THz Imaging Compressive sensing Hadamard spectroscopy THz Laser DMD Hadamard matrices Experimental model v1.0 Experimental model v2.0 Experimental model v3.0
Hadamard matrices
Mathematically, the multiplexed spectrometry is nothing more than a direct and inverse transformation: the spectrum is transformed by a linear binary transform and recovered by inverse transformation. The direct transform is performed in the optical domain and the inverse one in the numeric domain by a processor. Between the direct and inverse transformation, there is a detector that senses the transformed spectrum. More precisely, the coefficients are sensed one by one, A/D converted and provided to the processor that recovers the initial spectrum. Along this process, the spectrum undergoes a series of distortions and is corrupted by noise, making from the recovered version only an approximation of the original spectrum.
The advantage of performing an optical transform instead of directly sensing the spectral components is a better SNR (the detector introduces an additive noise that is significant in the case of THz detectors). The need for binary transform is dictated by the implementation of the transform, more precisely of the basis vectors, that can be either mechanical masks or mirrors, both with two states.
Measured spectrum
Measuring a part of the He lamp spectrum by Hadamard spectroscopy.

Hadamard spectrum:

...and the reconstructed spectrum:

Construction of a Hadamard matrix
In the case of size N = 2n, the Hadamard matrix can be obtained by the following algorithm:

The procedure is called doubling and allows to construct Hadamard matrices of size 2n.
  1. N. Sloan, Multiplexing Methods in Spectroscopy, Mathematics Magazine, Vol.52, No. 2, 71-80 (1979)
  2. E. D. NELSON, Hadamard Spectroscopy, JOURNAL OF THE OPTICAL SOCIETY OF AMERICA, VOLUME 60, NUMBER 12, 1664-1669 (1970)
  3. L. Streeter, G .R. Burling- Claridge, M.J.Cree and R . Kunnemeyer, Comparison of Hadamard Imaging and Compressed Sensing for Low Resolution Hyperspectral Imaging, IVCNZ 2008. 23rd International Conference, 1-6 (2008)
∴ Facts
Title: Compressive THz Imaging and Hadamard Spectroscopy for Space Applications
Project No: 17/19.11.2012
Project type: CDI STAR Project
Starting date: Nov 2012
Duration: 36 months
Partners: 2
INFLPR - Coordinator
CEO Space Tech - Partner
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