However, the results show that the performance met the smile requirements despite of the contamination.

4. Conclusion

The newly proposed method using the Field Identifier is used efficiently to measure the keystone and the smile in one measurement while the Shack-Hartmann Sensor is used to verify the WFE performance at the same time with the minimum change of the measurement setup. The periodically sampled measurement method makes the alignment setup simple and efficient enough to measure the characteristics simultaneously. As a result, the VNIR channel is aligned so as that the maximum keystone is measured to be approximately 0.08 pixel, the maximum smile 0.13 pixel and WFE 0.34 λ(@650 nm). In case of SWIR channel, the maximum keystone is to be 0.10 pixel, the maximum smile to be 0.04 pixel and the WFE 0.22 λ(@1300 nm) respectively.

References and links

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Item	Requirements
Item	VNIR	SWIR
Spectral Range [nm]	400~900	900~1700
Number of bands	More than 100 in each channel
Keystone [pixel]	0.15
Smile [pixel]	0.15
Worst WFE [Wave number]	0.39 λ (@650 nm)	0.25 λ (@1300 nm)

Channel	M1	M2 (DG)			M3	Image		F Number	Magnification
Channel	RoC [mm]	RoC [mm]	Groove Width [um]	Blaze Angle [°]	RoC [mm]	Spatial Length [mm]	Spectral Length [mm]	F Number	Magnification
VNIR	−147.10	−74.78	8.00	2.31	−147.10	15.36	4.80	2.55	1
SWIR	−115.78	−57.65	12.50	3.2	−121.81	15.99	4.00	2.55	1.04

Item	Requirements
Item	VNIR	SWIR
Spectral Range [nm]	400~900	900~1700
Number of bands	More than 100 in each channel
Keystone [pixel]	0.15
Smile [pixel]	0.15
Worst WFE [Wave number]	0.39 λ (@650 nm)	0.25 λ (@1300 nm)

Channel	M1	M2 (DG)			M3	Image		F Number	Magnification
Channel	RoC [mm]	RoC [mm]	Groove Width [um]	Blaze Angle [°]	RoC [mm]	Spatial Length [mm]	Spectral Length [mm]	F Number	Magnification
VNIR	−147.10	−74.78	8.00	2.31	−147.10	15.36	4.80	2.55	1
SWIR	−115.78	−57.65	12.50	3.2	−121.81	15.99	4.00	2.55	1.04

Efficient method to measure the spectral distortions using periodically distributed slit in hyperspectral imager

Abstract

1. Introduction

2. Optical Design of the VNIR and the SWIR channels

3. Periodically Sampled measurement of keystone and smile

3.1 The accuracy of the periodically sampled measurement

3.2 Assembly and Alignment Procedure with the FI

3.3 Keystone and smile measurement

3.4 Performance of the VNIR and SWIR channel

4. Conclusion

References and links

Cited By

Figures (13)

Tables (5)

Equations (13)

Optics Express

Item	Measured Value
Item	VNIR Channel	SWIR Channel
Keystone [Pixel]	0.08	0.10
Smile [Pixel]	0.13	0.04