Blind deconvolution: errors, errors everywhere

attempt to reconstruct a true spectrum from an observed one. The problem we're considering is sometimes called blind deconvolution, because we're trying to unravel not only the spectrum, but the function that caused the blurring. These problems also arise in image deblurring. Spectroscopy Consider the data in Figure 1, which represents counts measured by a spectrometer. Suppose we have particles whose energy ranges from e lo to e high and define some intermediate energy levels e lo = e 0 < e 1 < … < en b–1 < e n b = e high. This creates n b bins, where the count for the jth bin is the number of particles determined to have energies between ej –1 and ej. Our spectrometer records n b counts, one for each bin; in the figure, we've passed a curve through these counts. Ideally, the count in bin j is exactly the number of particles with energies in the range [e j–1 , e j ]. But some blurring occurs due to the measurement process, and a particle in that energy range might instead be included in the count for a different nearby bin. The probability that a particle with energy e is assigned to bin j is often mod-eled as a normal distribution with mean (e j + e j–1)/2 and variance s j 2. One way to model this system is to try to determine the correct counts f j and the correct blurring given the measured counts g j , j = 1, …, n b and estimates of the values s j. This gives us a matrix equation (K + E) f » g + r, where E accounts for errors in modeling the spectrometer's blur, and r accounts for errors in counts. The matrix entry k jᐉ is computed as the probability that a particle whose energy is in the interval [e ᐉ–1 , e ᐉ ] is assigned to bin j (j, ᐉ = 1, …, n b). There are several sources of differences between the true spectrum and the recorded spectrum: • We effectively assign energy (e j + e j–1)/2 to all particles in bin j, which isn't correct. • A count's value depends on the number of particles with the energies that it represents, but there is some smearing , so it also depends on the number of particles with nearby energies. • …