[tutorial] multispectral imaging, a case study

hlee — Thu, 09 Oct 2008 20:28:21 +0000

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.)

Why Gaussianity?

hlee — Wed, 10 Sep 2008 14:15:03 +0000

Physicists believe that the Gaussian law has been proved in mathematics while mathematicians think that it was experimentally established in physics — Henri Poincare

Couldn’t help writing the quote from this article (subscription required).^[1]

Why Gaussianity? by Kim, K. and Shevlyakov, G. (2008) IEEE Signal Processing Magazine, Vol. 25(2), pp. 102-113

It’s been a while since my post, signal processing and bootstrap from IEEE signal processing magazine, described as tutorial style papers on signal processing research and applications. Because of its tutorial style, the magazine delivers most up to date information and applications to people in various disciplines (their citation rate is quite high among scientific fields where data are collected via digitization except astronomy. This statement is solely based on my experience and no proper test was carried out to test this hypothesis). This provoking title, perhaps, will drag attentions about advances in signal processing from astronomers in future.

A historical account on Gaussian distribution, which goes by normal distribution among statisticians is given: de Moivre, before Laplace, found the distribution; Laplace, before Gauss, derived the properties of this distribution. The paper illustrates the derivations by Gauss, Herschel (yes, astronomer), Maxwell (no need to mention his important contribution), and Landon along with these following properties:

the convolution of two Gaussian functions is another Gaussian function
the Fourier transform of a Gaussian function is another Gaussian function
the CLT
maximizing entropy
minimizing Fisher information

You will find pros and cons about Gaussianity in the concluding remark.

Wikiquote said it’s misattributed. And I don’t know French. My guess could be wrong in matching quotes based on french translations into english. Please, correct me.

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[tutorial] multispectral imaging, a case study

Why Gaussianity?