Understanding the Impact of Evaluation Metrics in Kinetic Models for Consensus-based Segmentation

Understanding the Impact of Ev aluation Metr ics in Kinetic Mo dels f or Cons ensus-based Segmen tation R.F.Cabini ∗ , H. T e tt aman ti † , M. Zanella ‡ Abstract In this article we extend a recently introduced kinetic mod el for consen- sus-based segmen tation of images. In particular, we wi ll interpret the set of pixels of a 2D image as an interacting p article system which evolves in time in view of a consensus-typ e pro cess obtained by interactions b etw een pixels and external n oise. Thanks to a kinetic form ulation of the intro- duced mod el we derive the large time solution of the mo del. W e will sho w that the choice of parameters defining the segmentation task can b e chosen from a plurality of loss fun ctions characterising th e ev aluation metrics. 1 In tro duction The primary ob jective of image s egmentation is to partition an image into dis - tinct pixel r egions that ex hibit homogeneous characteristics, including spatial proximit y , intensit y v a lues, c olor v ariatio ns, texture patterns, br ightness levels, and contrast differences, thereby ena bling more effectiv e analysis and inter- pretation of the visual data. The application of image segmentation metho ds plays an impo rtant role in clinica l resea rch by facilitating the study of ana tom- ical str uctures, highlightin g regio ns of int erest, and measuring tissue volume [ 1 , 5 , 1 5 , 36 , 3 9 , 51 ]. In this context, the accurate reco gnition of area s a ffected by patholog ies can hav e great impact for mo re precise e arly diagnosis and mon- itoring in a great v a riety of dis ease that ra nge from brain tumor s to skin les io ns. Over the past decade s , a v ariety of computational strategies a nd mathemat- ical approaches have b een develop ed to addr e ss image segmentation challenges. Among these, deep lea rning techniques and neura l ne tw orks hav e emer ged as one o f the most widely used metho ds in contempora r y image segmentation ta sks [ 27 , 28 , 3 3 , 3 2 , 49 , 5 6 , 5 7 , 34 , 3 5 ]. Leveraging a set of examples, these techniques are capable of approximating the complex no nlinear relationship b etw een inputs and desired outputs. While deep learning models exce l in complex segment ation problems, their dep endence on large annotated datasets remains a significant challenge, particular ly in fields such as biomedical imaging, where data av ailab- ilit y is limited a nd ma nual lab eling can be b oth ex pe nsive and time-consuming. ∗ Euler Institute, U nive rsit` a della Svizzera Italiana; raffaella.fiamma.cabini@usi .ch † Departmen t of Mathematics ”F. Casorati”, Unive rsity of P avia; hora- cio.tettaman ti01@univ er sitadipa vi a.it ‡ Departmen t of Mathematics ”F. Casorati”, U ni ve r sity of Pa via; mattia.zanella@unip v.it 1 A different appro a ch is ba sed on clustering metho ds [ 14 , 24 , 30 , 47 , 46 , 31 ]. These metho ds gr oup pixels with similar c ha racteristics , effectively partitioning the ima ge into distinct regio ns. Clustering-bas ed metho ds offer an attractive alternative to deep lea r ning techniques, as they do not requir e sup ervis ed tr a in- ing and therefore, ca n b e us ed on small, unlab eled datasets. In this dir ection , a kinetic approach for unsuperv ised clustering pr oblems for image segmentation has b e en introduced in [ 9 , 26 ]. In these works, microscopic consensus-type mo d- els have be en co nnec ted to image s egmentation tasks b y co nsidering the pixels of an image a s a int eracting sy stem where each par ticle is characteris ed by its space p o sition and a feature determining the g ray le vel. A virtual in ter action betw een particles will then determine the asymptotic formation a finite num b er of clusters. Hence, a seg ment ation mask is genera ted by assigning the mean of their gray levels to each cluster o f particles and by applying a binary thres hold. Among the v arious nonlinear compromise terms that ha ve b een prop osed in the literature, we will co nsider the Hegselmann-Kr ause mo del describ ed in [ 25 , 48 ] where it is supp osed that each a gent may only interact with o ther ag ent that are sufficiently close. This type of interaction is clas sically known as b ounded confidence in ter action function. As a re sult, tw o pixel will interact based on their distance in space and their gray level. The appr oach developed in [ 9 ] is based on the methods of kinetic theory for consensus formatio n. In the la st dec- ades, after the first mo del developed in [ 17 , 16 , 22 , 52 ], several a ppr oaches have bee n designed to inv estiga te the emergence of patterns and collective s tr uctures for la rge systems of a gents/particles [ 8 , 12 , 38 , 23 ]. T o this end, the flexib- ilit y of kinetic-type equations hav e been of paramount impo rtance to link the microscopic scale and the macrosc o pic obser v a ble sc ale [ 2 , 10 , 20 , 21 , 43 , 42 , 54 ]. In order to c o nstruct a data-oriented pip eline, we calibrate the resulting mo del by exploiting a family of existing ev aluation metrics to obtain the relev- ant informa tion from a ground truth imag e [ 4 , 13 , 18 , 29 , 37 , 53 ]. The main developmen t of this study , compared to the one descr ib ed in [ 7 ], relies in the fact that we ev aluate multiple metrics to quantify segmentation erro r, which is cru- cial for the optimization of the internal model parameters. In pa rticular, we will concentrate o n the Sta ndard V olumetric Dice Similarity Co efficien t (V olumetric Dice), a volumetric measure base d on the quo tient b etw een the intersection o f the o btained seg mented images a nd their total volume, the Surfac e Dice Sim- ilarity Co efficient, which is analo gous to the V olumetric Dice but exploits the surface of the segment ed images [ 7 ]. F urthermore, we test the Jac card index, which is an alter na tive option to ev aluate the volumetric s imila rity betw een t wo segmentation masks, and the F β -measure, which is a perfo r mance metrics which allows a balance b etw e e n precisio n and se ns itivity . In this pap er we describ e these metrics in detail and we analyz e ho w these choices of ev alua tion metric influence the para meter optimization pro cess. F ur ther more, w e discus s the most suitable metrics for the fina l assessment o f the pr o duced segmentations. This expanded ev a luation provides novel insig hts in to the impact of ev aluation met- rics on model perfo rmance a nd enhances o ur understanding o f how to efficiently optimize the introduced segmentation pip eline. In more detail, the manuscript is o r ganized a s follows: In Section 2 we in- tro duce a n extensio n of the Hegselma nn- Krause mo del in 2D and we present the s tr ucture of the emer g ing s tea dy states for different v alues of the mo del parameters . Next, we present a des cription o f the mo del based o n a kinetic- t y p e appro ach. F urthermore, w e s how how can this mo del b e extended and 2 apply for the image segmentation problem. In Section 3 we present a Direct Sim ula tion Monte Carlo (DSMC) Metho d to approximate the evolution of the system and we introduce p ossible optimization methods to pro duce segment- ation masks for given images. T o this end, we introduce the definition of the principal optimization metr ic s used in the cont e x t of bio-medical images and their principal characteristics. In Sec tio n 4 w e s how the re s ults for a simple case o f segmenting a geometrica l image with a blurry background a nd co mpare the r esults obtained for different choices o f the diffusion function. Finally , we present the results obtained for v ario us Br ain T umor Images and discuss how the ch oice of different metrics may affect the final r esult. W e show that the F β − measure do es not pro duce consistent re sults for different v a lues of β . W e repro duce the exp ected rela tionship b etw een the V olumetric Dice Co efficient and Jacca rd Index and s how tha t b oth metr ics plus the Surface Dice Co efficient yield similar re s ults. Nev er theless, we arg ue that for this t yp e of images the Surface Dice Co efficien t pr o duces more accur ate loss v alues and its definition is mo re representative co mpare to the V o lumetric Dice Co efficient and Jacca rd Index. 2 Consensus mo delling and applications to im- age segmen tation In recent y ea rs, there has been growing interest in explor ing co nsensus formation within opinion mo dels to gain a deep er understanding of how so cial forces affect nonlinear aggr e gation pro cesses in multiagen t systems. T o this end, v ar ious mo dels hav e been pr op osed considering different scenario s and hypo thesis on how the pairwise in ter actions ma y lead to the emergence of a p ositio n. F or a finite num b er of pa rticles, the dynamics is usually defined in terms o f firs t or der differential equations having the ge neral for m d x i dt = 1 N N X j =1 P ( x i , x j )( x j − x i ) , (1) where x i ( t ) ∈ R d , d ≥ 1, c har acterise the po sition of the agent i = 1 , . . . , N at time t ≥ 0, a nd P ( · , · ) ≥ 0 tunes the interaction b etw een the agents x i , x j ∈ R d , see e.g. [ 3 , 1 2 , 25 , 3 8 , 40 ]. In addition to micr oscopic agent-based models , in the limit of an infinite nu m be r of a g ents, it is p ossible to derive the evolution of distribution functions characterising the collective b ehavior of in tera cting systems. These approaches, t y pically gr ounded in kinetic-type partial differential equations (PDE s), are cap- able of bridging the gap b e tw een microscopic forces and the emerging prop erties of the system, see [ 42 ]. 2.1 The 2D b ounded confidence mo del W e now consider the bidimensiona l case, d = 2, a nd we sp ec ify the int e r action function based on the so-called b ounded confidence mo del. In mor e detail, we consider N ≥ 2 a gents and define their opinion v a riable thro ug h a vector x = ( x i ( t ) , y i ( t )) ∈ R 2 , characterised by initial states { x 1 (0) , . . . , x N (0) } . Agen ts will mo dify their o pinion as a re s ult of the interaction with o ther agents only if 3 | x i − x j | ≤ ∆ , wher e ∆ ≥ 0 is a given confidence lev e l. Hence, w e can write ( 1 ) as follows d dt x i = 1 N N X j =1 P ∆ ( x i , x j )( x j − x i ) , (2) where P ∆ ( x i , x j ) = χ ( | x i − x j | ≤ ∆) : R 2 → { 0 , 1 } , being χ ( A ) the characteristic function of the set A ⊆ R 2 . W e can easily observe that the mean p os itio n of the ensemble of agents is conser ved in time, indeed d dt N X i =1 x i = 1 N N X i,j =1 χ ( | x i − x j | ≤ ∆)( x j − x i ) = 0 , (3) thanks to the symmetry of the considered bounded-c o nfidence int eraction func- tion. The bo unded confidence mo del conv er ges to a steady co nfiguration, mean- ing that the systems reaches consensus in finite time. The str ucture of the steady state de p ends on the v alue of ∆, see [ 43 ]. F urther more, to account for ra ndom fluctuations giv en b y exter nal fac to rs in the opinion of age nts we may consider a diffusion co mpo nent as follows d x i = 1 N N X j =1 P ∆ ( | x i − x j | ≤ ∆)( x j − x i ) dt + √ 2 σ 2 d W i (4) where { W i } N i =1 is a set of independent Wiener pro cesses. The impact of the diffusion is weight e d by the v ar iable σ 2 > 0. T o visualise the interplay b etw een consensus forces a nd diffusion we depict in Figure 1 the stea dy configur ation of the mo del ( 4 ) for differe nt combinations of the mo del parameter s. F or σ 2 = 0, the system for ms a finite num b er of cluster s dep ending on the v alue o f ∆ > 0, as illustrated in Figure 1 (a). F or v alues o f the diffusion co efficien t σ 2 > 0, the nu m be r of clusters of the system v ar ies as depicted in Figur e 1 (b). The right panel of Figur e 1 (b) shows the scenario in whic h the diffusion effect beco mes comparable to the tendency of agents to clus ter. Finally , in Figure 1 (c), for σ = 0 . 05 , the diffusion effect dominates the gro uping tendency , resulting in a homogeneous steady-sta te distribution. 2.2 Kinetic mo dels for consensus dynamics In the limit N → + ∞ it can b e shown that the empirica l density f ( N ) ( x , t ) = 1 N N X i =1 δ ( x − x i ( t )) of the system of particles ( 4 ) converges to a contin uous density f ( x , t ) : R 2 × R + → R + solution to the following mean-field equation ∂ t f ( x , t ) = ∇ x · [Ξ[ f ]( x , t ) + σ 2 ∇ x f ] f ( x , 0 ) = f 0 ( x ) (5) where Ξ[ f ]( x , t ) is defined as follows Ξ[ f ]( x , t ) = Z R 2 P ∆ ( x i , x j )( x − x ∗ ) f ( x ∗ , t ) d x ∗ , (6) 4 (a) (b) (c) Figure 1: Lar ge time distribution of the 2D b ounded confidence mo del for dif- ferent parameters characterising the compromise prop ensity and the diffusion for N = 10 5 particles in [0 , T ] with T = 100 and ∆ t = 0 . 01. In (a) the final state conv erg es to a n umber of clusters dep ending on the v alue o f ∆. As we r educe the rang e of interaction more clusters are cr eated. In rows (b) and (c) w e ca n see the interplay b e t ween the tendency of particles to ag grega te and the diffu- sion. O n the fir st column we see that the steady state conv erg es to a Gaussia n Distribution with a s tandard devia tio n given by σ 2 . In the second co lumn for (b) and (c) we see tha t the final states differ gr eatly in their structure. Finally , the last column shows the final states in the ca s e whe r e the diffusion sur passes considerable the aggr egation tendency . 5 see e.g. [ 11 ]. W e ca n der ive ( 5 ) using a kinetic appro ach b y writing x := x i ( t ) and x ∗ := x j ( t ) for a generic pair ( i, j ) of in tera cting ag ents/particles and we approximate the time deriv a tive in ( 4 ) in a time step ǫ = ∆ t > 0, thro ugh a n Euler-Mary ua ma approach, in the s ame spirit as [ 10 , 45 ]. Hence, we recov er the binary interaction rule x ′ = x + ǫ P ∆ ( x , x ∗ )( x ∗ − x ) + √ 2 σ 2 η x ′ ∗ = x ∗ + ǫ P ∆ ( x ∗ , x )( x − x ∗ ) + √ 2 σ 2 η ∗ , (7) where x ′ = x i ( t + ǫ ), x ′ ∗ = x j ( t + ǫ ) and η , η ∗ are tw o indep endent 2D centered Gaussian distribution random v aria ble such that h η i = h η ∗ i = 0 h η 2 i = h η 2 ∗ i = ǫ (8) where h·i denotes the in tegration with resp ect to the distribution η . F urther- more, in ( 7 ) we shall co nsider P ( x , x ∗ ) = χ ( | x − x ∗ | < ∆). W e c a n r emark that, if σ = 0, since P ∆ ∈ [0 , 1] and ǫ ∈ (0 , 1 ) we ge t h x ′ + x ′ ∗ i = x + x ∗ + ∆ t ( P ∆ ( x , x ∗ ) − P ∆ ( x ∗ , x ))( x ∗ − x ) = = x + x ∗ (9) since the int e raction function P ∆ is symmetric, consistently with ( 3 ). This shows that the mean p o sition is conser ved at every in ter action. Finally , we have |h x ′ i| 2 + |h x i| 2 = | x ′ | 2 + | x | 2 − 2 ∆ tP ∆ | x ′ − x | 2 + o (∆ t ) (10) and the mean energy is dissipated at each int e raction, since P ∆ ≥ 0. Hence, we co ns ider the distribution function f = f ( x , t ) : R 2 × R + → R + , such that f ( x , t ) d x represe nts the fraction of agents/particles in [ x 1 , x 1 + dx 1 ) × [ x 2 , x 2 + dx 2 ] at time t ≥ 0. The evolution of f as a r esult o f binary-interaction scheme ( 7 ) is obtained by a Boltzmann-type equa tion which reads in weak form d dt Z R 2 ϕ ( x ) f ( x , t ) d x =  Z R 4 ( ϕ ( x ′ ) − ϕ ( x )) f ( x , t ) f ( x ∗ , t ) d x d x ∗  , (11) being ϕ ( · ) a test function. As obser ved in [ 54 ], when ∆ t = ǫ → 0 + we ca n ob- serve that the binar y scheme ( 7 ) b ecomes quasi-inv ar ia nt and we ca n introduce the following expansio n h ϕ ( x ′ ) − ϕ ( x ) i = h x ′ − x i· ∇ x ϕ ( x )+ 1 2  ( x ′ − x ) T H [ ϕ ]( x ′ − x )  + R ǫ ( x , x ∗ ) (12) being R ǫ ( x , x ∗ ) a reminder term and H [ ϕ ] the Hessian matrix. Hence, sc a ling τ = ǫt a nd the distribution f ǫ ( x , τ ) = f ( x , τ /ǫ ), we may plug ( 12 ) in ( 11 ) to g et d dτ Z R 2 ϕ ( x ) f ǫ ( x , t ) d x = 1 ǫ Z R 4 h x ′ − x i · ∇ x ϕ ( x ) f ǫ ( x , τ ) f ǫ ( x ∗ , τ ) d x d x ∗ + 1 2 ǫ Z R 4 h ( x ′ − x ) T H [ ϕ ( x ]( x ′ − x ) i f ǫ ( x , τ ) f ǫ ( x ∗ , τ ) d x d x ∗ + 1 ǫ Z R 4 R ǫ ( x , x ∗ ) f ǫ ( x , τ ) f ǫ ( x ∗ , τ ) d x d x ∗ 6 ( x i , y i , c i ) Figure 2 : A schematic r epresentation of the pro p o sed mo del where ea ch pixe l can b e in terpr eted as a particle ( x i , y i , c i ), b eing c i a s ta tic feature in the interv al [0 , 1]. F o llowing [ 9 ], see also [ 42 ], we can prove that Z R 4 R ǫ ( x , x ∗ ) f ǫ ( x , τ ) f ǫ ( x ∗ , τ ) d x d x ∗ → 0 + , as ǫ → 0 + . Hence, integrating back b y parts the first tw o terms we obtain ( 5 ). In more detail, we can pr ove that f ǫ conv erg es, up to extraction o f a subsequence to a pr obability density f ( x , τ ) that is w eak solutio n to the nonlo cal F okker-Planck equation ( 5 ). 2.3 Application t o Image Segmen tation An a pplication of the Hegselman-K rause mo del for da ta-clustering pro blems ha s bee n prop o sed in [ 26 ]. The idea is to extend the 2D mo del b y characterising each pa rticle with an internal feature c i ∈ [0 , 1] that represents the gray c o lor of the i th pixel. Therefor e, we interpret each pixel in the image as a par ticle characterized by a p osition v ecto r and the static feature c as shown in Figure 2 . T o a ddress the segmentation ta sk, we can define a dynamic feature for the system of pixels throug h an in ter action function that acco unt s for a lignment pro cesses among pixels with sufficiently similar features . In particula r, let us consider the following P ∆ 1 , ∆ 2 ( x i , x j , c i , c j ) = χ ( | x i − x j | ≤ ∆ 1 ) χ ( | c i − c j | ≤ ∆ 2 ) . (13) Therefore, the time-contin uous evolution for the sy s tem of pixels is given b y d dt x i = 1 N N X j =1 P ∆ 1 , ∆ 2 ( x i , x j , c i , c j )( x j − x i ) d dt c i = 0 (14) In this cas e w e introduced t wo confidence b ounds ∆ 1 ≥ 0, ∆ 2 ≥ 0 taking int o account the p osition and the g ray level of the pixels, resp ectively . In this wa y , the in ter actions b etw e e n the pixels will gener ate a large time dis tribution which is c ha r acterized by s everal clusters, dep ending on the v alue of ∆ 1 and ∆ 2 . Hence, coherently with k-means methods , see e.g. [ 37 ], a pixel b elong s to a 7 cluster C µ = { x i : k x i − µ k ≤ α } , b eing α > 0 the pixel size, if it is sufficiently close to the lo cal q uantit y µ ∈ R 2 . W e highlight how we a r e only interested in clustering with resp ec t to the space v aria ble. This dynamics is repr esented in Figure 3 . t 1 t 2 t 3 t 4 Figure 3: Represe ntation of the evolution of pixels as they tend to agg regate in different clusters. Biomedical ima g es ar e often sub ject to am big uities aris ing from v arious sources o f uncertaint y r elated to clinical facto rs and p otential b ottlenecks in data a cquisition pro cesse s [ 5 , 3 2 ]. These uncerta int ies ca n b e broadly categor - ized into ale atoric uncertaint y , stemming fro m inher ent sto chastic v ar iations in the da ta co llection pr o cess, and epis temic uncertaint y , relates to uncer ta int ies in mo del parameters a nd can lea d to dev ia tions in the res ults. Alea toric uncer- tainties p o ses significant challenges in image segmentation, as image pr o cessing mo dels must contend with limitations in the raw acquisition data. Addr e ssing these uncertainties is c ritical, and the study o f uncertaint y quantification in image seg mentation is an ex panding field aimed at developing robust segmenta- tion a lgorithms c a pable of mitigating er r oneous outcomes . T o this end, in [ 9 ], it has be e n pro p o sed an extensio n of ( 14 ) to consider segmentation of biomedical images. In particular, the particle mo del ( 15 ) has bee n integrated a nonc o n- stant stochastic par t to take in to account a le atoric uncer tainties arising from the data acquisition pro ces s. These uncertainties may include factors suc h a s motion a rtifacts or field inhomoge ne ities in magnetic r esonance imaging (MRI). They mo dified equatio n 14 as follows: d x i = 1 N N X j =1 P ∆ 1 , ∆ 2 ( x i , x j , c i , c j )( x j − x i ) dt + p 2 σ 2 D ( c ) d W i d dt c i = 0 (15) where { W i } N i =1 is set of indep endent Wiener proc e sses, P ∆ 1 , ∆ 2 ( · , · , · , · ) ∈ [0 , 1] is the interaction function defined in ( 13 ), and D ( c ) ≥ 0 quantifies the im- pact of diffusion related to the v alue of the feature c ∈ [0 , 1 ]. Since the alea t- oric uncertainties a re expected to app ear far a way from the static feature’s bo undaries, only diffusion functions that are maximal at the center a nd satisfy D (0) = D (1) = 0 are considered. Similarly to ( 7 ), we ma y in tro duce the fol- lowing binar y in tera ction scheme by writing ( x , x ∗ ) := ( x i ( t ) , x j ( t )) a r andom 8 couple of pixels having features ( c, c ∗ ) := ( c i ( t ) , c j ( t )). W e get x ′ = x + ǫ P ∆ 1 , ∆ 2 ( x , x ∗ , c, c ∗ )( x ∗ − x ) + p 2 σ 2 D ( c ) η x ′ ∗ = x ∗ + ǫ P ∆ 1 , ∆ 2 ( x ∗ , x , c ∗ , c )( x − x ∗ ) + p 2 σ 2 D ( c ∗ ) η c ′ ∗ = c ∗ c ′ = c, (16) where ( x ′ , x ′ ∗ ) := ( x i ( t + ∆ t ) , x j ( t + ∆ t )) and ( c ′ , c ′ ∗ ) := ( c i ( t + ∆ t ) , c j ( t + ∆ t )). A t the s ta tistical level, as in [ 9 ], we may follows the a pproach descr ib ed in Section 2.2 . Hence, we introduce the distribution function f = f ( x , c, t ) : R 2 × [0 , 1] × R + → R + , such that f ( x , t ) d x r epresents the fraction o f agents/particles in [ x 1 , x 1 + dx 1 ) × [ x 2 , x 2 + dx 2 ] character ized b y a feature c ∈ [0 , 1] a t time t ≥ 0. The evolution of f whos e interaction follow the binary scheme ( 16 ) is given by the following Boltzmann- t yp e eq uation d dt Z 1 0 Z R 2 ϕ ( x , c ) f ( x , c, t ) d x dc = * Z [0 , 1] 2 Z R 4 ( ϕ ( x ′ , c ) − ϕ ( x , c )) f ( x , c, t ) f ( x ∗ , c ∗ , t ) d x d x ∗ dc dc ∗ + , (17) Hence, since the feature is not evolving in time, we can pr o ceed as in Section 2.2 to derive in the quas i- inv ariant limit for ǫ → 0 + the cor resp onding F okker- Planck-t yp e PDE ∂ t f ( x , c, t ) = ∇ x · h Ξ[ g ] ∆ 1 , ∆ 2 ( x , c, t ) f ( x , c, t ) + σ 2 D ( c ) ∇ x f ( x , c, t ) i (18) where Ξ[ g ] ∆ 1 , ∆ 2 ( x , c, t ) = Z 1 0 Z R 2 P ∆ 1 , ∆ 2 ( x , x ∗ , c, c ∗ )( x − x ∗ ) f ( x ∗ , c ∗ , t ) d x ∗ dc ∗ . 3 Ev aluation metrics and parameters estimation In this section we present classical Direct Simulation Monte Car lo (DSMC) metho ds to numerically approximate the evolution o f ( 17 ) as quasi-inv ar iant approximation of the F okker-Pla nck eq uation ( 18 ). The resulting numerical al- gorithm is fundamental to estimate consis tent parameters from MRI imag es. T o this end, we present several loss metrics with the aim to compare the result of our model-ba sed a pproach with ex is ting metho ds for biomedica l image seg - men tation. In this work we fo cus exclusively o n binar y metrics . F o r ev a luation of seg ment a tion with m ultiple lab els we p o int the reader to [ 53 ] for a detailed presentation of v arious metrics. 3.1 DSMC algorithm for image segmen tation The numerical approximation of Boltzmann-type equations has b een deeply in- vestigated in the rece nt deca des, se e e.g. [ 19 , 41 ]. The approximation of this class of equations is particularly challenging due to the curse of dimensio nal- it y bro ught up by the multidim ensional integral of the collision oper ator, and 9 the presence of multiple s c ales. F urthermor e , the preserv ation of relev ant phys- ical q uantities a re ess ential fo r a co rrect descr iption of the underlying physical problem [ 44 ]. In view o f its co mputational efficiency , in the following we w ill ado pt a DSMC approach. Indeed the computational cost of this metho d is O ( N ) wher e N is the nu m be r of particle s . Next, w e descr ibe the DSMC metho d based on a Nanbu- Bav osk y scheme [ 41 ]. W e b egin b y randomly selecting N / 2 pairs of particles and making them evolv e following the binary s cheme pr esented in ( 7 ). W e consider a time interv al [0 , T ] which we divide in N t int e r v als of size ∆ t > 0. The DSMC approach for the introduced kinetic eq ua tion is based o n a first order forward time discretization. In the following w e will alwa ys consider the ca se ∆ t = ǫ > 0 such that all the particles a re going to interact, see [ 41 ] for mo r e details. W e int r o duce the sto chastic ro unding o f a p os itive real nu mber x as: S round ( x ) = ( ⌊ x ⌋ + 1 with probability x − ⌊ x ⌋ ⌊ x ⌋ with probability 1 − x + ⌊ x ⌋ (19) where ⌊ x ⌋ is the integer pa r t o f x . The random v ariable η is sampled from a 2 D Gaussian Distribution centered at zero and a diago nal cov ar iance matr ix. Algorithm 1 DSMC algo rithm for B o ltzmann equation 1: Giv en N particle s ( x 0 n , c 0 n ), with n = 1 , . . . , N computed fro m the initial distribution f 0 ( x , c ); 2: for t = 1 to N t do 3: set n p = Sround( N / 2); 4: sample n p pairs ( i, j ) uniformly without rep etition among all p ossible pairs of particles at time step t ; 5: for ea ch pair ( i, j ), sample η , η ∗ 6: for ea ch pair ( i, j ), compute the da ta change: ∆ x t i = ǫP ∆ 1 , ∆ 2 ( x t i , x t j , c 0 i , c 0 j )( x t j − x t i ) + q 2 σ 2 D ( c 0 i ) η ∆ x t j = ǫP ∆ 1 , ∆ 2 ( x t j , x t i , c 0 j , c 0 i )( x t i − x t j ) + q 2 σ 2 D ( c 0 j ) η ∗ (20) compute x t +1 i,j = x t i,j + ∆ x t i,j (21) 7: end for 3.2 Generation of a mo del-orien ted segmentation masks In this s e ction we pr esent the pro cedure to estimate the Segmentation Mask o f Brain T umor Images. The pro cedure describ ed in this section closely follows the methodo logy presented in [ 9 ]. F or a given image, we define the fea ture’s v a lues in re la tion to the gray level of e ach pix el. In more detail, for a given pixel i ∈ { 1 , . . . , N } we define c i = C i − min i =1 ,...,N C i max i =1 ,...,N C i − min i =1 ,...,N C i ∈ [0 , 1 ] , 10 being C i , i = 1 , . . . , N , the gr ay v alue of the or iginal image. Therefore, the v a lue c i = 1 repr esents a white pixel and c i = 0 represe nts black pixel. In particular, for this work we used the br ain tumor dataset that consists of 3D in multi-parametric MRI o f pa tients affected by glio blastoma or low er -gra de glioma, publicly av aila ble in the context of the Brain T umor Image Segmentation Challenge http:/ /medi caldecathlon.com/ . The ac quisition sequences include T 1 -weigh ted, p ost-Gado linium contrast T 1 -weigh ted, T 2 -weigh ted and T 2 Fluid- A ttenuated Inversion Recov ery volumes. E ach MRI scan is acco mpanied by corres p o nding g round truth segmentation ma sk, which is a binary image where anatomical regions of in ter est are highlighted as white pixels while a ll other areas are repr esented as black pixels. These ground truth segmentation masks w er e manually delineated b y exp erienced radiologists a nd sp ecifica lly identify three structures: ”tumor cor e”, ”enhancing tumor” a nd ”whole tumor”. W e ev alua te the p e r formances of the DSMC algo rithm fo r t wo different segmentation tasks: the ”tumor cor e” and the ”whole tumor” annotations. F o r the first task we us e a single slice in the axial plane o f the p ost-Gado linium contrast T 1 -weigh ted scans while for the second task we use a sing le slice in the axial plane of the T 2 -weigh ted scans . The pro cedure to genera te the seg mentation masks is as follows: 1. W e b egin by asso cia ting e ach pixel with a po sition vector ( x i , y i ) and with static feature c i . W e scale the vector p osition to a domain [ − 1 , 1] × [ − 1 , 1] and the static fea ture to [0 , 1]. 2. W e apply a DSMC approach a s describ ed in Algo rithm 1 to numerically approximate the la rge-time s olution of the Bo ltzma nn-type mo del defined in ( 11 ). This appro ach enables pixels to aggr egate in to cluster s based on their E uclidean distance and gray colo r level. 3. The segmentation masks are g enerated by a ssigning to the original p osition of each pixel the mean v alue of the cluster s they belo ng to. In this wa y w e generate a multi-lev el mask comp osed of a num b er o f homogenous r egions. 4. Finally we obtain the bina r y mask by defining a thre s hold ˜ c such that: c i = ( 1 i f c ≥ ˜ c 0 i f c < ˜ c (22) F o r all the following e x p e riments, ˜ c is defined a s the 10th perce ntile o f pixels in the imag e that b elo ng to the region of interest. This p erc e nt ile was chosen as an optimal v alue for brain tumor images ; how ever, it could also b e co ns idered as a parameter to be optimized within the proce ss outlined in Se c tio n 3.2.1 . F o llowing this pro cedure, we apply tw o morpholog ical refinement steps to remov e small regions that hav e b een misclassified as foregr ound parts a nd to fill small r egions tha t have b een incorr ectly catego rized as background pix els. W e beg in by lab eling all the connected pixels in the fo regro und and rea s signing them to the bac k ground those whose num b er of pixels is less than a c ertain thres ho ld. Then w e rep e at the same pro cedure but for the pixels in the background. T o this end, we use the scikit-imag e python library that detects distinct ob jects of a bina ry imag e [ 55 ]. This allows us to o btain mo re precise s egmentation masks by reducing small imper fections. This entire pro cess is illustrated in Figur e 4 . 11 3.2.1 P arameters optimi zation In this section, we outline the pro cedure for optimizing the par ameters ∆ 1 > 0, ∆ 2 > 0 and σ 2 > 0 that best approximate the ground truth segmentation masks. The goa l is to identif y the par ameter co nfiguration tha t minimizes the discrepancy b etw een the computed and gr ound truth masks, meas ured through a pre defined loss metric. T o achiev e this, we solve the following minimization problem: min ∆ 1 , ∆ 2 ,σ 2 > 0 Loss ( S g , S t ) = min ∆ 1 , ∆ 2 ,σ 2 > 0 1 − M etr ic ( S g , S t ) (23) where S g is the Ground T ruth seg mentation ma sk and S t is the segment - ation mas k computed by the mo del. The differen t Loss metrics quan tify the discrepancy be tw een the masks, with low er v a lues indicating greater similarity . Accordingly , the M etri c function, detailed in Sectio n 3.3 , measur es the simil- arity b etw een the tw o ma sks, with hig her v alues indicating b etter agreement. The re la tionship Loss = 1 − M e tri c is satisfied when the M etr i c is defined to take a v a lue of 1 for per fect agr e e ment and 0 for complete mis ma tch. T o solve the optimization problem ( 23 ), we used the Hyp er opt pack age [ 6 ]. This optimization metho d randomly samples the par ameter configur ations from predefined distributions and selects the co nfiguration that minimizes the Loss metric. This sampling pro cess is r ep eated for a pre defined num b er o f itera- tions. In this work, we sample the v alues o f o ur par ameters from the following distributions: ∆ 1 ∼ U (∆ x, 0 . 7) ∆ 2 ∼ U (0 . 05 , 0 . 3 ) σ 2 ∼ log- uniform( e − 5 , 1) (24) where ∆ x r epresents the distance betw een the initial p ositions o f the pixels at t = 0. W e p er form 300 iterations of the o ptimization pro cess. T o ensure repro ducibility and correctly c ompare the different r esults o btained, the random seed fo r parameter sampling is fixed. Figure 4: Summar y o f the seg ment ation pro ce s s. The first ima g e s hows the input image. By means of the Algorithm 1 we gener ate the Multi-level mask where we reass ign each picture ’s gray level to the mean v alue of the cluster they ar e assigned to. The bina ry mask is pro duced as r e sult of the binarizatio n pro cess. And the Final ma sk is the result after the t wo mo rpholog ic al refinements steps hav e b een a pplied. 12 3.3 Segmen tation Metrics Next, w e in tr o duce the principal optimizatio n metrics use d for ev alua ting a bin- ary seg mentation mask . W e define { S 0 g , S 1 g , S 0 t , S 1 t } where S 0 g and S 1 g represent the s et of pixels that belo ng to the background and foreground o f the gro und truth segmentation ma sk r esp ectively . Sa me applies for S 0 t , S 1 t but for the binary mask we wan t to ev aluate. O ne co uld also wish to a sses the v a lidity o f a seg- men tation mask with multiple lab els, we refer to [ 53 ] for an intro duction to the sub ject. Fig ur e 5 pr esents a summary of the k ey terms used in the definitio ns of metrics. 3.3.1 V olume tri c and Surface Dice Indexes The V olumetric D ic e index, also known as the Standar d V olumetric Dice Simil- arity Co efficient, first int r o duced in [ 18 ], is the most used metric when ev aluating volumetric segmentations. It is defined as follows: DICE = 2 | S 1 g ∩ S 1 t | | S 1 g | + | S 1 t | (25) where, | · | indica tes the total num b er of pixels of the conside r ed region. This metric is equal to one if there is a p e r fect overlap b etw e en the t wo segmentation masks and n ull if b oth segmentations are co mpletely disjoint. Since the V olu- metric Dice co efficient is the mos t commonly us ed metric for segmentations, esp ecially in the biomedical field, the r esults are highly in ter pr etable and can be co mpared with those obtained in other studies. Howev er, whe n as sessing s ur- face segmentation masks, the V olumetric Dice co efficien t can yie ld s ub o ptimal results. This limitatio n arise s b eca use the V olumetric Dice co efficien t ev alua tes the similarity betw een segmentation masks bas ed on pixel ov er lap without con- sidering the spatial accuracy of the b ounda r ies. Sp ecifically , it treats all pixel displacements equally , without co nsidering how fa r a segmentation er ror might be from the tr ue b ounda ry of the o b ject. This means that segmentations with minor erro rs spread ac r oss m ultiple ar eas and those with a ma jor err or in a single area might receive simila r scores. T o addr ess this limitation, the Surfac e Dice Similar ity Co efficient was presented in [ 39 ] as a metric that can assess the accuracy o f segmentation masks by considering the simila rity of their b oundar - ies. W e define ζ : I → R 2 as a parameterization of ∂ S i , the b oundary of the segmentation mask S i . The b order regio n B ( τ ) i , which is a regio n ar ound the bo undary ∂ S i with tolerance τ , is defined as: B ( τ ) i = n x ∈ R 2 / ∃ y ∈ I s.t. || x − ζ ( y ) || ≤ τ o (26) where, τ is a po sitive real num be r that defines the maximum allowable distance from the b ounda r y ∂ S i for a p oint x to b e considered par t of the b o rder r egion B ( τ ) i . The Surface Dice Similarity Co efficient b etw e e n S t and S g with tolerance τ is defined a s : 13 S 1 t S 1 g S 1 t ∩ S 1 g (TP) (a) Intersection area or true p ositiv e (TP). S 1 t S 1 g S 1 t ∪ S 1 g (b) Union area. S 1 t S 1 g S 1 t − S 1 g (FP) (c) F alse p ositive (FP). S 1 t S 1 g S 1 g − S 1 t (FN) (d) F alse negative (FN). S 1 t S 1 g B τ =0 t ∩ B τ =0 g (e) Intersection of boun d- aries at τ = 0. S 1 t S 1 g B τ > 0 t ∩ B τ > 0 g (f ) Intersection of b ound- aries at τ > 0. Figure 5 : Repre sentation o f the relev ant area s b etw een the predicted S 1 t and ground truth S 1 g segmentation ma sks. B τ t and B τ g represent the cor r esp onding bo undaries with a τ thres hold. R ( τ ) g,t = 2    B ( τ ) g ∩ B ( τ ) t       B ( τ ) g    +    B ( τ ) t    (27) 14 R ( τ ) g,t , ranges from 0 to 1. A score of 1 indicates a p erfect ov erla p b etw een the t wo surfa c es, while a sco re of 0 indicates no ov erla p. A larger v alue of τ results in a wider bor der r egion, mak ing the metric mo re tolerant to s mall deviations in the b oundary . 3.3.2 Jaccar d Index The Jaccard Index (JAC) [ 29 ], similar to the V o lumetric Dice co efficient, mea s- ures the similar ity b etw een t wo segmentations b y quantifying the ov erlap b etw een the computed mask and the g round truth. It is defined as the ratio b etw een the int e r section and the union of the foregr ound’s seg mentation masks JAC = | S 1 g ∩ S 1 t | | S 1 g ∪ S 1 t | . (28) The J AC Index and the V olumetr ic Dice co efficient ar e closely r elated since we hav e JAC = DICE 2 − DICE DICE = 2JAC 1 + JAC . (29) F r o m ( 29 ) we get the r e lationship b etw een the JA C index and the V olumetric Dice co efficient. While bo th are widely used for measur ing segmentation simil- arity , they ca n pro duce slight ly different results. T o understand the implications of these differences , we can analyze how their absolute and relative error s are related. Definition 3. 1 (Absolute Approximation) . A similarity S is absolutely appr oximate d by ˜ S with err or ǫ ≥ 0 if t he fol lowing holds for al l y and ˜ y : | S ( y , ˜ y ) − ˜ S ( y , ˜ y ) | ≤ ǫ Definition 3.2 (Relative Approximation) . A similarity S is r elatively appr oximate d by ˜ S with err or ǫ ≥ 0 if the fol lowing holds for al l y and ˜ y : ˜ S ( y , ˜ y ) 1 + ǫ ≤ S ( y , ˜ y ) ≤ ˜ S ( y , ˜ y ) · (1 + ǫ ) . The following result holds Prop ositi on 3.1. J AC and V olumetric Dic e appr oximate e ach other with a r elative err or of 1 and an absolute err or of 3 − 2 √ 2 . W e p oint the rea der to [ 7 ] for a deep er co mparison betw een the Jacca rd and V olumetr ic Dice Index. 15 3.3.3 F-measure The F β − measure is c ommonly used as an info r mation retriev al metr ic [ 50 , 13 ]. T o define this metric, we first in tro duce t wo terms : P ositive Pr edicted V alue (PPV) and T rue P o sitive Rate (TPR), which are also known as Pr ecision and Sensitivity , resp ectively . The Pr e cision metric quantifies the prop or tion of cor - rectly predicted foregr ound pixels (true p ositives, TP) out of all pixels predicted as foregr ound (TP + false p o sitives, FP). The Sensitivity measure s the prop or- tion of actual foregro und pixels (TP ) correctly identified by the mo del out of all a ctual foregro und pixels (TP + false negatives, FN). These t wo metrics can be expressed as follows Precisio n = PP V = TP TP + FP Sensitivity = TPR = TP TP + FN (30) The Pr e cision metric indicates how many of the predicted foreground pixels are actually corr e ct. The Sen sitivity , on the other hand, mea sures how many of the actual foregr ound pixels were cor rectly pr edicted by the mo del. W e ca n define the F β − measure as a combination of P recision and Sensitiv- it y , with a parameter β that controls the trade-off b etw een these tw o metrics. Spec ific a lly , the F β − measure is g iven by FMS β = ( β 2 + 1 ) · PPV · TPR β 2 · PP V + TPR (31) W e may observe that if β = 1 we obtain the V olumetric Dice metric. T o understa nd the impa ct of β in the F β − measure, we can subs titute the definitions of PPV and TPR into ( 31 ), which results in the following FMS β = ( β 2 + 1)TP 2 ( β 2 + 1 )TP 2 + TP ( β 2 FN + FP ) (32) If β > 1 the F β − measure empha sizes minimizing F alse Negatives (maxim- izing Sensitivity), which can lead to more F alse Positives (low er Precision). If β < 1 the F β − measure fo cuses o n minimizing F als e Positives (maximizing Pr e- cision), potentially increasing the n umber of F a lse Negatives (low er Sensitivity). F urther more, it c an be noticed that lim β →∞ ( β 2 + 1 )TP 2 ( β 2 + 1 )TP 2 + TP( β 2 FN + FP ) = Sensitivity = TP TP + FN (33) 16 since for β ≫ 0 we neglect the contribution of the F a lse Positives by cons id- ering only the co ntribution of the F als e Neg atives where we re-obtain the TPR metrics defined in ( 30 ). In summary , thanks to the β par ameter, the F β -measure offers a flexible wa y to ev a luate segmentation mo dels by allowing for a tunable balance betw ee n Pre- cision and Sensitivity . It provides a useful metric when dealing with class im ba l- ances, esp ecially in the field of medical ima g ing, where the relative imp or ta nce of false po sitives and false negatives can v ar y according to eac h s egmentation task. 4 Numerical Results 4.1 Impact of differen t diffusion functions In this s e c tion we study the impact o f choo sing differen t diffusion functions D ( c ) in images c onsisting o f a blurry background and a g eometric s ha p e in the center, as shown in Figur e 7 . The ob jectiv e is to detect the shap e of the Geometric Figure and to c ompare how the choice of different diffusion functions affect the v a lue of the mo del par ameters ∆ 1 , ∆ 2 and σ 2 where the o ptimization pro cess is identical to the one introduce in Section 3.2.1 . T o this end, w e chose the following diffusion functions: D 1 ( c ) = c (1 − c ) D 2 ( c ) = 4 c 2 (1 − c ) 2 D 3 ( c ) = ( c 2 if c ≤ 0 . 5 c 2 (1 − c ) if c > 0 . 5 D 4 ( c ) = 64 c 4 (1 − c ) 4 . (34) W e point the r e a der to Figur e 6 for a summary of the v a r ious introduced diffusion functions in ( 34 ). F o r bo th the squar e and circle images, the Sur face Dice Co efficient was used to o ptimize the pa rameters with a tolera nce equal to the length of 1 pixel. Both images ha ve a s ha p e of (256 , 2 56) pixels. The final time w as set to T = 20 0 with ∆ t = 0 . 1. The r esulting binary mask w a s the same for all choices of diffusion functions, o btaining the same loss function v a lue. The r esults ar e shown in Figure 7 . In the ca se o f the squa re Fig ure 7 (a) we can see from T a ble 1 that for D 1 ( c ) and D 3 ( c ) the v a lues of ∆ 1 do no t differ greatly for this t wo diffusion functions. In the case of ∆ 2 we obtain a slight ly smaller v alue for D 1 ( c ) compar e d to the one obtained for D 3 ( c ) a nd a lar ger v alue of the parameter σ 2 > 0 fo r D 3 ( c ) compar e d to the one obtaine d fo r D 1 ( c ). If we lo ok at Figure 6 we no tice tha t D 1 ( c ) ≥ D 3 ( c ). Therefor e, a larger v alue of the diffusion functions is balanced by a smaller v alue of σ 2 to o bta in a s imilar diffusion effect. This holds also for D 1 ( c ) and D 3 ( c ) for the cir cle Fig ure 7 (b). 17 Figure 6: Diffusion functions defined in ( 34 ) to a ssess the v aria bilit y r elated to a g iven feature’s level. 18 (a) (b) Figure 7: Images used to test differen t diffusion functions . The first column displays the or ig inal images, the second column presents the e xp ected segment- ation mask, and the third column shows the res ulting binary mask. Each picture consists of (25 6 , 256) pix els. F or the o ptimization pro cedure we set T = 20 0 and ∆ t = 0 . 1 . W e define the num b er of iterations at 50. Row (a) shows the imag e with a square on a blurr y background, while row (b) displays a s imila r image, but with a circle. Only one resulting binar y mask was rep orted for ea ch of the images b eca use all the tests describ ed in this section obtain the same segment- ation mask. 19 F urther more, c o mparing D 2 ( c ) and D 4 ( c ) for the square imag e we ca n se e that the resulting parameters are smaller fo r D 2 ( c ) in co ntrast to the one obtained with D 4 ( c ). This is consistent b ecaus e, ag ain, we ca n see from Figur e 6 that D 2 ( c ) ≥ D 4 ( c ). If we no w compare D 2 ( c ) a nd D 4 ( c ) for the circle image we can see that the v alue of σ 2 is similar in this case. Nevertheless, in this case the difference is given by the v alues of ∆ 1 and ∆ 2 , which ar e b oth s maller for D 2 ( c ). This indicates that, for differen t diffusio n functions, the optimal parameters adjust to yield similar results. A very straight wa y is to obtain similar v alues o f ∆ 1 and ∆ 2 and a low er v alue of σ 2 for the diffusion function that has a higher v alue as in the cas e of the sq ua re imag e. How ever, the exa mple of the circle image shows that us tha t w e can also obtain different combinations of parameter s so as to counter the effect o f a bigg er diffusion function. T a ble 1: Parameters obtained for different Diffusion F unctions for the Square and Circle Images. The loss metric used to obtain these parameters was the Surface Dice Co efficient with a toler ance equa l to the length of 1 pixel. Square ∆ 1 ∆ 2 σ 2 D 1 ( c ) 0.884 0.310 0.889 D 2 ( c ) 0.351 0.054 0.047 D 3 ( c ) 0.817 0.407 1.341 D 4 ( c ) 0.442 0.081 0.624 Circle ∆ 1 ∆ 2 σ 2 D 1 ( c ) 0.435 0.341 1.829 D 2 ( c ) 0.013 0.160 2.717 D 3 ( c ) 0.408 0.268 2.693 D 4 ( c ) 0.154 0.228 2.572 F r o m T able 2 we can se e the pa rameters obtained by minimizing thre e differ- ent optimization metrics using as a diffusion function D 1 ( c ) for the square image. F o r all the cases the resulting Surfac e Dice was equal to one indicating a p erfect ov erla p betw een the co mputed a nd the g round truth s egmentation masks. The resulting binary mask obtained w ere the same for the three examples a nd ar e equiv alent to the o nes shown in Figure 7 . F o r the V o lumetric and Sur face Dice Co efficient we ca n see tha t the parameter s obtained where identical. Nev erthe- less, for the Jaccard Index, the resulting para meters differed, b eing smaller in this case. The lo ss is null in b oth c ases, coherently with the r elationship ( 29 ). 20 0 25 50 75 100 125 150 175 200 time 0.00 0.02 0.04 0.06 0.08  Figure 8 : Evolution of T where the kinetic density is the o ne co nsidered in Figure 7 (a). The image consists o f (256 , 2 56) pixels. W e can observe how T decreases until condition T < δ is reached with δ = 0 . 0 05. T a ble 2 : Parameters obtained for the Square Imag e by minimizing the Ja ccard Index and the V olumetric and Surface Dice Co efficient. F or the Surfa ce Dice Co efficient the toler ance w a s s et to the length of 1 pixel. The lo ss obtained w a s zero for the three cases. Square ∆ 1 ∆ 2 σ 2 V ol. Dice 0.884 0.310 0.889 Surf. Dice 0.884 0.310 0.889 JAC 0.442 0.081 0.624 4.2 Determining the final time In this section we sp ecify the criter ion tha t we implemented to determine the final time T > 0. As defined in Section 3.1 , w e approximate the solution of ( 18 ) through a DSMC approach, even though w e hav e no analytica l insigh t on the form of the steady state. The ob jectiv e is to find the v alues of the final time T > 0 such that a numerical s teady state can be defined. W e str e s s that the time ta ken to reach the equilibrium state for different initial conditions is not the s a me so w e need to de ter mine the time para meters for a ll the images we wan t to analyze. T o this end, if f n ( x , c ) is the approximation of the dens ity at 21 time t n = n ∆ t , we define T = Z R 2 × [0 , 1] | f n +1 ( x , c ) − f n ( x , c ) | d x dc, (35) which r epresents an index of v a r iation b etw een tw o suc c essive time steps of the reco nstructed kinetic density . As the so lution evolv es , this qua ntit y de- creases and tends to zero as the eq uilibrium state is rea ched, as illustrated in Figure 8 for the case of the Squar e Image with a blurr y background. Hence, we may introduce a brea king cr iter ion based on the co ndition T < δ , for some δ > 0. When this condition is sa tisfied, the reco nstructed density is co nsidered an a pproximation of the steady sta te. The sa me pro ce dur e was done for all imag es prese nt e d in this work so as to fulfill with the condition prese nt e d in this section. 4.3 Optimization Metrics for T umor I mage Analysis In this section we study the impact that the differe nt optimization metrics have on the r esulting binar y mask for the Core and Whole T umor. W e also analy ze the par a meters obtained for the different o ptimization metr ics. Both brain tu- mor images consist of N = (240 , 240 ) pixels. F or the optimization pro cedure we determine T = 100 a nd ∆ t = 0 . 01. F o r each segmentation mask generated we ev a luated 3 00 differ e nt com binations of pa rameters. Figure 9 s how the Seg - men tation Ma sks obta ined fo r b oth the Whole and Core T umo r by optimizing the Jaccard I ndex a nd the V olumetric Dice Co efficient. In T a ble 3 the re s ulting parameters and the lo s s obtained for b oth optimization metrics are pres e nt ed, in this case the loss is equa l to 1 for a perfect ov erlap and 0 if the images are totally disjoint. First, we ca n o bserve that the lo s s obtained with b oth metrics satisfy ( 29 ) as expec ted. It can b e noticed that fo r b o th segmentation masks the loss obtained is gr e ater for the V olumetric Dice Co efficient. F urthermore, the parameter ∆ 1 obtained with both optimization metrics is similar fo r b oth the Core and Whole T umor. Nevertheless, we can see that for the Whole T umor the ∆ 2 parameter obta ine d with the J accard Index is bigger than the one obtained with the V olumetric Dice Co efficient. F or the case of the Core T umor instead the ∆ 2 parameter is bigg er for the V o lumetr ic Dice Co efficien t. If we co mpare this to the v alues obtained fo r σ 2 in b oth case s for b o th metrics we ca n see that a bigger diffusion v alue is co unt e red b y a sma ller v a lue of ∆ 2 so as to obtain similar Segmentation Masks as seen fro m Figur e 9 . F o r the Surface Dice Co efficient, the tolera nce τ was set to the leng th of 1 pixel, b oth when used as the optimization loss and when used as the e v a luation metric. In Fig ure 10 shows the resulting bina ry mas k obtained with the Surface Dice Co efficien t a nd the V o lumetr ic Dice Co efficient for the Cor e and Whole T umor . In the cas e of the Whole T umor the lo ss obtained with Surface Dice 22 Original Gr ound T ruth Mask J A C - Binary Mask DICE - Binary Mask (a) Original Gr ound T ruth Mask J A C - Binary Mask DICE - Binary Mask (b) Figure 9: Segmentation Masks obtained by minimizing the Jaccard Index and the V olumetric Dice Co efficien t. (a) shows the res ults for the Core T umor and (b) shows the results for the Whole T umor . Both images consist of 24 0 × 240 pixels. F or the o ptimization pro cedure we set T = 100 and ∆ t = 0 . 01 . In bo th cases we co nsidered 300 iterations of the o ptimization algo rithm. In b oth cases the loss re po rted b y the Jaccar d Index was smaller compared to the one obtained with the V olumetric Dice Co efficient. F urthermor e, it can b e noticed that the los ses rep orted satisfy rela tion 29 as exp ected. F rom the v a lues of the parameters we c an observe that a bigger v alue of the diffusion is countered by a s maller v alue of ∆ 2 . 23 T a ble 3: Parameters obtained for the Whole and Co re T umor using the V olu- metric Dice Coefficie nt , Jac c a rd Index and the Surface Dice Co efficient. The loss rep orted is 1 for p erfect overlap and 0 for complete deviation. Whole T umor Opt. F unction ∆ 1 ∆ 2 σ 2 Loss V ol. Dice 0.4972 0.0888 2.6867 0.9292 JAC 0.5075 0.1187 2.3631 0.8672 Surf. Dice 0.6383 0.0579 2.6504 0.7447 Core T umor Opt. F unction ∆ 1 ∆ 2 σ 2 Loss V ol. Dice 0.3795 0.1254 2.1808 0.9360 JAC 0.3823 0.1004 2.7001 0.8796 Surf. Dice 0.6841 0.0760 1.4155 0.8727 Co efficient is smaller than the o ne obtaine d with the Ja ccard Index and the V olumetr ic Dice Coefficient. F or the Co re T umo r the loss obtained with the Surface Dice Co efficient is similar to the one rep orted by the Jacca rd Index and bo th are smaller than the obtained with the V olumetr ic Dice Co efficient. F or the Whole T umor we can see that the r esulting parameters are simila r for all the optimization metrics . Nevertheless, for the Co re T umor we can no tice that the para meters obtained with the Surface Dice Co efficient differ compare d to the ones obtaine d with the Jaccar d Index and the V olumetr ic Dice Co efficient. In pa rticular, we o btained a smaller v alue for σ 2 and slight ly bigger v alue for ∆ 1 . This indica tes tha t a smaller v alue for the diffusion of the particle s is comp ensated by allowing the par ticles to a ggreg ate with others that are slig ht ly more separ a ted than in the case of the V olumetric Dice and Jaccard Index. Given that b oth the V olumetric Dice Co efficient a nd the Jaccar d Index are a measure of the super po sition betw een t wo volumes (in this ca se t wo surfa ces) they do not represent the proximit y b etw ee n t wo surfaces making the Surface Dice Co efficient mor e suitable to use as a loss metric when comparing t wo different surfaces . F o r the F β − measure we ca n s ee in Figure 11 the binar y masks obtained for different v alues of β for the Co re and Whole T umor . F or the case of the Co re T umor we can observe that for β = 0 . 25 we obta in are as of misclassifie d pixels in the tumor r e gion. This can a lso be seen from T able 4 where the num b er o f F a ls e Negatives is bigger and the num b er of F alse Positives is s ma ller compare d to the results o btained for bigger v a lues of β . If we reca ll ( 31 ) w e can see that for low v a lues of β the F a lse Negatives are m ultiplied by a factor of β 2 th us having a smaller weight compared to the F alse P o sitives. As we increase the v a lue of β we can notice fr om b oth T able 4 and Fig ure 11 tha t mo difying the v a lue of β has no impact on the r esulting binary mask. This als o holds true 24 Original Gr ound T ruth Mask Surface Dice - Binary Mask DICE - Binary Mask (a) Original Gr ound T ruth Mask Surface Dice - Binary Mask DICE - Binary Mask (b) Figure 10 : Segmentation Masks o btained by minimizing the Surface and V olu- metric Dice Coe fficie nt . (a ) shows the results for the Core T umor and (b) shows the res ults for the Whole T umor . Both imag es consist of 2 40 × 24 0 pixels. F or the optimization pro cedure w e set T = 100 and ∆ t = 0 . 01. In b oth cases we considered 300 iter ations of the optimization algorithm. F or the Surface Dice Co efficient we set the tolerance τ equal to the leng th of 1 pixel. Given tha t bo th the V olumetric Dice Co efficient and the Ja ccard Index are a measure of the sup er p o sition be t ween the tw o surfaces and do no t ac c ount for the proxim- it y betw een the t wo surfaces at every given p oint the Surface Dice Co efficen t represents a more suitable metric when co mpa ring tw o different surfaces. 25 for the Whole T umor as no difference ca n be noticed in the results obtained for different v alues of β . Fina lly in Fig ure 12 we see the loss rep o rted for different v a lues of β where the loss equal to 1 represe nt s a p erfect ov er lap. First, it can be noticed that w e o btain the higher v a lue of the loss for β = 0 . 2 5, mea ning that this sho uld be the most a ccurate result, which is anywa y balanced by the fact that w e o btain the larger num b er of F alse Neg atives. Again, we s aw that this can be obtained from ( 31 ), where low v alues of β reduce the impact of a large num b er of F alse Nega tives on the resulting loss. Secondly , we o bserve tha t the loss decreases for la rger v alues of β . This b e havior arises b ecause the loss is inv ersely prop ortio nal to β , while the res ulting segment ation masks remain unch anged, as shown in T able 4 . This shows tha t the F β -measure may not b e a r e liable metric for these types of segmentation masks and this segmentation metho d, and that mo difying the v alue o f β provides no adv antage. T a ble 4: Parameters obta ined for the F β -measure for different v alues of β . The loss rep o rted is 1 for p er fect o verlap and 0 for co mplete deviation. The nu m be r of F alse Positives (FP), F alse Negatives (FN) and T rue Positives (TP) are presented for the resulting seg mentation mask for ea ch v a lue o f β . Whole T umor ∆ 1 ∆ 2 σ 2 FP FN TP Loss β = 0 . 25 0.6873 0.1707 2.23 95 134 347 3170 0.9559 β = 0 . 5 0.3351 0.1080 2.70 51 134 350 3167 0.9470 β = 0 . 75 0.5939 0.2304 2.67 18 134 350 3167 0.9373 β = 1 . 5 0.5316 0.1092 2.71 05 136 349 3168 0.9179 β = 5 . 0 0.5662 0.1225 2.70 43 136 349 3168 0.9032 β = 10 . 0 0.6061 0.2835 2.12 43 136 349 3168 0.9013 Core T umor ∆ 1 ∆ 2 σ 2 FP FN TP Loss β = 0 . 25 0.6575 0.2725 0.02 57 9 20 6 849 0.9 7 63 β = 0 . 5 0.3989 0.0637 1.80 94 25 107 948 0.9 582 β = 0 . 75 0.4073 0.0942 1.69 72 25 105 950 0.9 460 β = 1 . 5 0.5444 0.2077 2.35 45 25 105 950 0.9 220 β = 5 . 0 0.5587 0.1742 2.68 64 25 105 950 0.9 032 β = 10 . 0 0.6137 0.2425 1.97 57 25 105 950 0.9 012 Conclusions In this pap er we presented a consensus - based kinetic metho d and show how can this mo del can b e applied for the problem of image s egmentation. The pixel in a 2 D image is in terpr eted a s a par ticle that interacts w ith the rest through a conse nsus-type pro cess, which allows us to identify differe nt clusters and gener a te an ima ge seg mentation. W e develop ed a pro cedure that allows us to approximate the Ground T ruth Segmentation Mas k of different Br ain T u- mor Images . F urthermore, we presented a nd ev aluated differen t optimization metrics a nd study the impact on the results obtained. In particular we found 26 β = 0. 25 β = 0.5 β = 0. 75 β = 1.5 (a) β = 0. 25 β = 0.5 β = 0. 75 β = 1.5 (b) Figure 11: Segmentation masks obtained for the F β − Loss metr ic. (a) Shows the segmentation ma s ks obtained for β = 0 . 2 5 , 0 . 5 , 0 . 75 , 1 . 5 for the Cor e tumor and (b) Shows the seg ment a tion masks obtained using the same v alues of β for the Whole T umor. Both images co nsist o f 2 40 × 240 pixels. F or the optimization pro cedure we set T = 300 and ∆ t = 0 . 01. In b oth cases w e co nsidered 300 iterations of the optimization algorithm. In (a) we can obs e rve that for β = 0 . 25 the resulting seg ment ation masks display area s o f miscla s sified pixe ls while for bigger v alues of β the resulting segmentation ma s k does not differ. In (b) no zo omed area is shown as the segmentation ma sks display no visible differences for the differen t v alues of β . This can b e seen also from T able 4 by o bserving the num b er of F alse Positiv es (FP), F als e Nega tives (FN) a nd T rue P o sitives (TP) obtained for b oth images . 27 Figure 12: Relationship b etw een the F β − lo ss v alue and the β v alue for b oth the Core and Whole T umor Imag es. As β incr eases, the F β − loss decreas es showing that for low er v alues of β we should obta in a mor e precise segmentation mas k as the loss indica ted in this Figure is 1 for p erfect ov er lap. Nevertheless, the resulting binary mask is less accura te for low er v alues of β showing that this is not an appropr ia te metric for optimizing the co nsensus-bas ed mo del. that the Ja ccard Index and the V olumetric and Surface Dice Co efficient are appropria te metric to optimize our mo del. Nevertheless, given that the Sur- face Dice Coe fficie nt is meas ure of discrepancy betw e en the boundar ies of tw o surfaces it is a be tter representation compared to the Ja ccard Index and the V olumetr ic Dice Co efficient as they acc ount only for absolute difference s and do not attain to p oint wis e differences. F urthermore, we ass e ssed the use of the F β -Loss as a p otential optimization metric. W e found that b oth the loss v a lues and the corre s p o nding res ults were difficult to interpret, as low loss v alues often corres p o nded to low accuracy , making this metric challenging to a pply effect- ively for optimization in this context. F uture resea r ches will fo cus on the ca se of multidimensional features and p otential training metho ds for the introduced mo del. Ac kno wledgmen ts M.Z. is mem b er o f GNFM (Grupp o Na zionale di Fisica Matematica) o f INdAM, Italy and a ckno wledge s supp or t of PRIN20 22PNRR pro ject No.P20 2 2Z7ZAJ , Europ ean Unio n - NextGeneration EU. M.Z. ackno wledg es pa rtial suppor t by ICSC – Cen tr o Nazio na le di Ricer c a in High Performance Computing, Big Data and Q uantum Computing, funded by Eur op ean Union – NextGener ationEU 28 References [1] Agosti, A., Shaq iri, E., Paoletti, M., Solazzo , F., Bergsla nd, N., Colelli, G., Savini, G., Muzic, S., Santini, F., Deligianni, X. & Others Deep lea rning for automatic segmentation of thigh a nd leg muscles. Magn. R eson. Mater. Phys. Biol. Me d. . 35 pp. 4 67-48 3 (2021). [2] Albi, G., Paresc hi, L., T o scani, G., & Zanella, M. Recent adv anc e s in opin- ion modeling : control and so cial influence. 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