Detecting Topological Defects in 2D Active Nematics Using Convolutional Neural Networks

1 INTRODUCTION . 1.1 ACTIVE MATTER AND ACTIVE NEMATICS . Active materials encompass a broad range of systems that convert chemical energy into mechanical work . These systems self-organize and exhibit structure on time and length scales that exceed that of the constituent particles . One important class of active materials is active-nematics composed of anisotropic particles that exert extensile or contractile stresses on neighboring particles . Natural examples of active nematics include cultures of dividing E. coli and animal cells Doostmohammadi et al . ( 2016 ) ; Segerer et al . ( 2015 ) ; Duclos et al . ( 2014 ; 2017 ) . In active nematics , a single particle does not impart a net force into the system and therefore does self-propel ; however , the collective motion of many particles create non-trivial material flows , Marchetti et al . ( 2013 ) . Understanding the fundamental physics of collective motion of nematics can yield insight wound healing , tumor growth , and bacterial film dynamics , Doostmohammadi et al . ( 2018 ) . 1.2 2D CONFINED ACTIVE NEMATICS AND TOPOLOGICAL DEFECTS . In this work we study a model quasi-2D active nematic system composed of microtubules ( MT ) and motor proteins with tunable mechanical and dynamical properties developed by the Dogic group Henkin et al . ( 2014 ) ; DeCamp et al . ( 2015 ) . In this system , a suspension of microtubules and motor proteins is sedimented to a surfactant-stabilized oil-water interface creating a dense , liquid-crystalline nematic phase characterized by local orientational order . Motor proteins drive neighboring MTs to slide anti-parallel to one another , creating extensile stresses.The extensile stress makes the MT nematic inherently unstable to bend fluctuations . Undulations in the alignment of the MTs therefore grow in time , eventually saturating into localized topological disclinations shown in figure 1 . The defects are characterized by their winding number Kamien ( 2002 ) . Unlike defects in passive liquid crystals , these defects create fluid flows . The comet-shaped +1/2 defects are selfpropelling while the three-fold symmetry of the -1/2 defects create active flows that result in to net translation of the structure ; active flows are shown by the yellow arrows in figure 1 . The hydrodynamic interactions between multiple defects create chaotic , turbulent-like flows . By further confining the 2D nematic to microfluidic wells , these flows can be tamed into circulating vortices Woodhouse & Goldstein ( 2012 ) ; Norton et al . ( 2018 ) . Defect dynamics play a key role in moderating the transition from bulk-like turbulence to regular circulation . Quantifying their dynamics will help develop a better understanding of the fluid dynamics governing the behavior of this complex fluid and perhaps give insights into generic features of motile defects in biological systems . This requires researchers to analyze a large volume of videos of the 2D confined active nematics . 1.3 TRADITIONAL DEFECT DETECTING ALGORITHM AND LIMITATIONS . Remarkably utilizing the defining feature of a topological defect – its winding number , researchers developed an image processing algorithm which first generates a direction field from each image and then calculate the winding number at regions across the whole image to search for singularities . This algorithm performs reasonably well in detecting both +1/2 and -1/2 defects while being limited by two factors : various visual quality and noises caused by imperfect experimental settings . Parameters in the algorithm need to be fine-tuned for datasets with different visual qualities which would require huge amount of human labor as well as knowledge of image processing . The traditional algorithm also suffers from noises in the data including overexposure in some region , slightly unfocused image , materials exceeding above the 2D confinement plane , etc. , which tend to ruin the direction fields extracted from the images resulting poor detecting results . Aside from precision , this algorithm involves intensive calculation when processing one image . Fine-tuning of parameters and slow processing speed greatly limits the efficiency of data analysis in the experiments , while the algorithm ’ s lack of robustness compromising the detecting results . 1.4 PROPOSAL OF SOLUTION . To address the problems existing in the traditional defect detecting algorithm , we propose to apply deep learning , specifically deep convolutional neural network to replace the above defect detecting algorithm that is based on traditional image processing techniques . Our tasks include : • Label the positions of defects in representative experimental data . • Train a YOLO network with bounding boxes generated according to the positions of the labelled defects . • Test the performance of the trained defect detection model on experimental data with various visual qualities and sizes The changing visual quality , which greatly challenges the the traditional algorithm , improves the performance of our method . Adding more defects in various lighting conditions and image resolutions improves both robustness and precision of our algorithm . Apart from the visual quality , the diversity of the defects configuration causes troubles in traditional detecting algorithm . However , with defects in different shapes , sizes , configuration , and sometimes occlusion , our algorithm is able to generalize its model better to fit in these various samples , compromising the effect of overfitting . Another important advantage is our detection algorithms utilizes a unified , end-to-end neural network which enables efficient training and testing process . 2 PATTERNS IN DATASET . The labeled dataset consists of 9 typical videos from different experimental settings . In each experiment , active nematics are confined by a circular plate . Due to the constant input of energy , the active nematics continuously flows in a turbulent-like manner . Figure 2 and 3 shows a snapshot from each video . Specifically , the first testing dataset is from the same video as the first training dataset but are separated by 500 frames so that the first image of testing dataset has completely different appearance than the last image of the training dataset . Figure 4 shows the moving trajectory of a typical +1/2 defect from nucleation to annihilation . In the first frame , a pair of +1/2 and -1/2 defects nucleates with +1/2 defect moving away from -1/2 defects . In the next six frames , the +1/2 defect rotates around the center while being pushed outwards by inner materials , encountering a -1/2 defects . In the last frame , the +1/2 and -1/2 defects annihilate concurrently , balancing the topological charge of the region . 3 METHODS . When dealing with defects detection , the most intuitive way is to find the characteristics of a defect which differ it from its neighborhood area and make use of these characteristics to locate the regions containing defects . The defining difference between a defect region and a non-defect one is the topological winding number . A region containing a defect has a winding number of +1/2 or -1/2 around its boundary while the winding number associated to a non-defect region is strictly 0 . Previously , researchers made use of this definition and developed an effective algorithm by finding the direction field – the “ grains ” of an image , and calculating the winding number of different regions . As a result , this algorithm requires preprocessing to extract the direction field from the original image . Meanwhile , the performance of the detecting results relies heavily on the quality of the direction field extracted from the original image . Due to the imperfections and noises in the images , preprocessing is challenging . In some cases , the defects are occluded by overexposure area or are unfocused , leading to a problematic direction field near the defect . In addition , in the experimental video , materials such as hairs and dirts are observed which sometimes causes false detection . To resolve these problems , we propose to treat this task as an object detection task where we consider defects as the objects we are trying to detect in an image . We train a convolutional neural network with images containing defects whose locations are provided . During detection , our model generates a bounding box around defects as detecting results . In this method , the problems in the previous paragraph , namely increasing the size of dataset from various visual quality , diversity of defects configuration and size , help our model generalize a defect ’ s features better . Compared to the traditional algorithm which requires fine-tuning for different datasets , our method could efficiently provide reliable detecting results on data across various visual quality . 3.1 DATA COLLECTION . To create a training dataset , we manually labeled the positions of positive defects and negative defects in 8800 images . In an image , we labeled a comet-like ( +1/2 ) defect at the point where materials change orientation most sharply . For a trefoil-like ( -1/2 ) defect , we labeled them at the center of the triangular material-devoid region . Figure 5a is an example of labeled image with +1/2 defects labeled by red stars and -1/2 defects labeled by green circles . 3.2 SUB-CLASSIFICATION FOR +1/2 DEFECTS . The experimental datasets we work on are challenging to perform object detection . One main challenge is the varying sizes of +1/2 defects . In Figure 5b , there are 4 +1/2 defects by definition but defect 1 looks drastically different from defect 2 , 3 , and 4 . We call +1/2 defects similar to defect 1 “ hollow ” +1/2 defects . Assigning bounding boxes with the same size for all four defects in this image is problematic . A small bounding box can not include enough local information to recognize defect 1 as +1/2 defect , while big bounding boxes might include features of other defects which confuse the model . For defect 3 and 4 , big bounding boxes will include the -1/2 defects next to them . To resolve this issue , since hollow +1/2 defects are commonly seen in the video , we created a separate class for these defects when we labeled the data and assigned slightly bigger bounding boxes for this class . After labeling all the data , we assigned a bounding box to each labeled position . Since more contextual information is needed to detect hollow +1/2 defect than to detect normal +1/2 or -1/2 defects , we assigned the sizes of the bounding boxes with a ratio of 3:3:4 for +1/2 , -1/2 , hollow +1/2 , respectively . 3.3 IMAGE AUGMENTATION . Aside from normal image augmentation such as jitter and scaling , we added a random flipping and random rotation to the training images . Since the region of a defect is anisotropic , rotating the image effectively increase the robustness of the model . 3.4 COMPUTER VISION PIPELINE . In recent years , after the wide application of convolutional neural network in computer vision , the state-of-the-art object detection algorithm can be divided into two main categories . In the first category , input images are first processed by a method named selective search Uijlings et al . ( 2013 ) which generates thousands of region proposals that are likely to contain an object . These region proposals along with the input images will be passed to a CNN which performs feature extractions and image re-sampling . The results will eventually be generated by a classifier . This category of algorithms is the first to combine a selective search method with deep CNN where selective search perform the task of object localization and CNN performs the task of classification . RCNN series ( including R-CNN Girshick et al . ( 2014 ) , Fast R-CNN Girshick ( 2015 ) , and Faster R-CNN Ren et al . ( 2015 ) ) belongs to this catogary . The second category is YOLO Redmon et al . ( 2016 ) , which implements an end-to-end deep neural network that performs classification and localization all at once . The input images are first separated by 13 by 13 grid cells where each grid cell is responsible for detecting the object falling into it . The output of the network is forced to be a tensor which includes the coordinates of bounding boxes and a confidence value . YOLO outperforms its counterpart in speed but lack in average precision . In this project , we chose to implement YOLO because it is fast and easy to optimize due to its endto-end design . While implementing the YOLO algorithm , we mainly used the original setting in its paper except for some changes in the pipeline : • Changed the number of filter in the last convolutional layer to fit in our dataset . • Decreased the jitter value from 0.3 to 0.05 to avoid occlusion or missing of defects located near the boundary • Since our objects-defects are relatively smaller than objects in the PASCAL VOC dataset , we increased the non-object coefficient from 0.5 to 0.8 to increase the loss from confidence predictions for boxes that do not contain objects • Took out the image augmentation regarding hue , exposure , and saturation since our images are binary

Summary: In this paper the authors apply methods developed in computer vision towards the identification of topological defects in nematic liquid crystals. Typically, defects are identified using a costly algorithm that is based on numerically computing the winding number at different locations in the image to identify defects. The authors demonstrate that a deep learning approach offers improvement to both the identification accuracy and rate at which defects can be identified. Finally, the authors do some work investigating the limitations of the model and show that breakdown occurs near the edge of the field of view of the microscope. They show that this also happens with a conventional approach.