Fig 1.
Introduction of the Okada filter.
A. Schematic illustration for data processing of the Okada filter. At the focused time t, xt is compared with the preceding and following values xt–1 and xt+1, respectively. If xt is the median among xt–1, xt, and xt+1, then xt is not modified (top). If xt is not the median, then xt is substituted with the mean of xt–1 and xt+1, i.e., (xt–1 + xt+1)/2 (bottom). B. The substitution procedure in A is serially conducted from t = 2 to the number of data points − 1. In each step, the value updated in the preceding step is used as xt–1. C. The Okada filter is expressed in a single equation based on the logistic function. The coefficient α determines the steepness in the transition at (xt − xt–1) (xt − xt+1) = 0. If α is more than one order of magnitude higher than the maximal value in {xt}, the transition can be regarded as a digital-like jump. D. Simultaneous cell-attached unit recording (top) and fMCI from a CA3 pyramidal cell (middle). The fMCI trace was filtered using the Okada filter with α = 100 (bottom). Dashed lines represent the times of action potentials detected in cell-attached trace.
Fig 2.
The Okada filter does not burr the signal onsets.
The original fMCI trace is shown in the top (black), and the black arrowhead indicates the true onset of the calcium transient. In the bottom, each trace after the Okada (orange), median (blue), binomial (red), or Savitzky-Golay (green) filter were superimposed onto the original trace (gray). The colored arrowheads indicate the "detected" onsets of the calcium transient in the filtered traces. Note that both the binomial and Savitzky-Golay filters caused an erroneous shift in the onset time.
Fig 3.
Application of the Okada filter to fMCI data.
A. In a confocal image of the CA3 region of cultured hippocampal slices loaded with OGB1-AM (top), the cell bodies of neurons were identified (bottom). B. Left 20 traces are raw fluctuations in the OGB1 fluorescence intensities in the cell bodies numbered in A and were Okada-filtered (right). C. Comparison of the mean signal, the mean amplitude of calcium transients that occurred in individual cells, before (abscissa) and after Okada filtering (ordinary). Each dot indicates a single cell. D. The same as C, but for the background noise level, the SDs in fluorescence intensities during the baseline period in the absence of calcium transients. E. The same as C, but for the S/N ratio, the mean signal divided by the background noise level. P was determined using Wilcoxon signed rank test for 100 cells.
Fig 4.
Application of the median filter to fMCI data.
A. Left 20 traces are raw fluctuations in the OGB1 fluorescence intensities in the cell bodies numbered in Fig 3A and were median-filtered (right). B-D. Comparison of the signal (B), the background noise level (B) and its ratio (D) before and after median filtering. Each dot indicates a single cell. P was determined using Wilcoxon signed rank test for the same 100 cells as those used in Fig 3C–3E.
Fig 5.
Application of the binomial filter to fMCI data.
A. Left 20 traces are raw fluctuations in the OGB1 fluorescence intensities in the cell bodies numbered in Fig 3A and were binomial-filtered (right). B-D. Comparison of the signal (B), the background noise level (C) and its ratio (D) before and after binomial filtering. Each dot indicates a single cell. P was determined using Wilcoxon signed rank test for the same 100 cells as those used in Fig 3C–3E.
Fig 6.
Application of the Savitzky-Golay filter to fMCI data.
A. Left 20 traces are raw fluctuations in the OGB1 fluorescence intensities in the cell bodies numbered in Fig 3A and were Savitzky-Golay-filtered (right). B-D. Comparison of the signal (B), the background noise level (C) and its ratio (D) before and after Savitzky-Golay filtering. Each dot indicates a single cell. P was determined using Wilcoxon signed rank test for the same 100 cells as those used in Fig 3C–3E.
Fig 7.
Comparison of S/N improved by the Okada and the median, binomial, and Savitzky-Golay filters.
A. The S/N is compared between Okada-filtered (abscissa) and median-filtered traces (ordinary). Each dot represents a single cell. Wilcoxon signed rank test. n = 100 cells. B. The ratios of the S/N in the Okada-filtered traces to those in median-filtered traces are plotted against the S/N in the original raw traces. Pearson’s correlation coefficient r was negative, and the regression line crossed y = 1 at the original S/N of 1.19. C. Same as A, but compared to the binomial-filtered traces. D. Same as B, but for binomial-filtered traces. Pearson’s correlation coefficient r was negative, and the regression line crossed y = 1 at the original S/N of 1.04. E. Same as A, but compared to Savitzky-Golay-filtered traces. F. Same as B, but for Savitzky-Golay-filtered traces. Pearson’s correlation coefficient r was negative, and the regression line crossed y = 1 at the original S/N of 0.90.
Fig 8.
Comparisons of the frequency responses of the Okada, median, binomial, and Savitzky-Golay filters.
A. The mean ± SD power spectra of the 100 cells data same as used in Fig 3 are shown in black, where those of Okada-filtered traces are shown in color. B-D. Same as A, but for the median (B, blue), binomial (C, red) and Savitzky-Golay (D, green) filtered traces. E. The mean frequency response of each filter is shown relative to the mean powers of the original traces.
Fig 9.
Comparison of computational speed of the Okada and median, binomial, and Savitzky-Golay filters.
The statement of each left box is the MATLAB command lines executed in the corresponding filter. Note that the statements i) and ii) and the statements iii) and iv) are theoretically identical but are differently described. The bar graph indicates the mean ± SD times spend in computing in these MATLAB codes. *P = 2.1×10−17, **P = 3.9×10−18; Wilcoxon signed rank test. n = 100 cells.
Fig 10.
Application of the Okada filter with different window lengths to fMCI data.
The original fMCI trace (top) was filtered by the Okada filters with window lengths of three, five and seven frames (left). The same fMCI data as Fig 3 were filtered. The S/Ns in the filtered traces were compared to the original S/Ns (middle), and the effect of the Okada filter with the windows of five and seven frames was compared to that with three frames (right).
Fig 11.
Application of the modified Okada filters to fMCI data.
A. The modified Okada filter is expressed in the equation similar to that shown in Fig 1C, but the coefficient β in the denominator can take a value different from 2. B. The original fMCI trace (top) was filtered by the Okada filter (β = 2) and the modified Okada filters (β = 3–10). The same fMCI data as Fig 3 were filtered. The S/Ns in the filtered traces were compared to the original S/Ns (middle), and the effect of the modified Okada filter was compared to the Okada filter (right). C. The mean ± SEM S/Ns of the 100 filtered traces are plotted as a function of β.
Fig 12.
Application of the Okada and median filters to electrophysiological data.
A raw EPSC trace acquired using whole-cell voltage-clamped recording at 2 kHz (top) was repeatedly filtered up to five times using the Okada (left) and median filters (right). Note that the background noise is reduced even more when the Okada filtering is repeated more.
Fig 13.
Application of the Okada and median filters to photographs.
A. A single frame of fMCI (top) was Okada-filtered (middle) and median-filtered (bottom). B. The PSNR was compared between the original image and the Okada-filtered or median-filtered images shown in A. *P = 1.8×10−5, Z = 4.29, Wilcoxon signed rank test (F23,23 = 2.13, 2.20, F-test of equality of variances); **P = 4.5×10−12, t23 = 13.0, paired t-test (F23,23 = 1.03, F-test of equality of variances), mean ± SEM of 24 subregions. C. The same as A, but for a photograph to which shot noise was artificially added. D. The PSNR was compared between the original image and the Okada- or median-filtered images shown in C. *P = 1.03×10−20 and 2.44×10−21, respectively, t15 = 32.5, 34.6, paired t-test (F15,15 = 1.43, 1.08, F-test of equality of variances), mean ± SEM of 16 compartments.