The impact of K and the training for K ₂ estimation. In order to learn the impact K (implicit Laplacian smoothing coefficient) and determine the K value for a give batch of image sample, we constructed a training database and developed a non-linear estimation model. (a) The influence of different K to rendered threshold image and binary result by ILEE. A low K renders a threshold image that assimilates the object sample with finer filament structure (preserved high frequency information), but tends to underestimate the thickness of bright and thick filament; as K increases, less local detail are preserved, and the rendered binary image losses thin and faint filaments but get more accurate for thick and bright filament. There is a trade-off over the performance over faint/thin and bright/thick for a single K. Therefore, we used the full-outer-join result of a small K₁ and a high K₂ (Fig. 3 b and corresponding main text). (b) Construction of K₂ training database. Initially, we planned using the 7 images with hand-portrayed binary ground truth of filament fraction, but these sample have relatively limited range of filament size by pixel, so we decided to expand out sample pools using these data. Each sample is bicubically interpolated into the resolution of 0.5-, 1,5-, 2-, 2.5-, and threefold of the original and added to the database. Their corresponding binary imaged are converted to float data type with 0.0 and 1.0 to process the same bicubic interpolation, with the pixels over 0.42 are defined as True and False if not. Using this approach, the judgement of matching between the ground truth and ILEE result did not have significant change, as shown. Finally, each sample will be processed by ILEE with a single K₂ ranging from 1 to 3000, rendering totally 336 binary images as the training database. (c and d) Training algorithm. In (c), we converted the ILEE samples to a feature value—estimated K₂ with −0.2 of average deviation rate of pixels with top 5% DT in binary ground truth. Specifically, for each original or interpolated image sample (shown by independent lines, where different color represents different fold of interpolation), ILEE binary results using various single K₂ are compared with corresponding ground truth image to calculate the deviation rate, defined as the fold of difference of Euclidian distance transformation (DT) of ILEE result vs ground truth relative to ground truth. The averaged deviation rate of pixels with top 5% DT are taken as a feature value to represent thick filament. Next this feature was correlated to the corresponding K₂, and linear regressions were conducted to estimate the log₁₀ of the optimal K₂ that renders −0.2 for average deviation rate for each sample. Finally, the estimated optimal K₂ and the average DT of pixels with top 5% DT of each sample were utilized (shown in d) to generate an exponential regression model. This model covers the vast majority (if not all) of possible highest filament thickness rendered by confocal microscope images of cytoskeleton. The average DT of pixels with top 5% DT estimated by Niblack thresholding were used as an independent variable for optimal K₂ estimation.