## Characterization of a spectrograph based hyperspectral imaging system |

Optics Express, Vol. 21, Issue 10, pp. 12085-12099 (2013)

http://dx.doi.org/10.1364/OE.21.012085

Acrobat PDF (2683 KB)

### Abstract

A significant part of the uniformity degradation in the acquired hyperspectral images can be attributed to the coregistration distortions and spectrally and spatially dependent resolution arising from the misalignments and the operation principle of the spectrograph based hyperspectral imaging system. The aim of this study was the development and validation of a practical method for characterization of the geometric coregistration distortions and position dependent resolution. The proposed method is based on modeling the imaging system response to several affordable reference objects. The results of the characterization can be used for calibration of the acquired images or as a tool for assessment of the expected errors in various hyperspectral imaging systems.

© 2013 OSA

## 1. Introduction

2. F. W. L. Esmonde-White, K. A. Esmonde-White, and M. D. Morris, “Minor Distortions with Major Consequences: Correcting Distortions in Imaging Spectrographs,” Appl. Spectrosc. **65**(1), 85–98 (2011). [CrossRef] [PubMed]

3. L. Guanter, K. Segl, B. Sang, L. Alonso, H. Kaufmann, and J. Moreno, “Scene-based spectral calibration assessment of high spectral resolution imaging spectrometers,” Opt. Express **17**(14), 11594–11606 (2009). [CrossRef] [PubMed]

6. D. Schlapfer, J. Nieke, and K. I. Itten, “Spatial PSF Nonuniformity Effects in Airborne Pushbroom Imaging Spectrometry Data,” IEEE Trans. Geosci. Rem. Sens. **45**(2), 458–468 (2007). [CrossRef]

## 2. Methodology

*θ*,

*λ*), where

*θ*is a spatial position and

*λ*a wavelength. Firstly, the geometric properties of the HIS are captured by mapping the true object into the image space by a forward tuning function

**I**(

*u*,

*w*) of size U × W pixels can be estimated within error

**ε**as:where * denotes the convolution. Assuming a uniform response within the pixel borders, the signal (

**I**(

*u*,

*w*)) sampled by equally spaced finite size pixels

*D*(

*u*,

*w*) of size δu × δw. Using the two images, shown in Fig. 1, where each contains information varying either in the direction of the SRF or the SiRF, Eq. (1) can be expressed as a system of two equations, each describing the spatial or the spectral direction:

**S**

_{A}) of the spectral reference object (

*g*

_{S}) provides a good definition of the properties in the spectral direction and is vastly insensitive to the distortions in the spatial direction, while the opposite holds true for the image (

**Si**

_{A}) of the spatial reference object (

*g*

_{Si}). In general, the SRF and SiRF are aligned with the spatial and the spectral axis of the spectrograph and not necessarily with the columns and rows of the acquired image. However, to lower the computational complexity of the proposed method, the alignment of the spatial and spectral axes between the spectrograph and the camera is assumed. The assumption holds if the initial geometric coregistration errors are small, which is realistic, since the proposed method is used after the proper adjustments are made by other, sufficiently accurate procedures [1]. Furthermore, the SRF and SiRF are position variant. Therefore, the Eq. (2) and Eq. (3) should be directly computed only for small regions, where the local invariance can be adopted.

*υ*,

*ω*) = (

*T*

_{S}(

*θ*,

*λ*),

*T*

_{Si}(

*θ*,

*λ*)) and of the corresponding inverse tuning function (

*θ*,

*λ*) = (

*Z*

_{S}(

*υ*,

*ω*),

*Z*

_{Si}(

*υ*,

*ω*)). The first is used to estimate the image of a given object, while the second is required for a physically meaningful interpretation of the acquired image. The tuning functions are determined by establishing a correspondence between the features in the object space and the corresponding positions in the acquired image. For this purpose, the feature locations have to be extracted from the acquired image. Furthermore, the quality of HIS is also defined by its resolution, therefore the position dependent response functions SRF(

*υ*,

*ω*) and SiRF(

*υ*,

*ω*) are also determined. With respect to the HIS properties and technological limitations of the reference object design, solving the two equations, namely Eq. (2) and Eq. (3), requires a different approach.

### 2.1 Spectral feature extraction and SRF characterization

*ω*-

*w*

_{0};

*γ*

_{L},

*γ*

_{G}), defined as a convolution of the centered Gaussian and Lorentzian broadening functions. Therefore, spectral response of the HIS to a single spectral line, centered at a position

*w*

_{0}=

*T*

_{S}(

*θ*,

*λ*

_{0}), is completely defined by two spectrally and spatially variant SRF parameters (

*γ*

_{L},

*γ*

_{G}). Using a fast approximation formula proposed in [13

13. E. E. Whiting, “An empirical approximation to the Voigt profile,” J. Quant. Spectroc. Rad. **8**(6), 1379–1384 (1968). [CrossRef]

*unique spectral lines, were simultaneously modeled. A given*

_{l}*j*-th spectral line in the spectrum of

*l*-th reference lamp is defined by its wavelength λ

_{0,}

_{l}_{,}

*and intensity. By modeling the spectral line with delta function, transforming it by forward spectral tuning function, and modeling intensity*

_{j}*α*

_{l}_{,}

*, the*

_{j}*T*

_{s}(

*g*

_{S}(

*θ*,

*λ*)) in Eq. (2) is substituted by

*α*

_{l}_{,}

*δ*

_{j}∙*(*

_{u}*ω-w*

_{0,}

_{l}_{,}

_{j}_{,}

*). Using the Voigt profile as a SRF model in Eq. (2), and taking into account the convolution rule of SRF and shifted delta function, the modeled spectrum*

_{u}*S*

_{M,}

_{l}_{,}

*of the*

_{u}*l*-th reference lamp corresponding to a spatial channel

*u*of

**S**

_{A,}

*can be written as a sum of responses to the individual spectral lines, i.e. shifted Voigt profiles*

_{l}*α*

_{l}_{,}

*V(*

_{j∙}*ω*-

*w*

_{0,}

_{l}_{,}

_{j}_{,}

*,*

_{u}*γ*

_{G}(

*u,w*

_{0,}

_{l}_{,}

_{j}_{,}

*),*

_{u}*γ*

_{L}(

*u,w*

_{0,}

_{l}_{,}

_{j}_{,}

*)). Approximating the integral in Eq. (2) over a rectangular area of size δu × δw by the rectangle rule [14], and adding a polynomial model of the baseline shift, the final spectrum model can be written as:where*

_{u}*β*

_{l}_{,}

_{k}_{,}

*are the K + 1 polynomial coefficients of the baseline. Since the baseline is mostly due to the thermal drift of the camera and the majority of the SRF is attributed to the spectrograph and lens, the baseline term is not convolved by the SRF. The goal of the spectral characterization algorithm is finding the accurate image positions of the spectral lines and the parameters of the SRF by matching the modeled and the acquired spectrum. Unfortunately, the spectral lines of affordable atomic emission reference sources are usually clustered in non-uniformly distributed fields, which lead to ill-conditioned optimization problem due to poor separation of the instrument response to the individual spectral lines. Furthermore, accurate intensities of the individual spectral lines in the light reaching the optical sensor are elusive, due (but not limited to) to the variability of the reference source properties within manufacturing tolerances, operating environment, and the spectrally and spatially dependent absorption of the utilized diffusing element. Problem was mitigated by imposing additional physically meaningful constraints: (1) the SRF exhibits smooth variation with respect to wavelength, (2) the relation between the true (λ*

_{u}_{0,}

_{l}_{,}

*) and the recorded (*

_{j}*w*

_{0,}

_{l}_{,}

_{j}_{,}

*) wavelengths of the spectral lines is smooth and continuous, (3) intensities of the spectral lines are always non-negative (*

_{u}*α*

_{l}_{,j}≥0). First constraint (1) was imposed by the global parameterization of the SRF

*γ*

_{G}(

*u,w*

_{0,}

_{l}_{,}

_{j}_{,}

*) = f(*

_{u}*w*

_{0,}

_{l}_{,}

_{j}_{,}

*;*

_{u}**k**,

_{B}**b**(

_{G}**k**

_{B},*u*)) and

*γ*

_{L}(

*u,w*

_{0,}

_{l}_{,}

_{j}_{,}

*) = f(*

_{u}*w*

_{0,}

_{l}_{,}

_{j}_{,}

*;*

_{u}**k**,

_{B}**b**(

_{L}**k**

_{B},*u*)). Where f is a global, cubic spline based model, parameterized by

**b**and

_{G}**b**defined at K

_{L}_{B}fixed uniformly spaced set of breaks

**k**. Since similar properties are expected of the second constraint (2), analogous parameterization

_{B}*w*

_{0,}

_{l}_{,}

_{j}_{,}

*= f(λ*

_{u}_{0,}

_{l}_{,}

*;*

_{j}**k**,

_{G}**b**(

_{S}**k**

_{G}_{,}

*u*)) with K

_{G}breaks was used. It should be noted that K

_{B}and K

_{G}are significantly smaller than the total number of the modeled spectral lines. The third constraint (3) was imposed by the constrained optimization method. Search for the parameter set, defining the modeled spectral characteristics of the HIS, was performed separately for each spatial position by minimization of the dissimilarity (

*χ*

_{S}) between the modeled and the acquired spectrum {

**b**,

_{G,Opt}**b**,

_{L,Opt}**b**} = argmin(

_{S,Opt}*χ*

_{S}(

**b**,

_{G}**b**,

_{L}**b**,

_{S}**α**

_{l}_{,}

*,*

_{j}**β**

_{l}_{,}

_{k}_{,}

*)). The dissimilarity was defined by combining the weighted correlation coefficient (wcc) and the weighted sum of squared errors (wsse):where the weight array*

_{u}**µ**

*(*

_{l}*w*) was used to emphasize the more informative portions of the spectral range. The weighted correlation coefficient [15

15. P. R. Griffiths and L. Shao, “Self-Weighted Correlation Coefficients and Their Application to Measure Spectral Similarity,” Appl. Spectrosc. **63**(8), 916–919 (2009). [CrossRef] [PubMed]

**s**

_{M}(

*w*) =

**S**

_{M,}

_{l}_{,}

*(*

_{u}*w*) and the acquired

**s**

_{A}(

*w*) =

**S**

_{A,}

*(*

_{l,u}*w*) spectra in arbitrary spatial channel

*u*can be written as

*s*

_{M}> = Σ

**µ**(

*w*)∙

**s**

_{M}(

*w*) / Σ

**µ**(

*w*) and <

*s*

_{A}> = Σ

**µ**(

*w*)∙

**s**

_{A}(

*w*)/ Σ

**µ**(

*w*) are the weighted means of the corresponding arrays. In this notation,

*w*is considered as the index of pixel centers and all summations are from

*w*= 1 to

*w*= W. Wsse can be expressed as the sum of squared differences between the modeled and acquired spectra multiplied by the normalized weights:

**b**

_{G},

**b**

_{L}

_{,}b_{S}) were optimized by outer Broyden-Fletcher-Goldfarb-Shanno (BFGS) [17] nonlinear optimization method. This gradient-based iterative quasi-Newton method exhibits quadratic convergence and employs the observed behavior of the dissimilarity

*χ*

_{S}and its gradient for construction of the Hessian matrix approximation and step size selection. In each BFGS iteration the optimal auxiliary parameters

**α**

_{l}_{,}

*and*

_{j}**β**

_{l}_{,}

_{k}_{,}

*, capturing the unknown experimental variations (HIS independent), were computed by the constrained linear least-squares imposing the non-negativity of the spectral line intensities. Initial approximation of*

_{u}**b**

_{S}were computed by a linear fit between the real and the acquired positions of the distinct spectral peaks (Hg:546.10 nm, Ne:671.70 nm, Ar:763.51 nm, 965.78 nm). While the initial SRF parameters

**b**

_{G}and

**b**

_{L}were set to values fitting a single Ar peak at the central part of the acquired image (Ar:763.51 nm). Prior to optimization the parameters were scaled to improve convergence and aid the proper selection of stopping criterion. Furthermore, convergence to local minima was prevented by iteratively increasing the number of breaks employed by the subsequent runs of the optimization. The number of breaks K

_{B}and K

_{G}act as regularization parameters. For values less than four the model exhibited underfitting. Between five and seven the optimization problem was properly regularized and best results were achieved. Further increasing the number of breaks led to overfitting. Since the computation time increases with the number of breaks, both K

_{B}and K

_{G}were set to five. Optimal parameters for a single spectrum were computed in approximately 11 s on a current generation personal computer.

**µ**

*(*

_{l}*w*), corresponding to the lamp specific informative spectral range were set to unit and left zero otherwise. In the HIS under study, the baseline shift was mainly contributed by the thermal drift of the camera sensor. The approximate shape of the baseline was assessed by recording an image of white reference material illuminated by a halogen broadband light and a series of images corresponding to the camera dark current. The frequency at which the dark frames were captured was low enough that the camera drift was measurable. For each frame a difference between the corresponding dark frame and the first dark frame in the series was computed. This difference was further divided by the difference between the white frame and the first dark frame in the series. The resulting series of images represented the time-lapse of baseline drift for each spatial channel. By fitting each baseline in the series it was determined that a baseline can be approximated by second order polynomial. Since the weights were nonzero only within narrow spectral bands, baseline was locally modeled by a first order polynomial (K = 1).

### 2.2 Spatial feature extraction and nonparametric SiRF characterization

*w*, may be interpreted as a sequence of blurred edges of alternating orientations. The model of the single edge with transition located at the image position

*u*

_{0}is written as a Heaviside step function H(

*υ*-

*u*

_{0}) modulated by a multiplicative

*c*

_{M}and an additive term

*c*

_{A}, arising from uneven illumination, temperature drift of the sensor and the reflected light. Therefore, in Eq. (3), the term

*T*

_{Si}(

*g*

_{Si}(

*θ*,

*λ*)) is substituted by the edge model at the wavelength

*λ*where orientation of the given edge is expressed by integer

*τ*taking values ± 1. The complexity of a realistic edge blurring was captured by modeling the SiRF as a normalized mixture of M Gaussian functions, each weighted by a nonnegative

*ρ*:Each Gaussian function in the mixture is parameterized by its center

_{m}*µ*

_{G,}

*and standard deviation*

_{m}*σ*. The convolution of a single Gaussian and a Heaviside function results in a cumulative distribution function (cdf) for the normal distribution, which can be expressed by the Gauss error function (erf). Considering the convolution properties (distributivity and scalar multiplication associativity) the convolution of SiRF and the Heaviside edge model can be considered as a weighted sum of M separate convolutions and therefore as a weighted sum of M cdfs, scaled by

_{m}*c*

_{M}and shifted by

*c*

_{A}. Taking into account the function normalization Σ

*ρ*= 1, the convolution of edge model and SiRF can be written as:The acquired image

_{m}**S**i

_{A}(

*u*,

*w*) is sampled with optical elements of finite size. Therefore, to model the recorded intensity of a pixel located at (

*u*,

*w*), Eq. (10) has to be integrated over the rectangular area of size δu × δw. Using the rectangle rule of integral approximation and taking into account the wavelength invariant intensity, the acquired profile of a single

*e*-th edge of the spectral channel

*w*can be modeled within error

**ε**

_{Si}as:The ambiguity of the parameters is resolved by imposing constraints arising from the SiRF normalization and by setting the SiRF center of gravity to zero:

_{R}of small (2∙∆u

_{R}+ 1) × (2∙∆w

_{R}+ 1) sized image regions R

*centered at (*

_{r}*u*

_{R,}

*,*

_{r}*w*

_{R,}

*). A region and corresponding variables are shown in Fig. 3. The two keys for selecting the region denoted by index*

_{r}*r*were capturing the information relevant to the convolution of the SiRF and keeping the edge

*e*

_{r}_{,}approximately in the center of the region. The spatial region size ∆u

_{R}was chosen to be the rounded half of the average distance between two adjacent edge approximations

*u*

_{R,}

*of the*

_{r}*r*-th region was set to the rounded value of

*r*can be expressed as:

*η*ϵ [

_{r}*u*

_{R,}

*-∆u*

_{r}_{R},

*u*

_{R,}

*+ ∆u*

_{r}_{R}]. Size of the region in the spectral direction (∆w

_{R}) was chosen with respect to the balance between the influence of the noise and keeping the variation of the SiRF and edge position small within the region. Spectral centers

*w*

_{R,}

*were uniformly distributed across the spectral axis to accommodate non-overlapping and aligned stack of (2∙∆w*

_{r}_{R}+ 1) pixel wide regions. Taking into account the limited spatial scope of the region (

*u*=

*η*) around the edge

_{r}*e*=

*e*, and substituting

_{r}**S**i

_{A,e}with the mean region profile into Eq. (11), the optimal parameters defining SiRF and edge positions were assessed by minimizing the sum of squared fit errors:where arrays

**ρ**

*,*

_{r}**σ**

*,*

_{r}**υ**

*contain the M parameters specifying the corresponding SiRF,*

_{r}*u*

_{0,}

*is the location of the edge encompassed by the region, and*

_{r}*c*

_{M,}

*,*

_{r}*c*

_{A,}

*are the multiplicative and additive coefficients. The summation was performed over all the elements in the region*

_{r}*η*

_{r}.*u*

_{0}

*,*

_{,r}**ρ**

*,*

_{r}**σ**

*,*

_{r}**µ**

_{G}

*) was performed by a nonlinear constrained sequential quadratic programming (SQP) [18] algorithm, while the auxiliary parameters*

_{r}*c*

_{M,}

*and*

_{r}*c*

_{A,}

*were computed by a pseudo-inverse. At each iteration of the SQP method an approximation of the Hessian of the Lagrangian function is made using the BFGS update method. Step direction is determined by a quadratic programing solution of a subproblem, which is formed using the approximated Hessian of the Lagrangian. The step size is determined by a line search procedure. The smoothness of SiRF was implicitly enforced by setting M = 3 and imposing the lower limit of the parameters*

_{r}*σ*

_{m}_{,}

*>0.5. The optimization problem was further constrained by solving the Eq. (12) and limiting the weights to nonnegative values (*

_{r}*ρ*≥0). Initial values of the edge positions were set to match the corresponding region centers

_{m,r}*u*

_{0,}

*=*

_{r}*u*

_{R,}

*. while the parameters*

_{r}*σ*

_{m}_{,}

*,*

_{r}*ρ*and

_{m,r}*υ*

_{m}_{,}

*were determined by fitting the given edge with simplified Gaussian SiRF model. According to the earlier described region size selection criterion, ∆w*

_{r}_{R}and ∆u

_{R}were set to 16 and 2, respectively. Optimal parameters for a single region were computed in approximately 0.9 s.

### 2.3 Global modeling of the computed parameters

*u,w*

_{0,}

_{l}_{,}

_{j}_{,}

*). By assigning the values of the target properties to each point of the grid, three sets of data are formed: one for the spectral tuning function*

_{u}*λ*(

*u, w*

_{0,}

_{l}_{,}

_{j}_{,}

*), and two for the SRF shape parameters*

_{u}*γ*

_{G}(

*u,w*

_{0,}

_{l}_{,}

_{j}_{,}

*) and*

_{u}*γ*

_{L}(

*u,w*

_{0,}

_{l}_{,}

_{j}_{,}

*). By fitting 2D polynomial surfaces to each of the data sets, the interpolation functions (*

_{u}*Γ*

_{G},

*Γ*

_{L}) and the inverse spectral tuning function

*Z*

_{S}were computed:

*θ*

_{0,}

*) the correspondence pairs*

_{e}*θ*

_{0,}

*(*

_{e}*u*

_{0,}

*,*

_{r}*w*

_{R,}

*) were formed and the inverse spatial tuning function*

_{r}*Z*

_{Si}was obtained by a procedure analogous to the computation of the

*Z*

_{S}:Since the SiRF was in a non-parametric form, its value at a given point was set to the SiRF determined for the nearest image region. Using the computed inverse tuning function, the coordinates of both grids were computed in the object space. The forward tuning function was determined by fitting a 2D polynomial surface to the reversed data sets: Polynomial order of

*Γ*

_{G}and

*Γ*

_{L}was five, while all the functions describing the tuning functions were seventh order.

### 2.4 Hyperspectral imaging system and experimental setup

## 3. Results and validation of the proposed characterization method

*ω*and their corresponding wavelengths

*λ*can be described by a first order polynomial model

*λ = z*

_{S,1}∙

*ω*+

*z*

_{S,0}+

*E*

_{S}. In such case, the only contribution to the fit error would be the spatially and spectrally independent noise

*E*

_{S}. However, due to the coregistration errors, the first order polynomial model has to be expanded by a term that includes spectral and spatial dependence of the inverse tuning function:

*λ*=

*z*

_{S,1}∙

*ω*+

*z*

_{S,0}+

*E*

_{S}+ δZ

_{S}(

*υ*,

*ω*). The parameters

*z*

_{S,1}and

*z*

_{S,0}were assessed by a first order polynomial fit between the true wavelengths of the spectral lines λ

_{0,}

_{l}_{,}

*and their corresponding locations*

_{j}*w*

_{0,}

_{l}_{,}

_{j}_{,}

*in all spatial channels of the acquired image (*

_{u}*z*

_{S,1}= 0.8004,

*z*

_{S,0}= 524.3097). Example of the estimated spectral line positions is shown in Figs. 5(a) and 5(b). The position dependent residual δZ

_{S}(

*υ*,

*ω*) is shown in Fig. 5(c). Likewise, the relation between true spatial position

*θ*and the corresponding spatial position in the acquired image

*υ*can be written as

*θ*=

*z*

_{Si,1}∙

*υ*+

*z*

_{Si,0}+

*E*

_{0}+ δZ

_{Si}(

*υ*,

*ω*). The parameters

*z*

_{Si,1}and

*z*

_{Si,0}were assessed by a first order polynomial fit between the true edge positions θ

_{0,}

*and their corresponding locations*

_{e}*u*

_{0,}

*in all regions R*

_{r}*of the acquired image (*

_{r}*z*

_{Si,1}= 0.1525,

*z*

_{Si,0}= 0.6108). Distribution of the extracted edge positions and the residual δZ

_{Si}, are presented in Figs. 5(d)-5(f).

*w*

_{V,}

_{j}_{,}

*of*

_{u}*j*-th spectral line at spatial channel

*u*were then transformed into corresponding wavelengths

*λ*

_{V,}

*by the inverse tuning function. The wavelength prediction error was calculated as δ*

_{j,u}*λ*

_{V}

*=*

_{,j,u}*λ*

_{V,}

_{j}_{,}

_{u}**-**

*λ*

_{V,0,}

*, where*

_{j}*λ*

_{V,0,}

*are the true wavelengths of the spectral lines. For comparison, additional spectral calibration was performed according to the method proposed by Polder et.al [7*

_{j}7. G. Polder, G. van der Heijden, L. Keizer, and I. Young, “Calibration and characterisation of imaging spectrographs,” J. Near Infrared Spectrosc. **11**(1), 193–193 (2003). [CrossRef]

*θ*,

_{q}*λ*) is formed in the object space. Each pixel

_{q}*q*is then transformed by the forward tuning function into the acquired image space (

*υ*,

_{q}*ω*) = (

_{q}*T*

_{S}(

*θ*,

_{q}*λ*)

_{q}_{,}

*T*

_{Si}(

*θ*,

_{q}*λ*)). Finally, the intensities of the object space image are computed by bicubic interpolation. Results of the validation image calibration are displayed in Fig. 11.

_{q}## 4. Conclusion

20. Ž. Špiclin, M. Bürmen, F. Pernuš, and B. Likar, “Characterization and modelling of the spatially- and spectrally-varying point-spread function in hyperspectral imaging systems for computational correction of axial optical aberrations,” Proc. SPIE **8215**, 82150R, 82150R-9 (2012). [CrossRef]

21. J. Katrašnik, F. Pernuš, and B. Likar, “Deconvolution in Acousto-Optical Tunable Filter Spectrometry,” Appl. Spectrosc. **64**(11), 1265–1273 (2010). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | R. N. Jørgensen and F. Risø, |

2. | F. W. L. Esmonde-White, K. A. Esmonde-White, and M. D. Morris, “Minor Distortions with Major Consequences: Correcting Distortions in Imaging Spectrographs,” Appl. Spectrosc. |

3. | L. Guanter, K. Segl, B. Sang, L. Alonso, H. Kaufmann, and J. Moreno, “Scene-based spectral calibration assessment of high spectral resolution imaging spectrometers,” Opt. Express |

4. | M. Meroni, L. Busetto, L. Guanter, S. Cogliati, G. F. Crosta, M. Migliavacca, C. Panigada, M. Rossini, and R. Colombo, “Characterization of fine resolution field spectrometers using solar Fraunhofer lines and atmospheric absorption features,” Appl. Opt. |

5. | Y. Feng and Y. Xiang, “Mitigation of spectral mis-registration effects in imaging spectrometers via cubic spline interpolation,” Opt. Express |

6. | D. Schlapfer, J. Nieke, and K. I. Itten, “Spatial PSF Nonuniformity Effects in Airborne Pushbroom Imaging Spectrometry Data,” IEEE Trans. Geosci. Rem. Sens. |

7. | G. Polder, G. van der Heijden, L. Keizer, and I. Young, “Calibration and characterisation of imaging spectrographs,” J. Near Infrared Spectrosc. |

8. | J. Zadnik, D. Guerin, R. Moss, A. Orbeta, R. Dixon, C. G. Simi, S. Dunbar, and A. Hill, “Calibration procedures and measurements for the COMPASS hyperspectral imager,” Proc. SPIE |

9. | T. G. Chrien, R. O. Green, and M. L. Eastwood, “Accuracy of the spectral and radiometric laboratory calibration of the Airborne Visible/Infrared Imaging Spectrometer,” Proc. SPIE |

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12. | T. Skauli, “An upper-bound metric for characterizing spectral and spatial coregistration errors in spectral imaging,” Opt. Express |

13. | E. E. Whiting, “An empirical approximation to the Voigt profile,” J. Quant. Spectroc. Rad. |

14. | C. D. Cantrell, |

15. | P. R. Griffiths and L. Shao, “Self-Weighted Correlation Coefficients and Their Application to Measure Spectral Similarity,” Appl. Spectrosc. |

16. | A. Kramida and Y. Ralchenko, J. Reader, and N. A. Team, “NIST Atomic Spectra Database (version 5.0),” (National Institute of Standards and Technology, Gaithersburg, MD., 2012). |

17. | C. T. Kelley, |

18. | R. Fletcher, |

19. | R. A. Schowengerdt, |

20. | Ž. Špiclin, M. Bürmen, F. Pernuš, and B. Likar, “Characterization and modelling of the spatially- and spectrally-varying point-spread function in hyperspectral imaging systems for computational correction of axial optical aberrations,” Proc. SPIE |

21. | J. Katrašnik, F. Pernuš, and B. Likar, “Deconvolution in Acousto-Optical Tunable Filter Spectrometry,” Appl. Spectrosc. |

**OCIS Codes**

(110.3000) Imaging systems : Image quality assessment

(120.4800) Instrumentation, measurement, and metrology : Optical standards and testing

(120.6200) Instrumentation, measurement, and metrology : Spectrometers and spectroscopic instrumentation

(150.1488) Machine vision : Calibration

(110.4234) Imaging systems : Multispectral and hyperspectral imaging

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: March 18, 2013

Revised Manuscript: April 30, 2013

Manuscript Accepted: May 1, 2013

Published: May 10, 2013

**Citation**

Matjaž Kosec, Miran Bürmen, Dejan Tomaževič, Franjo Pernuš, and Boštjan Likar, "Characterization of a spectrograph based hyperspectral imaging system," Opt. Express **21**, 12085-12099 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-10-12085

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### References

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