## Sensor noise informed representation of hyperspectral data, with benefits for image storage and processing |

Optics Express, Vol. 19, Issue 14, pp. 13031-13046 (2011)

http://dx.doi.org/10.1364/OE.19.013031

Acrobat PDF (1131 KB)

### Abstract

Many types of hyperspectral image processing can benefit from knowledge of noise levels in the data, which can be derived from sensor physics. Surprisingly, such information is rarely provided or exploited. Usually, the image data are represented as radiance values, but this representation can lead to suboptimal results, for example in spectral difference metrics. Also, radiance data do not provide an appropriate baseline for calculation of image compression ratios. This paper defines two alternative representations of hyperspectral image data, aiming to make sensor noise accessible to image processing. A “corrected raw data” representation is proportional to the photoelectron count and can be processed like radiance data, while also offering simpler estimation of noise and somewhat more compact storage. A variance-stabilized representation is obtained by square-root transformation of the photodetector signal to make the noise signal-independent and constant across all bands while also reducing data volume by almost a factor 2. Then the data size is comparable to the fundamental information capacity of the sensor, giving a more appropriate measure of uncompressed data size. It is noted that the variance-stabilized representation has parallels in other fields of imaging. The alternative data representations provide an opportunity to reformulate hyperspectral processing algorithms to take actual sensor noise into account.

© 2011 OSA

## 1. Introduction

*photon noise*is signal-dependent, but well characterized and understood by the sensor designers. Therefore, good estimates of noise amplitude can be provided to image processing, based on the sensor physics. As will be shown here, information about sensor noise can also lead to reductions in data volume.

1. B. Penna, T. Tillo, E. Magli, and G. Olmo, “Transform coding techniques for lossy hyperspectral data compression,” IEEE Trans. Geosci. Rem. Sens. **45**(5), 1408–1421 (2007). [CrossRef]

11. G. P. Abouselman, M. W. Marcellin, and B. R. Hunt, “Compression of hyperspectral imagery using the 3-D DCT and hybrid DPCM/DCT,” IEEE Trans. Geosci. Rem. Sens. **33**(1), 26–34 (1995). [CrossRef]

14. T. Skauli, T. V. Haavardsholm, I. Kåsen, G. Arisholm, A. Kavara, T. O. Opsahl, and A. Skaugen, “An airborne real-time hyperspectral target detection system,” Proc. SPIE **7695**, 76950A, 76950A-6 (2010). [CrossRef]

15. A. S. Norsk Elektro Optikk, http://www.neo.no/products/hyperspectral.html.

16. T. Skauli, “Sensor-informed representation of hyperspectral images,” Proc. SPIE **7334**, 733418, 733418-8 (2009). [CrossRef]

## 2. Hyperspectral image recording process and information capacity

### 2.1 Basic signal model

19. Spectral Imaging, Ltd., http://www.specim.fi.

15. A. S. Norsk Elektro Optikk, http://www.neo.no/products/hyperspectral.html.

20. R. O. Green, M. L. Eastwood, C. M. Sarture, T. G. Chrien, M. Aronsson, B. J. Chippendale, J. A. Faust, B. E. Pavri, C. J. Chovit, M. Solis, M. R. Olah, and O. Williams, “Imaging Spectroscopy and the Airborne Visible/Infrared Imaging Spectrometer (AVIRIS),” Remote Sens. Environ. **65**(3), 227–248 (1998). [CrossRef]

*i*and spatial pixel index

*j*. For a pushbroom-scanning sensor, the pixel index

*j*indicates the position along the line-shaped instantaneous field of view. For a frame imaging sensor,

*j*indicates the pixel position in a two-dimensional field of view.

*λ*[

*i*] with width Δ

*λ*[

*i*], and an image pixel where the spectral radiance from the scene in band

*i*is

*L*[

*i*,

*j*] (power per unit wavelength, area and solid angle). In the notation here, index [

*i*] refers to a vector with one element for each band. Indices [

*i*,

*j*] refer to a matrix with elements corresponding to all bands and all spatial pixels seen simultaneously by the sensor. Sometimes the indices are omitted for simplicity.

*A*. For each spatial pixel

*j*, the sensor receives light over an instantaneous field of view

*Ω*(of course pointing in different directions for different pixels). Then for each spectral band the sensor selects light within a spectral range Δ

*λ*[

*i*] centered on the wavelength

*λ*[

*i*]. Light is collected over a time

*t*, known as the integration time (or exposure time). The amount of light that corresponds to this spatial, spectral and temporal selection can be expressed as the number of photons entering the sensor:where

*h*is Planck's constant and

*c*the speed of light. Note that Eq. (1) is a simplifying notation where dense spectral sampling is assumed, so that scene radiance and photon energy can be assumed to be constant within the band. Otherwise the multiplication by Δ

*λ*must be replaced by an appropriate integral over wavelength.

*photoelectrons*, are collected and become the electrical signal from the photodetector. The resulting signal can be expressed as a photoelectron count

*η*[

*i*,

*j*] is the

*quantum efficiency*, which accounts for signal losses and sensor response nonuniformity as discussed below. The last terms in Eq. (2) account for unwanted contributions to the signal: The

*dark current I*

_{d}[

*i*,

*j*] (electrons per unit time in a particular detector element) is apparent signal from effects such as thermally excited electrons or thermal radiation emitted within the sensor enclosure. The term δ

*N*accounts for the

*readout noise*, the total signal variability in the transfer and amplification of the photoelectron signal. The readout noise is assumed to be Gaussian with zero mean and standard deviation δ

*N*.

*G*, the photoelectron signal is digitized to raw digital numbers

*D*

_{raw}:where

*round*() represents the rounding of analog signals to discrete values in digitization.

*D*

_{0}represents a possible offset in the readout electronics. In the following, it will be assumed that

*D*

_{0}is zero, or that the offset has been subtracted.

*G*<1 so that each increment in

*D*

_{raw}corresponds to several photoelectrons. Some newer types of commercial photodetector array can work meaningfully with

21. J. Hynecek, “Impactron - A new solid state image intensifier,” IEEE Trans. Electron. Dev. **48**(10), 2238–2241 (2001). [CrossRef]

22. B. Fowler, C. Liu, S. Mims, J. Balicki, W. Li, H. Do, J. Appelbaum, and P. Vu, “A 5.5Mpixel 100 frames/sec wide dynamic range low noise CMOS image sensor for scientific applications,” Proc. SPIE **7536**, 753607, 753607-12 (2010). [CrossRef]

23. J. D. Beck, C.-F. Wan, M. A. Kinch, J. E. Robinson, P. Mitra, R. E. Scritchfield, F. Ma, and J. C. Campbell, “The HgCdTe electron avalanche photodiode,” J. Electron. Mater. **35**(6), 1166–1173 (2006). [CrossRef]

*full well capacity*, here denoted

*N*

_{max}. This effect sets an upper limit on the measurable signal range and on the range of

*D*

_{raw}. Stronger signals are said to be in

*saturation*, and will typically produce some maximum output value

*D*

_{max}.

*D*

_{raw}[

*i*,

*j*] values form a spectral image cube, or a part of it, depending on the sensor type. (Recall that

*j*indexes the spatial pixels that are seen simultaneously by the sensor.) Before processing, the data are usually first corrected for sensor effects and transformed to represent for example radiance values. The data correction and representation is a central topic of this paper.

### 2.2 Quantum efficiency

*η*, which describes signal losses and is of particular interest for a hyperspectral sensor. A part of the loss occurs in the optics, due to reflections or absorption as well as to inefficiencies in the spectral selection. Another part of the loss occurs in the photodetector element: Photons may get absorbed by unwanted processes that do not produce a photoelectron, and some photoelectrons may be lost and not become part of the signal. In Eq. (2), the quantum efficiency matrix

*η*[

*i*,

*j*] is defined to account for all losses through the system. For simplicity at this point, the matrix

*η*[

*i*,

*j*] also incorporates residual nonuniformities of gain and optical throughput, assumed to be small. Conventionally, quantum efficiency is defined as a property of the detector only, or lumped with other factors into a responsivity value. For the sensor model here, however, it is relevant to define

*η*for the entire sensor.

*η*is illustrated by the realistic example in Fig. 2 . The combined effects of the grating and photodetector wavelength dependencies lead to a very large relative variationof

*η*across the spectral range of the sensor.

### 2.3 Noise properties of raw data

*N*is a Poisson-distributed random variable. This random variation is the “photon noise”, which is a fundamental lower limit on the noise properties of a photodetector.

*N*the noise increases, but its relative amplitude becomes smaller.

*A*,

*Ω*,

*η, t*and Δ

*λ*pertaining to the particular sensor, since these parameters determine

*N*according to Eqs. (1) and (2). The noise performance can still be improved by modifying the sensor to increase the value of the

*AΩηt*Δ

*λ*product. However increasing

*AΩη*tends to be expensive and technologically difficult, while increasing

*t*and Δ

*λ*may not be permitted by the application.

*N*. The typical error in

*I*

_{d}[

*i*,

*j*] to the signal, according to Eq. (2), can also be assumed to follow a Poisson distribution. Thus in cases where the photoelectron signal is small, dark current can become the dominating source of noise.

*digitization noise*, expressed by the rounding in Eq. (3) has an RMS amplitude [17] of

*D*

_{raw}scale, corresponding to

20. R. O. Green, M. L. Eastwood, C. M. Sarture, T. G. Chrien, M. Aronsson, B. J. Chippendale, J. A. Faust, B. E. Pavri, C. J. Chovit, M. Solis, M. R. Olah, and O. Williams, “Imaging Spectroscopy and the Airborne Visible/Infrared Imaging Spectrometer (AVIRIS),” Remote Sens. Environ. **65**(3), 227–248 (1998). [CrossRef]

### 2.4 Example illustrating photon noise in raw data

^{16}electrons and the signal is read out as

*D*

_{raw}values with 12 bits of precision. Then each step in

*D*

_{raw}corresponds to 16 electrons. Figure 3 shows a set of simulated random

*D*

_{raw}values drawn over the full range of

*N*, illustrating how photon noise varies with signal level. For a signal level of

*N*=2

^{8}electrons, the standard deviation of photon noise is

*D*

_{raw}step size. Below this signal level, rounding errors in

*D*

_{raw}start to become non-negligible, and higher precision of the digitization may be needed to avoid degradation of the signal to noise ratio. At the other end of the scale, near the full well signal level, the standard deviation of photon noise is 2

^{8}electrons, which corresponds to 16 steps in

*D*

_{raw}. We see that for high signal levels, resolution is wasted on digitizing noise. Therefore, the raw data representation of hyperspectral images may be suboptimal with respect to either accuracy or data volume or both, depending on the resolution of the digitization.

### 2.5 Information rate of the hyperspectral sensor

*N*

_{max}then, in the limit of large

*N*

_{max}, an upper bound for the information capacity isbits per sample. In terms of imaging,

*electrons then the information capacity of the imaging system is*

^{W}*W*/2-1 bits per sample in the limit of large

*N*

_{max}. In the example of Fig. 3, the information capacity of the sensor is at most 7 bits per sample, consistent with the observation that 12-bit data have excess precision over most of the dynamic range. Thus the information-theoretic limit provides clear motivation to choose more efficient data representations.

## 3. Conventional and alternative representations for hyperspectral images

### 3.1 Raw data

_{raw}can only be processed with the aid of a large set of metadata to account for sensor properties, including the pixel-to-pixel variations described by the dark current and quantum efficiency matrices

*I*

_{d}[

*i*,

*j*] and

*η*[

*i*,

*j*]. Therefore it is generally difficult to process raw data directly. This representation of hyperspectral images is at best only suitable for systems where processing is very tightly integrated with the sensor.

### 3.2 Radiance representation

*D*

_{raw}are converted to estimated values for incoming radiance in a pre-processing step, based on sensor calibration data. In principle, the radiance estimates are found from Eqs. (1) to (3), but in practice the conversion to radiance is based on gain and offset correction matrices

*C*

_{1}[

*i*,

*j*] and

*C*

_{2}[

*i*,

*j*] measured during sensor calibration:were

*C*

_{1}[

*i*,

*j*] and

*C*

_{2}[

*i*,

*j*] and the gain

*G*, a data user can estimate the photon noise from image data represented as radiance. A somewhat inelegant aspect of Eq. (7) is that the data user needs the full calibration matrices of the sensor to estimate noise. The next section discusses ways to simplify the metadata needed for noise estimation.

*D*

_{max}corresponds to a very large radiance, as expressed by Eq. (6). Therefore the radiance representation must allow a larger range of values than for the raw data. As an example, consider the case in Fig. 2, where the quantum efficiency at the blue end of the spectrum is about 10% of its peak value. Then, roughly speaking, the blue band can measure a 10 times higher radiance value without saturating. If the data are represented as integers then the bit width must be increased by log

_{2}10=3.3 bits, compared to the raw data, to preserve the full dynamic range of the sensor. The example sensor with 12-bit raw data would then need a data size of 16 bits to store the data as radiance values without loss of precision. An obvious way around the increase in data size is to use a different scaling factor for each band for conversion of radiance values to integers. The following section introduces such a scheme, with the added benefit of easy noise estimation.

### 3.3 Radiance-linear “corrected raw data” representation

*I*

_{d}[

*i*,

*j*], is of little interest in data processing, but the average amount of dark current could be of interest for noise estimation. For the quantum efficiency

*η*[

*i*,

*j*], spatial variation within a band is usually small to moderate, while the spectral variation can be large, as discussed above. In the following, a transformation of the raw data is developed that compensates for small and uninteresting sensor effects. The transformation produces “corrected raw data” that can be seen as the output of an idealized sensor, for which the sensor physics is more easily accessible.

*η*[

*i*,

*j*] is split into two parts. Let

*η*[

*i*] (with only a band index) represent the average quantum efficiency over all spatial pixels for band

*i*. Thus for a sensor with

*P*spatial pixels,

*F*[

*i*,

*j*] be the residual responsivity nonuniformity error defined by

*F*is 1. Here,

*F*accounts for small differences in loss and gain between individual photodetector elements, as well as for residual variation in the optics throughput

*AΩ*which was taken to be constant in Eq. (1). The range of

*F*[

*i*,

*j*] values depends on the details of the sensor, but should be well within 0.5<

*F*[

*i*,

*j*]<2 in practice. With

*D*

_{max}being the full scale of raw data, let

*C*

_{max}>

*D*

_{max}be the new maximum data value for the corrected raw data. Define a scaling factor

*S*so that

*F*[

*i*,

*j*] and

*I*

_{d}[

*i*,

*j*] to correct the data for pixel-to-pixel nonuniformities and then scaling by S, we obtain the corrected raw data representation

*D*

_{C}:

*D*

_{C}[

*i*,

*j*] can be considered to be scaled photoelectron counts of an idealized sensor without pixel-to-pixel nonuniformities and with gain

*S*(noting that

*D*

_{C}data, with appropriate rescaling of spectrally varying algorithm parameters, since

*D*

_{C}[

*i*,

*j*] values are readily converted to radiance estimates:where

*K*[

*i*].

*F*[

*i*,

*j*] and dark current matrix

*I*

_{d}[

*i*,

*j*]. However, it can be noted that estimation of noise can normally be permitted to be less accurate than estimation of radiance. Therefore, it is generally acceptable to neglect the pixel-to-pixel variations of

*F*[

*i*,

*j*] (whose mean is 1) in estimation of photon noise. Also, variations in gain

*G*between photodetector elements is usually small and therefore neglected here. To estimate the total noise, dark current and readout noise should also be considered. Unless there are large nonuniformities of dark current

*I*

_{d}[

*i*,

*j*] across the detector array, the noise contribution from dark current can be represented by an average value. The variance contributions from dark current and readout noise δ

*N*can be combined in a parameter

*N*:

_{0}*I*

_{d}[

*i*,

*j*]. We can then calculate an effective photoelectron count which is actually an approximation to the total signal variance:where the nonuniformities described by

*F*[

*i*,

*j*] and

*I*

_{d}[

*i*,

*j*] have been neglected. Under this approximation, an estimate of the noise amplitude, in photoelectrons, is obtained from the estimated standard deviationThe equivalent radiance noise can then be estimated from

*D*

_{C}data can be straightforwardly converted to estimates of radiance

*and*noise, given the simple parameter set

*K*[

*i*],

*S*and, when needed,

*N*

_{0}. In place of

*K*[

*i*], the image metadata can preferably include explicit values for the factors entering in

*K*[

*i*]:

*η*[

*i*],

*t*,

*A*and

*Ω*. Of course, values for Δ

*λ*[

*i*] and

*λ*[

*i*] are also supplied with the image. Then the image metadata not only enable calculation of radiance and noise, but also provide the user with a first-order model for the sensor.

*D*

_{C}values. If

*F*

_{min}is the minimum value in the responsivity correction matrix

*F*[

*i*,

*j*], then the possible range of

*D*

_{C}values is

*D*

_{C}data to an

*n*-bit format, choose

*C*

_{max}=2

*-1, which determines*

^{n}*S*from Eq. (8). Observe that the

*D*

_{C}data range is unaffected by the quantum efficiency variations described by

*η*[

*i*], and that

*D*

_{C}[

*i*,

*j*] is well bounded, to a much smaller range than for radiance values at a comparable level of rounding error.

_{C}values introduces an error with RMS amplitude

*D*

_{C}scale [17]. On the same scale, the digitization error in the raw data is

*D*

_{C}data are stored in a format 1 bit wider than

*D*

_{raw}then

*C*

_{max}=2

*D*

_{max}and the rounding of

*D*

_{C}values increases the total rounding error by ≈12%. This slight increase should be unproblematic, assuming that digitization noise in the raw data is already small compared to the photon noise.

*F*[

*i*,

*j*.] The element with the highest value of

*F*[

*i*,

*j*], denoted

*F*

_{max}, experiences the largest additional rounding error, as seen from Eq. (9). The error due to rounding of

*D*

_{C}values remains less than the digitization noise in

*D*

_{raw}as long as

*C*

_{max}>

*F*

_{max}

*D*

_{max}. Note that as long as this condition holds, it is possible to reconstruct the raw data exactly, using the full calibration data, since the steps in

*D*

_{C}are smaller than the steps of

*D*

_{raw}on a radiance scale. Therefore, sensor data may be stored directly as

*D*

_{C}data during recording

*without any loss of information*, even if the actual raw data are not stored. For the example sensor above with 12-bit raw data, the

*D*

_{C}data could be stored losslessly in 13 bits even if

*F*

_{max}were as large as 2.

*D*

_{C}can be processed directly and are easily converted to radiance and to estimates of sensor noise. The required metadata are simple parameters of a first-order sensor model, in itself informative for the data user. With a very modest increase in data size over the raw data, on the order of 1 bit per sample, the

*D*

_{C}representation is fully lossless. This data representation has been implemented as an option for data storage in the HySpex line of commercial hyperspectral sensors [15

15. A. S. Norsk Elektro Optikk, http://www.neo.no/products/hyperspectral.html.

*D*

_{C}data in real time. This data stream is used for data processing in FFI's airborne real-time processing system [14

14. T. Skauli, T. V. Haavardsholm, I. Kåsen, G. Arisholm, A. Kavara, T. O. Opsahl, and A. Skaugen, “An airborne real-time hyperspectral target detection system,” Proc. SPIE **7695**, 76950A, 76950A-6 (2010). [CrossRef]

### 3.4 Image data representation with variance stabilization

*D*

_{C}representation, will be inefficient with respect to storage space, due to the signal-dependent noise. By applying a

*variance stabilizing transform*to the data, the noise can be made approximately signal-independent. A well known variance-stabilizing transform is the Anscombe transform [25]. For the Poisson-distributed quantity

*N*, the Anscombe transform is

27. R. L. White and J. W. Percival, “Compression and progressive transmission of astronomical images,” Proc. SPIE **2199**, 703–713 (1994). [CrossRef]

31. G. M. Bernstein, C. Bebek, J. Rhodes, C. Stoughton, R. A. Vanderveld, and P. Yeh, “Noise and bias in square-root compression schemes,” Proc. Astron. Soc. Pacific **122**(889), 336–346 (2010). [CrossRef]

34. B. Zhang, M. J. Fadili, J.-L. Starck, and J.-C. Olivo-Marin, “Multiscale variance-stabilizing transform for mixed-poisson-gaussian processes and its Applications in Bioimaging” in *Proceedings of IEEE International conference on image processing* (Institute of Electrical and Electronics Engineers, New York, 2007) pp. VI233–VI236.

*S*

_{R}, which will be used to rescale the data for storage, as

*S*does for

*D*

_{C}above. The statistical distribution of the transformed data

*R*is very close to Gaussian with variance

*R*representation: As opposed to radiance-linear representations, the nonlinearly transformed

*R*data conform to the often-used assumption of signal-independent additive Gaussian noise.

*R*data can be readily converted to radiance and to estimates of radiance noise. Following Eqs. (10) and (14),

*D*

_{C}representation, the user can reconstruct radiance and noise from the

*R*data with a simple set of metadata that also give a first-order description of the sensor.

*R*as integers, after rounding. The scaling factor

*S*

_{R}should be chosen so that the rounding error is negligible. Due to the nonlinear transformation, rounding error in R cannot be referred to digitization error in

*D*

_{raw}. Instead, it must be compared to the photon noise. The relative noise contribution of the

*S*

_{R}=1 gives

*r*=0.58. Then the RMS sum of photon noise and rounding error is 15% larger than photon noise alone, which may not be acceptable. Choosing

*S*

_{R}=2 gives

*r*=0.29 and a 4% higher noise, which should be tolerable in many cases: As pointed out in [31

31. G. M. Bernstein, C. Bebek, J. Rhodes, C. Stoughton, R. A. Vanderveld, and P. Yeh, “Noise and bias in square-root compression schemes,” Proc. Astron. Soc. Pacific **122**(889), 336–346 (2010). [CrossRef]

*S*

_{R}=2 can be compensated by an 8% increase in exposure time. Expressed this way, the consequences of any given choice of

*S*

_{R}can be readily assessed. The particular choice

*S*

_{R}=2 transforms the photon noise conveniently into a constant variance and standard deviation of 1.

31. G. M. Bernstein, C. Bebek, J. Rhodes, C. Stoughton, R. A. Vanderveld, and P. Yeh, “Noise and bias in square-root compression schemes,” Proc. Astron. Soc. Pacific **122**(889), 336–346 (2010). [CrossRef]

*R*values introduces a small bias in the reconstructed radiance corresponding to 1/12 photoelectron for

*S*

_{R}=2, which is negligible in most cases but can easily be corrected for.

*N*

_{max}=2

^{16}. We note that with

*S*

_{R}=1, the full-scale value of

*R*is 2

^{8}and the image data can be fitted in a 1-byte format. With

*S*

_{R}=2, the data can be stored in 9 bits per sample with only a 4% degradation of SNR. Thus the variance-stabilized

*R*representation is quite efficient with respect to data storage and approaches the minimum data size of 7 bits given by the information capacity (5).

## 4 Possible sensor-related refinements of the data representation

### 4.1 Low photoelectron counts

*N*in Eq. (14) incorporates the noise from dark current and readout so that the noise estimation in Eq. (15) is valid to very low signal levels. At the same time, this definition of

_{0}*R*is consistent with the recommended practice to keep the zero-signal point within the data range [18] to be able to detect offset drift errors. The

*D*representation, on the other hand, is defined to be proportional to radiance, and the dark current term is subtracted in Eq. (9). When the signal is comparable to or smaller than the dark current then the subtraction of a mean dark current may lead to a negative output value for some samples. In many cases these negative values can be truncated to zero for storage in an unsigned integer format. However such truncation would tend to skew the data distribution in a way that is not consistent with the sensor physics. An alternative definition of

_{C}*D*

_{C}would be to incorporate an offset term, for example

*N*

_{0}, so that zero signal is represented by a small positive

*D*value.

_{C}### 4.2 Large constant background level

*R*will vary over a small range relative to their mean. Then essentially the

*R*values have a large constant offset or pedestal value

*R*, which is undesirable for storage in a compact format. However this offset can be subtracted and supplied as metadata, resulting in modified data

_{min}### 4.3 Saturation

*D*

_{max}. Note that if this value is converted into any of the representations discussed above, it may result in an output value falling within the range of normal unsaturated data. Therefore the saturated light samples need to be treated specially.

*D*

_{C}representation in an

*n*-bit format, using

*C*

_{max}=2

*-2 guarantees that the maximum value will never be used for normal data.*

^{n}### 4.4 Bad sensor elements

### 4.5 Known upper limit for the signal spectrum

## 5. Summary and discussion

### 5.1 Noise and data processing

27. R. L. White and J. W. Percival, “Compression and progressive transmission of astronomical images,” Proc. SPIE **2199**, 703–713 (1994). [CrossRef]

34. B. Zhang, M. J. Fadili, J.-L. Starck, and J.-C. Olivo-Marin, “Multiscale variance-stabilizing transform for mixed-poisson-gaussian processes and its Applications in Bioimaging” in *Proceedings of IEEE International conference on image processing* (Institute of Electrical and Electronics Engineers, New York, 2007) pp. VI233–VI236.

### 5.2 Storage and compression

*R*data can be stored in a data volume comparable to the information capacity, depending on the desired accuracy. In the example case used throughout the text, the raw data are 12 bits,

*D*

_{C}data would require 13 bits per sample for lossless storage and radiance data would require 16 bits. In comparison, it was shown that variance-stabilized

*R*data could be stored in 9 bits, although with a small (4%) degradation of the signal to noise ratio. It is thus possible to achieve a near-lossless compression on the order of 50% just by transforming the data based on basic measurement physics. Of course, a much higher compression level can be achieved by proper compression algorithms that exploit redundancies in spectral, spatial or temporal image information. However, it appears reasonable to say that square-root transformation followed by generic lossless data compression is an appropriate baseline for calculation of data compression ratios.

### 5.3 The many virtues of the square-root transformation

36. M. Stokes, M. Anderson, S. Chandrasekar, and R. Motta, “A standard default color space for the internet - sRGB”, http://www.w3.org/Graphics/Color/sRGB (1996).

## 6. Conclusions

*D*and variance-stabilized data

_{C}*R*have been proposed here. These representations facilitate the use of physical noise estimates in processing of hyperspectral data. There is an obvious potential in reformulating many image processing algorithms to work on variance-stabilized data. The variance-stabilized representation also allows compact data storage approaching the information-theoretic limit, and is an appropriate baseline for evaluation of hyperspectral image data compression. If knowledge of sensor noise is not taken into account, there is an obvious risk that image processing results are suboptimal, and that data volumes are much larger than necessary.

## References and links

1. | B. Penna, T. Tillo, E. Magli, and G. Olmo, “Transform coding techniques for lossy hyperspectral data compression,” IEEE Trans. Geosci. Rem. Sens. |

2. | Q. Du and J. E. Fowler, “Hyperspectral image compression using JPEG2000 and principal component analysis,” IEEE Geosci. Remote Sens. Lett. |

3. | J. Wang and C. I. Chang, “Independent component analysis-based dimensionality reduction with applications in hyperspectral image analysis,” IEEE Trans. Geosci. Rem. Sens. |

4. | B. Penna, T. Tillo, E. Magli, and G. Olmo, “Progressive 3-D coding of hyperspectral images based on JPEG 2000,” IEEE Geosci. Remote Sens. Lett. |

5. | J. Mielikainen and P. Toivanen, “Clustered DPCM for the lossless compression of hyperspectral images,” IEEE Trans. Geosci. Rem. Sens. |

6. | B. Aiazzi, L. Alparone, and S. Baronti, “Near-lossless compression of 3-D optical data,” IEEE Trans. Geosci. Rem. Sens. |

7. | M. J. Ryan and J. F. Arnold, “Lossy compression of hyperspectral data using vector quantization,” Remote Sens. Environ. |

8. | M. J. Ryan and J. F. Arnold, “The lossless compression of AVIRIS images by vector quantization,” IEEE Trans. Geosci. Rem. Sens. |

9. | S. E. Qian, A. B. Hollinger, D. Williams, and D. Manak, “Fast three-dimensional data compression of hyperspectral imagery using vector quantization with spectral-feature-based binary coding,” Opt. Eng. |

10. | R. E. Roger and M. C. Cavenor, “Lossless compression of AVIRIS images,” IEEE Trans. Image Process. |

11. | G. P. Abouselman, M. W. Marcellin, and B. R. Hunt, “Compression of hyperspectral imagery using the 3-D DCT and hybrid DPCM/DCT,” IEEE Trans. Geosci. Rem. Sens. |

12. | Recent papers on hyperspectral compression were retrieved from the ISI Web of science database in January 2011, but not listed here for space reasons. References are available from the author. |

13. | J. W. Boardman, “Using dark current data to estimate AVIRIS noise covariance and improve spectral processing” in |

14. | T. Skauli, T. V. Haavardsholm, I. Kåsen, G. Arisholm, A. Kavara, T. O. Opsahl, and A. Skaugen, “An airborne real-time hyperspectral target detection system,” Proc. SPIE |

15. | A. S. Norsk Elektro Optikk, http://www.neo.no/products/hyperspectral.html. |

16. | T. Skauli, “Sensor-informed representation of hyperspectral images,” Proc. SPIE |

17. | E. L. Dereniak and G. D. Boreman, |

18. | P. C. D. Hobbs, |

19. | Spectral Imaging, Ltd., http://www.specim.fi. |

20. | R. O. Green, M. L. Eastwood, C. M. Sarture, T. G. Chrien, M. Aronsson, B. J. Chippendale, J. A. Faust, B. E. Pavri, C. J. Chovit, M. Solis, M. R. Olah, and O. Williams, “Imaging Spectroscopy and the Airborne Visible/Infrared Imaging Spectrometer (AVIRIS),” Remote Sens. Environ. |

21. | J. Hynecek, “Impactron - A new solid state image intensifier,” IEEE Trans. Electron. Dev. |

22. | B. Fowler, C. Liu, S. Mims, J. Balicki, W. Li, H. Do, J. Appelbaum, and P. Vu, “A 5.5Mpixel 100 frames/sec wide dynamic range low noise CMOS image sensor for scientific applications,” Proc. SPIE |

23. | J. D. Beck, C.-F. Wan, M. A. Kinch, J. E. Robinson, P. Mitra, R. E. Scritchfield, F. Ma, and J. C. Campbell, “The HgCdTe electron avalanche photodiode,” J. Electron. Mater. |

24. | A. Martinez, “Capacity bounds for the Einstein radiation channel,” Proc. Int. Symp. Inf. Theory, ISIT 2006, 9–14 July 2006, Seattle (USA), pp. 366–370 (2006). |

25. | F. J. Anscombe, “The transformation of Poisson, binomial and negative-binomial data,” Biometrika |

26. | J. A. Rice, |

27. | R. L. White and J. W. Percival, “Compression and progressive transmission of astronomical images,” Proc. SPIE |

28. | M. A. Nieto-Santisteban, D. J. Fixsen, J. D. Offenberg, R. J. Hanisch, and H. S. Stockman, “Data compression for NGST” in |

29. | R. A. Gowen and A. Smith, “Square root data compression,” Rev. Sci. Instrum. |

30. | R. L. Seaman, R. L. White, and W. D. Pence, “Optimal DN encoding for CCD detectors” in |

31. | G. M. Bernstein, C. Bebek, J. Rhodes, C. Stoughton, R. A. Vanderveld, and P. Yeh, “Noise and bias in square-root compression schemes,” Proc. Astron. Soc. Pacific |

32. | F. Murtagh, J.-L. Starck, and A. Bijaoui, “Image restoration with noise suppression using a multiresolution support,” Astron. Astrophys. Suppl. Ser. |

33. | J.-L. Starck and F. Murtagh, “Astronomical image and signal processing: looking at noise, information, and scale,” IEEE Signal Process. Mag. |

34. | B. Zhang, M. J. Fadili, J.-L. Starck, and J.-C. Olivo-Marin, “Multiscale variance-stabilizing transform for mixed-poisson-gaussian processes and its Applications in Bioimaging” in |

35. | C. Poynton, |

36. | M. Stokes, M. Anderson, S. Chandrasekar, and R. Motta, “A standard default color space for the internet - sRGB”, http://www.w3.org/Graphics/Color/sRGB (1996). |

**OCIS Codes**

(110.4280) Imaging systems : Noise in imaging systems

(100.4145) Image processing : Motion, hyperspectral image processing

(110.4234) Imaging systems : Multispectral and hyperspectral imaging

**ToC Category:**

Image Processing

**History**

Original Manuscript: January 19, 2011

Revised Manuscript: June 2, 2011

Manuscript Accepted: June 8, 2011

Published: June 22, 2011

**Citation**

Torbjørn Skauli, "Sensor noise informed representation of hyperspectral data, with benefits for image storage and processing," Opt. Express **19**, 13031-13046 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-14-13031

Sort: Year | Journal | Reset

### References

- B. Penna, T. Tillo, E. Magli, and G. Olmo, “Transform coding techniques for lossy hyperspectral data compression,” IEEE Trans. Geosci. Rem. Sens. 45(5), 1408–1421 (2007). [CrossRef]
- Q. Du and J. E. Fowler, “Hyperspectral image compression using JPEG2000 and principal component analysis,” IEEE Geosci. Remote Sens. Lett. 4(2), 201–205 (2007). [CrossRef]
- J. Wang and C. I. Chang, “Independent component analysis-based dimensionality reduction with applications in hyperspectral image analysis,” IEEE Trans. Geosci. Rem. Sens. 44(6), 1586–1600 (2006). [CrossRef]
- B. Penna, T. Tillo, E. Magli, and G. Olmo, “Progressive 3-D coding of hyperspectral images based on JPEG 2000,” IEEE Geosci. Remote Sens. Lett. 3(1), 125–129 (2006). [CrossRef]
- J. Mielikainen and P. Toivanen, “Clustered DPCM for the lossless compression of hyperspectral images,” IEEE Trans. Geosci. Rem. Sens. 41(12), 2943–2946 (2003). [CrossRef]
- B. Aiazzi, L. Alparone, and S. Baronti, “Near-lossless compression of 3-D optical data,” IEEE Trans. Geosci. Rem. Sens. 39(11), 2547–2557 (2001). [CrossRef]
- M. J. Ryan and J. F. Arnold, “Lossy compression of hyperspectral data using vector quantization,” Remote Sens. Environ. 61(3), 419–436 (1997). [CrossRef]
- M. J. Ryan and J. F. Arnold, “The lossless compression of AVIRIS images by vector quantization,” IEEE Trans. Geosci. Rem. Sens. 35(3), 546–550 (1997). [CrossRef]
- S. E. Qian, A. B. Hollinger, D. Williams, and D. Manak, “Fast three-dimensional data compression of hyperspectral imagery using vector quantization with spectral-feature-based binary coding,” Opt. Eng. 35(11), 3242–3249 (1996). [CrossRef]
- R. E. Roger and M. C. Cavenor, “Lossless compression of AVIRIS images,” IEEE Trans. Image Process. 5(5), 713–719 (1996). [CrossRef] [PubMed]
- G. P. Abouselman, M. W. Marcellin, and B. R. Hunt, “Compression of hyperspectral imagery using the 3-D DCT and hybrid DPCM/DCT,” IEEE Trans. Geosci. Rem. Sens. 33(1), 26–34 (1995). [CrossRef]
- Recent papers on hyperspectral compression were retrieved from the ISI Web of science database in January 2011, but not listed here for space reasons. References are available from the author.
- J. W. Boardman, “Using dark current data to estimate AVIRIS noise covariance and improve spectral processing” in Summaries of the Fifth Annual JPL Airborne Geoscience Workshop, Pasadena, CA 1995.
- T. Skauli, T. V. Haavardsholm, I. Kåsen, G. Arisholm, A. Kavara, T. O. Opsahl, and A. Skaugen, “An airborne real-time hyperspectral target detection system,” Proc. SPIE 7695, 76950A, 76950A-6 (2010). [CrossRef]
- A. S. Norsk Elektro Optikk, http://www.neo.no/products/hyperspectral.html .
- T. Skauli, “Sensor-informed representation of hyperspectral images,” Proc. SPIE 7334, 733418, 733418-8 (2009). [CrossRef]
- E. L. Dereniak and G. D. Boreman, Infrared Detectors and Systems (Wiley, New York 1996).
- P. C. D. Hobbs, Building Electro-Optical Systems: Making It All Work, 2nd ed. (Wiley, 2009).
- Spectral Imaging, Ltd., http://www.specim.fi .
- R. O. Green, M. L. Eastwood, C. M. Sarture, T. G. Chrien, M. Aronsson, B. J. Chippendale, J. A. Faust, B. E. Pavri, C. J. Chovit, M. Solis, M. R. Olah, and O. Williams, “Imaging Spectroscopy and the Airborne Visible/Infrared Imaging Spectrometer (AVIRIS),” Remote Sens. Environ. 65(3), 227–248 (1998). [CrossRef]
- J. Hynecek, “Impactron - A new solid state image intensifier,” IEEE Trans. Electron. Dev. 48(10), 2238–2241 (2001). [CrossRef]
- B. Fowler, C. Liu, S. Mims, J. Balicki, W. Li, H. Do, J. Appelbaum, and P. Vu, “A 5.5Mpixel 100 frames/sec wide dynamic range low noise CMOS image sensor for scientific applications,” Proc. SPIE 7536, 753607, 753607-12 (2010). [CrossRef]
- J. D. Beck, C.-F. Wan, M. A. Kinch, J. E. Robinson, P. Mitra, R. E. Scritchfield, F. Ma, and J. C. Campbell, “The HgCdTe electron avalanche photodiode,” J. Electron. Mater. 35(6), 1166–1173 (2006). [CrossRef]
- A. Martinez, “Capacity bounds for the Einstein radiation channel,” Proc. Int. Symp. Inf. Theory, ISIT 2006, 9–14 July 2006, Seattle (USA), pp. 366–370 (2006).
- F. J. Anscombe, “The transformation of Poisson, binomial and negative-binomial data,” Biometrika 35, 246–254 (1948) (Curiously, Anscombe credits the “Anscombe transform” to A. H. L. Johnson.).
- J. A. Rice, Mathematical Statistics and Data Analysis, (Duxbury press, 1995) p. 321.
- R. L. White and J. W. Percival, “Compression and progressive transmission of astronomical images,” Proc. SPIE 2199, 703–713 (1994). [CrossRef]
- M. A. Nieto-Santisteban, D. J. Fixsen, J. D. Offenberg, R. J. Hanisch, and H. S. Stockman, “Data compression for NGST” in Proceedings of Astronomical data analysis software and systems VIII, ASP Conference series 172, 137–140 (1999).
- R. A. Gowen and A. Smith, “Square root data compression,” Rev. Sci. Instrum. 74(8), 3853–3861 (2003). [CrossRef]
- R. L. Seaman, R. L. White, and W. D. Pence, “Optimal DN encoding for CCD detectors” in Proceedings of Astronomical Data Analysis Software and Systems XVII, ASP Conference Series, Vol. 411, 101 (2009).
- G. M. Bernstein, C. Bebek, J. Rhodes, C. Stoughton, R. A. Vanderveld, and P. Yeh, “Noise and bias in square-root compression schemes,” Proc. Astron. Soc. Pacific 122(889), 336–346 (2010). [CrossRef]
- F. Murtagh, J.-L. Starck, and A. Bijaoui, “Image restoration with noise suppression using a multiresolution support,” Astron. Astrophys. Suppl. Ser. 112, 179–189 (1995).
- J.-L. Starck and F. Murtagh, “Astronomical image and signal processing: looking at noise, information, and scale,” IEEE Signal Process. Mag. 18(2), 30–40 (2001). [CrossRef]
- B. Zhang, M. J. Fadili, J.-L. Starck, and J.-C. Olivo-Marin, “Multiscale variance-stabilizing transform for mixed-poisson-gaussian processes and its Applications in Bioimaging” in Proceedings of IEEE International conference on image processing (Institute of Electrical and Electronics Engineers, New York, 2007) pp. VI233–VI236.
- C. Poynton, A Technical Introduction to Digital Video (Wiley, 1996), Chap. 6.
- M. Stokes, M. Anderson, S. Chandrasekar, and R. Motta, “A standard default color space for the internet - sRGB”, http://www.w3.org/Graphics/Color/sRGB (1996).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.