Development of Multiplexing in Sensing

Multiplexing in sensing is the idea that each measurement sample is a physical combination of various parts of the analog signal-of-interest. Multiplexing is a powerful tool that can be exploited by the sensor designer to eliminate or relax SNR related trade-offs. A simple example which illustrates the usefulness of multiplex sensing is weighing objects. In this example, one needs to weigh 100 sheets of paper. Assume that the measurement error of the scale is insignicant. Isomorphic sensing means one would need to measure each sheet of paper individually, requiring 100 measurements. If the measurement error of the scale is on the order of the weight of a single sheet, measuring each sheet individually produces a large measurement error. In order to reduce the error to an acceptable SNR one needs to make several measurements per sheet. However, one can measure all 100 sheets at the same time. Since the weight of all 100 sheets is much larger than the measurement error of the scale, one can dramatically increase the precision of the measurement. If each sheet is the same weight, then the measurement process is nished. The weighing problem is analogous to the spectroscopy example. As discussed earlier in section 1.1, there is trade-off between light collection and spectral resolution. Increasing the slit-width to increase the amount of light has the effect of degrading the spectral resolution \sigma_\lambda. Around the late 1940’s and early 1950’s, several important papers and inventions demonstrated the effectiveness of multiplexing in spectroscopy. At the time the FPA was non-existant, so in the slit spectrometer shown in Figure 1.4, where the FPA is pictured, there was actually another slit. To record the intensity at each spectral channel, either the dispersive element or the exit slit had to be mechanically translated, making the measurements even slower by a factor of N_\lambda, the number of spectral channels of interest. Golay was the rst to propose multiplexing the slit spectrometer by creating a pattern of binary (1’s and 0’s) entrance and exit slits . In the Golay multi-slit spectrometer, the patterns of entrance and exit slits are matched based on mathematically useful properties. Intuitively, the ability to use multiple entrance and exit slits increases the optical throughput of the spectrometer. In communications theory, the process of structuring the data from the source to the receiver is referred to as coding. Similarly, in computational sensing, the transmission of information between an object signal-of-interest and the sensor is considered a coding problem. In the multi-slit spectrometer, the entrance slits act to code the spectrum while the exit slits decode the coded spectrum. Golay’s idea dramatically increased the optical throughput without degrading the spectral resolution. Another example that is pertinent to this dissertation is coded aperture imaging. Coded aperture imaging can be thought of as the multiplexed version of a pinhole camera.there is a trade-off between the throughput and spatial resolution. However in many elds, such as high-energy particle imaging, refractive lenses and re ective mirrors are non-existant or underdeveloped. By using multiple pinholes the throughput is increased without sacricing spatial resolution. However, the pattern of the pinholes (which is the code) must be carefully designed in order for the reconstruction to be feasible. Fenimore, Canon, and Gottesman were among the rst to create an elegant solution to coded aperture design called uniformly redundant arrays . The uniformly redundant array increases throughput without signicantly degrading the spatial resolution. In summary, multiplexing has the ability to eliminate classic trade-offs in isomorphic sensors: signal strength or resolution. Modern researchers are still actively developing novel ways to implement multiplexing to increase resolution and sensitivity in the spatial domain , spectral domain, and temporal domain. However, multiplexing is not without its own set of challenges. As I mentioned, the coding must often be designed to obtain feasible signal reconstruction. I now discuss inverse problems in computational sensing.

Figure 1.4: An isomorphic slit spectrometer with a 4F conguration. The slit limits the lateral extent of the object (or intermediate image). The light from the slit is collimated and separated into dierent angles based on the wavelength. A second lens then images wavelength shifted copies of the slit on the image plane where the detector is. As the slit size decreases, less light is allowed, but one gains spectral resolution by rejected light from neighboring locations on the object.

Figure 1.4: An isomorphic slit spectrometer with a 4F conguration. The slit limits the lateral extent of the object (or intermediate image). The light from the slit is collimated and separated into dierent angles based on the wavelength. A second lens then images wavelength shifted copies of the slit on the image plane where the detector is. As the slit size decreases, less light is allowed, but one gains spectral resolution by rejected light from neighboring locations on the object.

 

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