1 Dynamic detection and quantitative analysis of biological network structures from live images Alessandro Abate, Angelo Cenedese, and Alberto Silletti I. DESCRIPTION OF THE TECHNIQUE We will consider the three phases of the procedure and approach the methodological issues of interest in order to compute the structure of interest in the image. 1) Detection. The first available image frame is analyzed searching for an underlying network structure, which is then extracted leveraging the use of a random walk model to navigate the network edges combined with a network agent to organize the retrieved information into a complex structure. The first frame is taken as a reference frame, also for the warping function w and the outcome of this procedure is a network graph (vertices, edges) with spatial information. 2) Tracking. The networked structure is then deformed in time starting from the information given by the reference frame and according to the visual data of the following frames. This phase resorts to the computation of optimal (or sub-optimal) warping maps w for motion and deformation that account for the underlying dynamics of the deformable system and at the same time accommodate the dataflow. A set of time stamped network structures is produced, each uniquely associated to the corresponding image frame. 3) Registration. Once the salient structures are detected and tracked in time, is of fundamental importance to establish a quantitative relationship among them by matching networks from consecutive frames. A. Detection: random walk model and network model The rationale behind the detection procedure is that of finding topologically continuous paths over the image I exploiting the collective motion of a set of agents {A 1 , ..., A Nτ } exploring the digital field of the frame, and tracking 1 their paths along (see Fig. 1(a)), similarly to what proposed in [1], [4]. The motion of these exploring agents is a random walk model driven by an external input that is related to the features of the already traveled path and some prediction on the following steps, in order to A. Abate is with the Delft Center for Systems and Control, TU Delft, The Netherlands; a.abate@tudelft.nl. A. Cenedese is with the Department of Information Engineering, University of Padova, Italy; angelo.cenedese@unipd.it. A. Silletti is with the Department of Information Engineering, Univer- sity of Padova, Italy;silletti@dei.unipd.it. The research leading to these results has received funding from ... 1 In this context, the temporal coordinate τ is set by the pseudo-time of the detection algorithm. ensure a level of continuity in the track. On the other hand, the number of the agents N τ is not constant during the detection phase and determined by the complexity of the structures: At any bifurcation point one or more new agents are initialized with different heading directions in order to proceed towards the detection of new branches, and whenever a closed structure is completely detected the detecting agent ceases to exist. More formally, each random walk agent A i is character- ized at instant τ by a couple {x i ,θ i }(τ ), respectively the position on the image frame and the heading direction, and employs a motion law defined as: x i (τ + 1) = x i (τ )+ αe jθi(τ ) , (1) where α is the step size of the advancement and j is the imaginary unit. The heading direction θ i (τ ) ∈ Θ = [0, 2π) is the polar angle seen by the walker agent obtained from the following solution set {arg max θ∈Θ ik ( E I (θ, x i (τ )) |Θ ik s.t. E I (θ, x i (τ )) > ¯ E ) }, (2) where E I is a suitable energy function defined over the image in a x i neighborhood domain (see Fig. 1(b)), θ is the polar angle coordinate, and ¯ E is a suitable thresh- old value. Basically, if there is only one subdomain of Θ where E I exceeds ¯ E , be it Θ i1 , there is a unique possible heading direction θ i the agent keeps on travel- ing. Conversely, if the thresholding operation highlights more intervals {Θ i1 ,..., Θ iNΘ }, the extremization pro- cedure of Eq. 2 suggests a set of heading directions {θ i1 ∈ Θ i1 ,...,θ iNΘ ∈ Θ iNΘ } as local extremal points in the intervals of Θ (see Fig. 1(c)). In such a case, N Θ agent instances are initialized with the same position and different heading directions {{x i ,θ i1 }, ..., {x i ,θ iNΘ }}, and the procedure is able to accommodate path bifurcations. Loosely speaking, the energy function E gathers local in- formation by exploring the surroundings of current position x i (τ ) and the rˆ ole of this energy term is to drive the agent towards the salient structure, being strictly related to the image intensity function and the visual data appearance. In the specific case, the reticular structure appears as a light network on a darkish cluttered and noisy background: Hence, the energy term E I related to the image intensity I is chosen with respect to a reference value I 0 as: E I (θ, x i (τ )) =  Ωi (I (x) − I 0 (x)) 2 dx  Ωi dx , (3)