Landweber-based deconvolution optimization.
After measuring each MEMS mirror’s response to an impulse, iterative deconvolution can be used to determine a set of input voltages that will more closely produce the desired output scan pattern. We use a Landweber iteration to solve this inverse problem. The iteration has two major components: A forward operator (H), which takes a desired input and produces the expected result after convolution with the MEMS mirror impulse response, and a transpose operator (HT), which assigns blame to the input for disagreements between the expected response and the desired response. The forward operator consists of: i) a blurring step, in which the current set of input voltages V(n) is convolved with the impulse response, and ii) a cropping step, in which only the results in the scan regions are considered. Cropping is performed because constraining the procedure to defined scan regions allows for higher accuracy in these regions (see S2 Fig), and because it is difficult to define exactly what the "desired" result is in undefined regions. Practically we carry out the cropping operation by comparing the blurred voltages with a binary mask (defining the constrained scan regions), and concatenating the resulting masked regions. In addition to the constrained scan regions, there is a small constrained region at the end of each waveform to ensure that the mirror settles quickly to its original position. After producing the cropped, blurred voltages, we compare iii) the result to the desired (and similarly cropped) result to produce a residual iv). The transpose operator consists of a ‘crop transpose’ step v), where the residual is again compared to the binary mask and zero padded to restore the length of the original input voltages; and a ‘blur transpose’ step vi), where the padded residual is convolved with the time-reversed impulse response. This produces a ‘correction voltage’ which is multiplied by a relaxation factor λ and added to the original input voltage V(n) to produce a corrected input voltage V(n + 1), vii). Empirically, we find that λ = 0.004 and n = 5,000 iterations produce good results. For clarity, we have omitted units on the vertical (proportional to voltage) and horizontal (time or index) axes in all graphs.
