The offline phase provides a pdf to generate the entire set of pathologist’s navigations, even though during navigation the belief must be adjusted to a particular pathologist’s navigation. Adaptation of the found pdf to the cur- rent navigation is achieved through online learning by continuously adjusting the parameters to observations while predictions are performed. In this case the process is xt = (x1t, x2t, . . . , x7t) = (t0,t, D1,t, D2,t, µ1,t, µ2,t, σ1,t, σ2,t), so that parameters of a velocity peak for a time t and observations correspond to the velocity profile for this time zt = (vt).
The online adaptation problem consists in estimating the parameters of the velocity profile given a set of observed velocities, i.e., p(xt|z1, z2, . . . , zt). This estimation can be recursively found using a Bayesian filter [10] so that it is needed to define an estimation of the system states at the beginning of the whole process. The whole adaptation is governed by two equations: a dynamic equation which indicates how the system states are evolving and a equation of observations which relates the system states and observations. The recursive estimation requires two phases: a prediction step which uses the dynamic equation for estimating every possible state of the system in time t and an updating step which modifies those possible states for matching them to observations.
The parameters of the initial pdf come from the offline learning phase. In the dynamic Equation 6.4, let us assume that before the pathologist moves the mouse, he/she has a pre-programmed desired path. The previous hypoth- esis is valid only between separated WoIs and indeed this has been already proved in eye tracking studies [79, 137]. Small adjustments to the system are
6.2. MATERIALS AND METHODS
given by the visual feedback and are herein modeled by independent Brow- nian motions of each of the Plamondon parameters. Therefore, the hidden parameters defined in Equation 6.1 should be modified so that the predicted velocity can vary and properly approach the sequence of observations. For doing so, in the prediction phase, the hidden navigation parameters are mod- ified by a Gaussian perturbation. The noise produced by the variability of the sensed mouse positions is assumed to be additive, as well as independent of the positions of the mouse neighbors, whereby it can be modeled as a Gaussian. The initial pdf parameters come from the offline learning phase (Equation 6.3). The dynamic system equation indicates how the parame- ters change within a velocity peak, a phenomenon modeled as a Gaussian perturbation of the hidden parameters
p(xjt|x j t−1) = G(x j t| x j t−1, ˆσj2) 6.4 The updating equation is able to generate velocities using parameters of the velocity peak, this is modeled as the sum of the parametric expression (∆-lognormal) and a Gaussian noise:
p(zt|xt) = N(zt− D1∆(t; t0, µ1, σ12) + D2∆(t; t0, µ2, σ22), Σ2)
where, xt are the parameters for a navigation time t and ˆσj2, Σ2 are prede- fined parameters whose values were calculated by minimizing the error when predicting the velocity from observed velocities. The used loss function was:
f (xt, Σ, ˆσ1, ˆσ2, . . . , ˆσ7) =PKi=1 (zi t−D1,t∆(t;t0,t,µ1,t,σ21,t)+D2,t∆(t;t0,t,µ2,t,σ22,t))2 2Σ2 +P7 j=1 PK i=1 (xj,it−1−xi t)2 2 ˆσj2 where, K is the number of training samples, zi
t and x j,i
t−1 are the velocity samples and the navigation parameters used for training. In the loss function, the first term accounts for the error introduced by the mouse, while the second stands for the user adjustements during navigation. Minimization of equation was achiveve using the Levenberg−Marquardt method [91], with K = 10 training samples corresponding to velocity peaks randomly selected in the time t, while xj,it−1 parameters were non-linear least square estimated, as mentioned before.
Because of the non linear nature of the model of observations, there is no analytical solution for the proposed Bayesian filter so that the estimation of
the state is achieved using a sequential importance resampling (SIR) filter [10]. This technique approximates a pdf with a random set of samples, each with an associated importance weight. Using this discretization, the predic- tion step applies the dynamic equation to each sample while the importance weights are modified in the updating step for the sequence of samples match observations. Resampling aims to concentrate the samples around the areas with a high importance ratio so that some samples will disappear. How- ever, provided this a dynamic and stochastic process, importance can change within the navigation and some of the disappeared samples can become im- portant but they result completely unretrievable. Therefore, the sequential importance resampling uses an additional resampling process which avoids the weight of some particles early vanishes.
6.2.4
Evaluation
To study the performance and the accuracy of the proposed prediction method, we have implemented two classical methods for forecasting: an expected weighted moving average (EWMA) and an elliptical predictor based on the Kalman filter (elliptical model) [19]. The algorithms were written in C++ and run under windows with an Intel Centrino processor of 1.7 GHz and 1 GB in RAM. Time performance was assessed by calculating the mean run- ning time for each algorithm. We also studied the prediction error of the three methods as follows: firstly, the prediction error for each method was computed by comparing with the recorded navigations. Secondly, we study the degree of dependence between the results provided by the proposed pre- diction method and the pathologist, i.e., whether or not the predicted results are independent of the pathologist. Statistical significance was determined using the Barlett’s test [126]. The error for each navigation step was mea- sured in pixels as the root mean squared error (RMSE) between the recorded position and the predicted position provided by each strategy. The predicted displacement vector was calculated as the velocity times the mouse sampling time (0.1 s).
Parameter values were tuned for each of the three predictors using four different navigations, corresponding to two pathologists navigating on two different mega-images. The EWMA’s time window size and the vector of weights were set as the least error found in the test images using an extensive numerical search, namely the time window size varied between 1 to 10, using a step increment of 1 while weights were varied between 0 and 1, with step
6.3. RESULTS
increments of 0.01. The parameters of the elliptical model were obtained from a non-linear optimization process, as previously described in [19].
6.3 Results
A total of 40 navigations were used either for training or assessing. Patholo- gists were instructed to navigate until they could achieve a diagnosis about the organ or the particular pathology. Overall, these mega-images were parts of full histological slides, a relative size which varied between 10 % to a 30 % of the whole histological sample. This means that pathologists never had the entire information as to have comparable levels of coincidence on the diagnosis and many times they just gave up. It is worthy to strength out that the main objective of this study was to determine the coincidence on the velocity patterns rather than the sequence of events for diagnosis or even the diagnosis. Finally, the time used for navigating varied between 20 s and 2 min, depending on the image contents. In general, navigations were shorter for images stained with inmunofluorescent techniques, for which the islets of Langerhans constitute the main information on which diagnosis lies. On the contrary, inflammatory pathologies stained with Hematoxylin-Eosin took larger exploration intervals and very rarely they were able to achieve a correct diagnosis.