#### [tutorial] multispectral imaging, a case study

Without signal processing courses, the following equation should be awfully familiar to astronomers of photometry and handling data:

$$c_k=\int_\Lambda l(\lambda) r(\lambda) f_k(\lambda) \alpha(\lambda) d\lambda +n_k$$

Terms are in order, camera response (c_k), light source (l), spectral radiance by l (r), filter (f), sensitivity (α), and noise (n_k), where Λ indicates the range of the spectrum in which the camera is sensitive.

Or simplified to $$c_k=\int_\Lambda \phi_k (\lambda) r(\lambda) d\lambda +n_k$$

where φ denotes the combined illuminant and the spectral sensitivity of the k-th channel, which goes by augmented spectral sensitivity. Well, we can skip spectral radiance r, though. Unfortunately, the sensitivity α has multiple layers, not a simple closed function of λ in astronomical photometry.

Or $$c_k=\Theta r +n$$

Inverting Θ and finding a reconstruction operator such that r=inv(Θ)c_k leads spectral reconstruction although Θ is, in general, not a square matrix. Otherwise, approach from indirect reconstruction.

Studying that Smile(subscription needed)

A tutorial on multispectral imaging of paintings using the Mona Lisa as a case study

by Ribes, Pillay, Schmitt, and Lahanier

IEEE Sig. Proc. Mag.Jul. 2008, pp.14-26

Conclusions:In this article, we have presented a tutorial description of the multispectral acquisition of images from a signal processing point of view.

- From the section
**Camera Sensitivity**: “From a signal processing point of view, the filters of a multispectral camera can be conceived as sampling functions, the other elements of φ being understood as a perturbation”. - From the section
**Understanding Noise Sources**:”The noise is present in the spectral, temporal, and spatial dimensions of the image signal”. … (check out the equation and the individual term explanation) … “the*quantization*operator represent the analog-to-digital (A/D) conversion performed before stocking the signal in digital form. This conversion introduces the so-called quantization error, a theoretically predictable noise”. (This quantization error is well understood in astronomical photometry.) - Understanding the sampling function φ is common for imaging and photometry but strategies and modeling (including uncertainties by error models) are different. Figures 3, 7, 8 tell a lot about usefulness and connectivity of engineers’ spectral imaging and astronomers’ calibration.
- Hessian matrix in regression suffers similar challenges corresponding to issues related to Θ which means spectral imaging can be converted into statistical problems and likewise astronomical photometry can be put into the shape of statistical research.
- Discussion of
**Noise**is personally most worthwhile.

I wonder if there’s literature in astronomy matching this tutorial from which we may expand and improve current astronomical photometry processes by adopting strategies developed by more populated signal/image processing engineers and statisticians. (Considering good textbooks on **statistical signal processing,** and many fundamental algorithms born thanks to them, I must include statisticians. Although not discussed in this tutorial, Hidden Markov Model (HMM) is often used in signal processing but from ADS, with such keywords, no astronomical publication is aware of HMM – please, confirm my finding that HMM is not used among astronomers because my search scheme is likely imperfect.)

## vlk:

This is indeed a commonly encountered astronomical situation, and not just for photometry. DEMs too, for example.

But one of the major problems with “inverting” that matrix equation is in the high-frequency instability. How do the signal processing people deal with that? If there is a way to tamp it down based on the characteristics of the error term, that may be very useful.

10-13-2008, 1:20 pm## hlee:

I don’t think I understood your question except the part about “matrix inversion.” Related to it, the topics on compressed sensing, random matrix, and sparse matrix are diversely discussed. Books and papers about statistical image/signal processing, or filter design, or solving nonlinear systems in engineering are related to getting feasible methods of inverting matrices. Once Prof. Babu said that it is hard to find unique research topics because one is likely to find similar problems already progressed in other fields. Only jargon distinguishes and differentiate them. The challenge is, as you always say, that it isn’t easy to make other scientists understood/reformulate your problems into their terminology. I’m very much overwhelmed.

10-13-2008, 2:30 pm