Fig 1.
Test rig for characterizing mirror performance.
Fig 2.
Point scanning microscope test rig.
Test rig for point scanning fluorescence microscopy.
Fig 3.
Naïve input voltages become inaccurate for high-speed MEMS operation.
At slow scan speeds (left column (a), 8 ms/sweep, 1.6 ms/turnaround between forward and backward sweeps), using an input directly proportional to the desired result produces a reasonably accurate output. However, as the speed of the scan increases (middle Column (b), 3 ms/sweep, 0.6 ms/turnaround; right Column (c), 0.6 ms/sweep, 0.12 ms/turnaround), residual error between the desired and achieved patterns also increase. At speeds desirable for many scanning applications (right column (c)), the scan pattern is unusable. The top row compares the desired scan pattern with the measured result. The bottom row shows the residual error (between the desired and achieved pattern).
Fig 4.
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.
Fig 5.
Using linear deconvolution to determine input voltage improves scan accuracy compared to naïve voltages.
The response of the mirror at high speed can be greatly improved by using Landweber deconvolution to determine the input voltage. However, nonlinearities in the mirror response still produce a non-trivial residual. As in Fig 3, the top row shows the desired scan pattern and measured result, and the bottom row shows the residual. Compare left (a) and right (b) columns to Fig 3b and 3c middle and right columns, respectively; residual data from Fig 3b and 3c is shown here in gray.
Fig 6.
Overview of iterative, measurement-based deconvolution method for precise MEMS mirror control.
Given the measured impulse response (top left) and desired output (top right), deconvolution (middle) provides an input that produces a measured output that approximates the desired output (right, m = 0). Nonlinearities in the mirror’s response lead to a difference (residual, left, m = 0) between expected and measured responses, especially at high speeds. However, the deconvolution algorithm can incorporate the measured residual, computing a modified input that reduces the residual error. Repeating this procedure over a few measurement cycles (examples shown after 3, 7 iterations) dramatically lowers the residual, producing the desired result with high accuracy.
Fig 7.
Response of the MEMS mirror after using iterative, measurement-based deconvolution.
The response of the mirror converges to the desired scan pattern (left) by iteratively measuring and incorporating the residual (difference between the desired and achieved pattern) into the deconvolution method. The residual in constrained scan regions is used to compute a set of correction voltages that will cancel out the remaining residual. These correction voltages, produced by deconvolution after each measurement cycle, eventually lower the residual (middle, right) to within 1% of the desired result. Shown is a scan pattern with 300 μs /sweep and 100 μs /turnaround, with linear deconvolution (blue diamond), m = 1 iteration (red square) and m = 5 iterations (yellow circle).
Fig 8.
Optimized illumination scanning improves fluorescence image quality.
Images of a plastic fluorescence slide (left column), mixed pollen grains (middle column), and submandibular gland (right column) were acquired by scanning the excitation focus across the field of view in a raster pattern and recording the fluorescence on a camera. Results obtained with a naïve raster waveform (top row) are compared to the optimized waveform (bottom row). The naïve results show pronounced intensity variation across the scan (especially obvious when comparing the middle of each scan to the periphery), warping of the overall raster pattern, and obvious “lininess” within each imaging field. These artifacts are corrected when using the optimized waveform.