RECOGNITION of online handwriting aims at finding the

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1 SUBMITTED ON SEPTEMBER A General Framework for te Recognition of Online Handwritten Grapics Frank Julca-Aguilar, Harold Moucère, Cristian Viard-Gaudin, and Nina S. T. Hirata arxiv: v1 [cs.cv] 19 Sep 2017 Abstract We propose a new framework for te recognition of online andwritten grapics. Tree main features of te framework are its ability to treat symbol and structural level information in an integrated way, its flexibility wit respect to different families of grapics, and means to control te tradeoff between recognition effectiveness and computational cost. We model a grapic as a labeled grap generated from a grap grammar. Non-terminal vertices represent subcomponents, terminal vertices represent symbols, and edges represent relations between subcomponents or symbols. We ten model te recognition problem as a grap parsing problem: given an input stroke set, we searc for a parse tree tat represents te best interpretation of te input. Our grap parsing algoritm generates multiple interpretations (consistent wit te grammar) and ten we extract an optimal interpretation according to a cost function tat takes into consideration te likeliood scores of symbols and structures. Te parsing algoritm consists in recursively partitioning te stroke set according to structures defined in te grammar and it does not impose constraints present in some previous works (e.g. stroke ordering). By avoiding suc constraints and tanks to te powerful representativeness of graps, our approac can be adapted to te recognition of different grapic notations. We sow applications to te recognition of matematical expressions and flowcarts. Experimentation sows tat our metod obtains state-of-te-art accuracy in bot applications. Index Terms Grapics recognition, online andwriting recognition, grap parsing, matematical expression, flowcart. 1 INTRODUCTION RECOGNITION of online andwriting aims at finding te best interpretation of a sequence of input strokes [1]. Rougly speaking, andwriting data can be divided into two broad categories: text and grapics. In text notation, symbols are usually composed of strokes tat are consecutive relative to a time or spatial order; and symbols temselves are also arranged according to a specific order, for example, from left to rigt. Te ordering of symbols defines a single adjacency, or relation type, between consecutive symbols. By contrast, grapics encompass a variety of object types suc as matematical or cemical expressions, diagrams, and tables. Symbols in grapics notation are often composed of strokes tat are not consecutive wit respect to neiter time nor spatial order. Furtermore, a diversified set of relations is possible between arbitrary pairs of symbols. See Figure 1, for instance, were a andwritten matematical expression illustrates some caracteristics of grapics notation. Due to te linear arrangement of symbols, text recognition can be modeled as a parsing of one-dimensional (1D) data. On te oter and, grapics are intrinsically two-dimensional (2D) data, requiring a structural analysis, and tere are no standard parsing metods as in te 1D case. Parsing depends on symbol segmentation (or, stroke grouping), symbol identification, and analysis of structural relationsip among constituent elements. Stroke grouping Frank Julca-Aguilar and Nina S. T. Hirata are wit te Department of Computer Science, Institute of Matematics and Statistics, University of São Paulo, Brazil. {faguilar, nina}@ime.usp.br Harold Moucère and Cristian Viard-Gaudin are wit Institut de Recerce en Communications et Cyberntique of Nantes, University of Nantes. {cristian.viard-gaudin, arold.moucere}@univ-nantes.fr above below k rigt rigt n = rigt 8 x 9 10 rigt subscript n z 12 subscript n Fig. 1. Handwritten matematical expression example. Top: A sequence of strokes were te order (indicated by numbers in blue) is given by te input time. Symbols and z are composed of non-consecutive strokes. Bottom: Te expression is composed of symbols and several types of spatial relations between tem. in texts is relatively simpler tan in grapics as already mentioned. Identification of segmented symbols include callenges suc as te possibly large number of symbol classes, sape similarity between symbols in distinct classes, and sape variability witin a same class (e.g. arrows in flowcarts migt include arbitrary curves, and be directed towards any orientation). Structural analysis involves te identification of relations between symbols and a coerent integrated interpretation. Te large variety of relations

2 SUBMITTED ON SEPTEMBER migt define complex ierarcical structures tat increments te difficulty in terms of efficiency and accuracy. Tere is a strong dependency among te tree tasks since symbol segmentation and classification algoritms must often rely on structural or contextual information to solve ambiguities, and structural analysis algoritms depend on symbol identification to build coerent structures. Altoug recognition of 2D objects is a subject of study since long ago [2], many of te efforts are still focused on solving specific aspects of te recognition process (e.g., detection of constituent parts or classification of components and teir relations). A large number of works tat tackle te entire recognition problem is clearly emerging, but tey are often restricted to specific application domains and ave limitations [3], [4], [5]. Motivated by te problem of online andwritten matematical expression recognition, we ave examined issues related to te recognition process and identified tree features tat are desirable. Te first feature is multilevel information integration. By multilevel information integration we mean integrating symbol and structural level information to find te best interpretation of a set of strokes. In matematical expression recognition, metods tat seek information integration ave already been te concern of several works [6], [7], [8], but it is still one of te most callenging problems. Te second feature is related to model generalization. Existing metods often limit te type of expressions to be recognized (for instance, do not include matrices), consider a fixed notation (for instance, it adopts eiter n i=1 x i or n x i ), or i=1 limit te set of matematical symbols to be recognized. Any extensions regarding tese limitations may require major canges in te recognition algoritms. Te tird feature is computational complexity management. A general model often results in exponential time algoritms, making its application unfeasible. Existing models andle time complexity issues by adopting constraints tat limit te recognizable structures [8], [9]. To deal wit te issues described above, we ave elaborated a general framework for te recognition of online andwritten matematical expressions and ten sow its generality by building a flowcart recognition system using te same framework. We model a matematical expression as a grap, and represent te recognition problem as a grap parsing problem. Te recognition process is divided into tree stages: (1) ypoteses identification, (2) grap parsing, and (3) optimal interpretation retrieval. Te first stage computes a grap, called ypoteses grap, tat encodes plausible symbol interpretations and relations between pairs of suc symbols. Te second stage parses te set of strokes to find all interpretations tat are valid according to a predefined grap grammar, using te ypoteses grap to constrain te searc space. Te parsing metod is based on a recursive searc of isomorpisms between a labeled grap defined in te grap grammar and te ones derived from te ypoteses grap. Te last stage retrieves te most likely interpretation based on a cost function tat models symbol segmentation, classification and structural information jointly. Conceptually, te valid structures are defined troug a grap grammar and likely structures in te input stroke set are captured in te ypoteses grap. Tus, te proposed framework enances independence of te parsing step wit respect to specificities of te matematical notation considered. As a consequence, we ave a flexible framework wit respect to different matematical notations. For instance, new expression structures can be included in te family of expressions to be recognized by just including te structures in te grammar rules. Similarly, te class of matematical symbols to be recognized can be extended by just including new symbol labels in te grammar and in te ypoteses grap building procedure. Wit respect to grapics in general, among tem tere is large difference in te set of symbols and relations between symbols. Tus, recognition tecniques are often developed for a specific family of grapics, introducing constraints tat not only limit teir effectiveness, but also teir adaptation to recognize different families of grapics. In spite of tese differences, grapic notations sare common concepts a set of interrelated symbols spread over a bidimensional space, organized in ierarcical structures tat are decisive to te interpretation. We argue tat te flexibility of te proposed framework encompasses oter families of grapics. Tis argument is supported by te fact tat graps as already proven adequate to model grapics in general. In addition, tere are examples tat sow tat families of grapics can be specified by means of a grap grammar [5], [10], [11]. Moreover, ypoteses graps can be built based on datadriven approaces. Te main contributions of tis work are tus twofold. First, we present a general framework in wic te parsing process is independent of te family of grapics to be recognized and a control of te computational time is possible by means of a ypoteses grap. Second, we demonstrate an effective application of te framework to te recognition of matematical expressions and flowcarts. Te remaining of tis text is organized as follows. In Section 2 we review some metods and concerns in previous works related to te recognition of matematical expression and flowcarts, as tese types of grapics served as te ground for te development of te metod described in tis manuscript. We also briefly comment on some works tat proposed grap grammars for te recognition of 2D data and influenced our work. In Section 3 we detail te proposed framework. Ten in Section 4 we describe ow te elements and parameters required by te framework ave been defined for te recognition of matematical expressions and flowcarts. In Section 5 we present and discuss te experimental results for bot applications, and in Section 6 te conclusions and future works. 2 RELATED WORK In tis section, we review some caracteristics of te recognition process in previous works, wit empasis on metods for matematical expression [12], [13], [14] and flowcart recognition [15], [16], [17]. Early works related to te recognition of matematical expressions were predominantly based on a sequential recognition process consisting of te symbol segmentation, symbol identification and structural analysis steps [18], [19],

3 SUBMITTED ON SEPTEMBER [20]. However, a weakness of sequential metods is te fact tat errors in early steps are propagated to subsequent steps. For instance, it migt be difficult to determine if two andwritten strokes wit sape ) and (, close to eac oter, form a single symbol x, or are te opening and te closing parenteses, respectively. To solve tis type of ambiguity, it may be necessary to examine relations of te strokes wit oter nearby symbols or even wit respect to te global structure of te wole expression. Tis type of observation as motivated more recent works to consider metods tat integrate symbol and structural level interpretations into a single process. Most of tem are based on parsing metods as described below. Given an input stroke set, te goal of parsing is to find a parse tree tat explains te structure of te stroke set, relative to a predefined grammar. From a ig-level perspective, parsing-based tecniques avoid sequential processing by generating several symbol and relation interpretations, combining tem to form multiple interpretations of te wole input stroke set, and selecting te best one according to a score (based on te wole structure). An important element in parsing based approaces is te grammar. A grammar defines ow we model a (grapics) language. For matematical expressions, most approaces [21], [22], [23], [24], [25] use modifications of context-free string grammars in Comsky Normal Form 1 (CNF). Suc grammars define production rules of te form A r BC, were r indicates a relation between adjacent elements of te rigt and side (RHS) of te rule. For instance, expression 4 2 can be modeled troug a rule T ERM superscript NUMBER NUMBER. However, as suc grammars impose te restriction of aving at most two elements on te RHS of a rule, structures wit more tan two components, like 2 + 4, or n x i, must be modeled as a recursive composition of pairs of components. MacLean et. al. [8] proposed fuzzy relational context free grammars to overcome tis limitation. Tey included production rules of te form: A r A 1 A 2... A k, were r indicates a relation between adjacent elements of te RHS of te rule. However, te model assumes tat te relation can only be of vertical or orizontal types. Celik and Yanikoglu [9] use grap grammars wit production rules of te form A B, were bot A and B are graps, and B represents te components of a subexpression as vertices and teir relations as edges. Grap grammar models offer more powerful representativeness compared to string grammars. However, te autors limit te grammars to ave specific structures (eac grap in a rule is eiter a single vertex grap, or a star grap a grap wit a single central vertex and surrounding vertices tat are connected only to te central one), largely restricting te set of recognizable expressions. Wit respect to parsing, most algoritms proposed in te literature for matematical expressions are based on te CYK algoritm [26]. Te CYK algoritm assumes tat te input (in our case) strokes form a sequence and te grammar is in CNF. Tose based on bottom-up approaces build a parse tree by first identifying symbols (leaves) from 1. In a CNF, all production rules eiter ave te form A a, or A BC, were a is a terminal and A, B, and C are non-terminals i single or groups of consecutive strokes, and ten combining te symbols recursively to form subcomponents (subtrees), until obtaining a component tat covers te wole input set. To adapt te CYK algoritm to te recognition of matematical expressions, Yamamoto et. al. [24] introduced an ordering of te strokes based on te input time. Oter approaces avoid te stroke ordering assumption, but introduce different constraints to satisfy te decomposition of te input into pairs of components [21], [22], [23], [25]. MacLean et. al. [8] proposed a top-down parsing algoritm tat does not assume grammars in CNF, but assumes tat te input follows eiter a vertical or orizontal ordering (te fuzzy relational context free grammars mentioned above). Metods tat use te CYK algoritm or oters borrowed from te context of string grammars must decompose te 2D input into a set of 1D inputs. As tere is no guarantee tat suc decomposition is possible, tese metods may present strong limitations wit respect to parsable 2D structures and be completely inappropriate for oter types of 2D data. On te oter and, metods tat consider grap grammars face computational complexity issues. A key step of any parsing algoritm is te definition of ow a stroke set can be partitioned according to te RHS of a rule. Let us consider a set of n strokes. Assuming stroke ordering and a CYK-based algoritm as in [21], [22], [23], [24], [25], rules ave at most two components in te RHS and terefore te number of meaningful partitions is O(n) we can assign te first i strokes to te first component and te rest for te second, wit i {1,..., n 1}. On te oter and, if we do not impose CNF, but keep te stroke ordering assumption as in [8], ten a rule may ave k symbols on its RHS, and te number of meaningful partitions is O ( n k), corresponding to k 1 split points on te sequence of n strokes. In grap grammars, witout any restriction and a rule wit k vertices in te RHS, te number of partitions is O(n k ) any non-empty stroke subset can be mapped to any vertex. Restricting te grap structures in te grammar, for instance to star grap structures as done by Celik and Yanikoglu [9], is a way to manage te parsing complexity. Note, owever, tat in tis case te set of recognizable expressions is constrained not only by te parsing algoritm but also by te grammar. Flowcarts in general ave a smaller symbol set tan matematical expressions. However, teir structure presents iger variance. For instance, te flowcart in Figure 2 includes two loops, and adjacent symbols can be located at any (vertical, orizontal, or diagonal) position relative to eac oter, regardless te relation type. In contrast, in matematical expressions, for a given relation type between two symbols (e.g. superscript) it is expected tat one symbol is located at some specific area relative to te oter (e.g. toprigt). Tus, for flowcarts it may be difficult to establis a spatial ordering of te input strokes. To cope wit te structural variance of diagrams, some approaces introduce strong constraints in te input, as requiring all symbols to ave only one stroke [27], or looplike symbols to be written by consecutive strokes [15]. Wit respect to symbol recognition, detection of texts (or text box) and arrow symbols are regarded as more difficult, as tey do not present a fixed sape. For instance, Carton et. al. [16] determine box symbols (like decision, and data structure)

4 SUBMITTED ON SEPTEMBER and ten select te best interpretations using a deformation metric. Text symbols are recognized only after box symbols. Bresler et. al. [17] also first recognize possible box and arrow symbols, and leave text recognition as a last step. After symbol candidates are identified, te best symbol combination is selected troug a max-sum optimization process. Fig. 2. Flowcart example. Strokes are colored according to te symbol type tey belong to. An interesting example of grap grammar use is described in [11]. Te autors propose an attributed grap grammar tat allow attributes to be passed from node to node in te grammar, bot vertically and orizontally, to describe a scene of man-made objects. Projection of rectangles are used as primitives. However, passage of attributes must be evaluated during parsing, making te parsing algoritm be context-dependent. In [10] entity-relationsip diagrams are modeled by a context-sensitive grap grammar wit te left-and side of every production being lexicograpically smaller tan its rigt-and side. A critical part of te parsing algoritm is to find matcings of te rigt-and side of a rule to replace te left-and-side, making it very complex. Te above review on some caracteristics related to te recognition of 2D data illustrates tat existing metods present several restrictions and limitations and clearly can not be easily transposed to te recognition of oter families of grapics. In te metod proposed in tis work, instead a CYKbased algoritm (tat assumes a grammar in CNF), we define a grap grammar and use a top-down parsing algoritm, similar to te one of [8], but witout assuming any ordering of te input strokes. To avoid context-aware algoritms during parsing, we consider stroke partitions drawn from a previously built ypoteses grap (see Section 3.4) to matc te rigt-and side of te rules. By doing tis, we decouple te parsing algoritm from te particularities of te family of grapics, and acieve independence of te target notation. In addition, it is important to note tat target domain knowledge can be fully exploited in te grap grammar definition and ypoteses grap building. Tis caracteristic makes te proposed metod general enoug to be applied to te recognition of a variety of grapic notations. 3 THE PROPOSED RECOGNITION FRAMEWORK Te proposed recognition framework is composed of tree main parts: (1) ypoteses grap generation, (2) grap parsing, and (3) optimal tree extraction. In te first part, stroke groups tat are likely to represent symbols, and a set of possible relations between tese stroke groups are identified and stored as a grap, called ypoteses grap. In te second part, valid interpretations (potentially multiple of tem) are built from te ypoteses grap by parsing it according to a grap grammar. Te interpretations found are stored in a parse forest. Ten, in te tird part an optimal tree is extracted from te parse forest, based on a scoring function. We first discuss te two main input data of te framework, a andwritten input grapic to be recognized (a set of strokes) and a grap grammar, and ten detail te tree parts, keeping an abstraction level suitable for te recognition of a variety of grapics in general. Concepts are illustrated using matematical expressions as examples. Implementation related details regarding te application of te framework to te recognition of matematical expressions and flowcarts are presented in Section Stroke set Online andwriting consists of a set of strokes. Eac stroke is, typically, a sequence of point coordinates sampled from te moment a writing device (suc as a stylus) touces te screen up to te moment it is released. We assume tat eac stroke belongs to only one symbol (tis assumption is common wen dealing wit andwritten grapics). Oterwise, a preprocessing step could be applied to split a stroke tat is part of two or more symbols. Tese concepts are illustrated in Figures 3a and 3b (a) 4 5 Fig. 3. Handwritten expressions representing x n. Eac expression n is composed of a set of strokes, were eac stroke is a sequence of bidimensional coordinates (dots in gray). In (a), stroke 5 belongs to two symbols. In (b), eac stroke belongs to only one symbol. 3.2 Grap grammar model A grap grammar [28] defines a language of graps. We denote a grap G as a pair (V G, E G ), were V G represents te set of vertices of G and E G represents te set of edges of G. A labeled grap is a grap wit labels in its vertices and edges. Hereafter we assume labeled graps, wit labels (b) 4 5 6

5 SUBMITTED ON SEPTEMBER defined by a function l tat assigns symbol labels (in a set SL) to vertices and relation labels (in a set RL) to edges. We define a family of grap grammars, called Grapic grammars, to model grapics as labeled graps. Definition 1. A grapic grammar is a tuple M = (N, T, I, R) were: N is a set of non-terminal nodes (or non-terminals); T is a set of terminal nodes (or terminals), suc tat N T = (for convenience we denote elements in T using te same names used for te labels in SL); I is a non-terminal, called initial node; R is a set of production (or rewriting) rules of te form A := B were A is a non-terminal node and B = (V B, E B ) is a connected grap wit label l(v) N T for eac v V B, and label l(e) RL for eac e E B. TRM sp TRM sp TRM ME OP OP OP ME TRM OP TRM := TRM TRM TRM sp := TRM TRM TRM sp := sp TRM Note tat M is a context-free grap grammar [28]. Te language defined by a grapic grammar M = (N, T, I, R) is a (possibly infinite) set of connected labeled graps and is denoted L(M). Similarly to string grammars, a labeled grap G belongs to L(M) if G can be derived (or generated) from te initial non-terminal node I by successively applying production rules in R, until obtaining a grap wit only terminal nodes. Figure 4 sows a grapic grammar tat models simple aritmetic and logical expressions. Eac production rule defines te replacement of a non-terminal, a single vertex grap G l at te left and side (LHS) of te rule, wit a grap G r at te rigt and side (RHS). sp TRM a sp b + + OP + := sp TRM Fig. 5. Generation of a grap tat represents te expression a b + c d. At eac rule application, te replacing grap nodes are depicted in dark gray. Edges tat link te replacing grap wit te ost grap are depicted wit dased arrows. Rule applications after te fourt one are not sown. c sp d a same production rule, and te graps generated for eac embedding. C Fig. 4. Grap grammar tat models basic matematical expressions. Te grammar is defined by non-terminals N = {ME, T RM, OP, }, relation labels RL = {sp, sb, }, terminals T = {+,, <, >, a,..., z, A,..., Z, 0,..., 9}, rules R = {r 1,..., r 73}, and ME at te left and side grap of rule r 1 is te initial node. Abbreviations: ME = matematical expression, sp = superscript, sb = subscript, = orizontal, T RM = term, OP = operator, = caracter. Figure 5 sows a grap generation process using te grammar of Figure 4. Rules are applied sequentially, starting wit non-terminal M E, until all elements in te generated grap are terminals. Dased arrows correspond to edges tat link te replacing graps wit te ost grap. Te definition of ow a replacing grap sould be linked to a ost grap G is called embedding [28], and it sould be specified for eac production rule. Formally, given a production rule G l := G r, its application consists in replacing a subgrap G l of G wit G r and te embedding defines ow G r will be attaced to G\G l. Te attacment may be defined by a set of edges tat link te replacing grap G r to G \ G l. For instance, Figure 6 sows two different embeddings for ɛ(r) = {(V, D) (V, A) G} {(D, V ) (A, V ) G} y C z B x D w E B y x r: z A A D E w := ɛ(r) = {(V, E) (V, A) G} {(E, V ) (A, V ) G} y C z B x E w D Fig. 6. Grap transformation wit two different embeddings. Te top grap is transformed troug rule r. Eac embedding defines edges between vertices tat are linked to vertex A of te top grap wit vertex D (left and side embedding) or E (rigt and side embedding) of te replacing grap. Dased arrows represent te edges defined by eac embedding. Te embedding specification depends on te desired language. It is possible to define a same embedding specification for all rules, as we do for matematical expressions (see Section 4). An embedding can also take spatial

6 SUBMITTED ON SEPTEMBER information into consideration, for example by including edges only between spatially close vertices. More detailed examples of embeddings are provided in Section 4, troug applications to te recognition of matematical expressions and flowcarts. To ensure tat te generated graps are connected, we assume tat every embedding is specified in suc a way tat its application generates connected graps. 3.3 Hypoteses grap generation Given a set of strokes S, we define a ypoteses grap as an attributed grap H = (V H, E H ), were V H is a set of symbol ypoteses and E H is a set of relation ypoteses computed from S. Eac symbol ypotesis v V H corresponds to a subset of S, denoted as stk(v), and as as an attribute a list L(v) = {(l i, s i ), i = 1,..., k v } of likely interpretations. Eac of tese interpretations (l i, s i ) consists of a symbol label l i SL and its respective likeliood score s i [0, 1]. Note tat a stroke may be sared by multiple symbol ypoteses. Relation ypoteses (edges in E H ) are defined over pairs of disjoint symbol ypoteses (i.e., ypoteses suc tat teir stroke sets are disjoint), and also ave as an attribute a list of likely relation interpretations denoted L(e). Relation labels are in RL. Figure 7 sows a andwritten matematical expression and a ypoteses grap calculated from it. Fig. 7. Hypoteses grap example. Vertices represent symbol ypoteses and edges represent relations between symbols. Te labels associated to symbols and relations indicate teir most likely interpretations. To build a ypoteses grap, macine learning metods are effective in identifying groups of strokes tat may form symbols and, similarly, relations among tem (see application example in Section 4). Since many stroke groups do not correspond to an actual symbol and many pairs of symbols are not directly related eac oter witin a grapic, rater tan training classifiers to identify only true ypoteses, tose tat do not represent any symbol or relation can be included as elements of an additional class, called junk. Training data can be extracted from witin te grapic, togeter wit surrounding context, in order to improve rejection of false ypoteses. As will become clear later, ypoteses graps play an important role to constrain te searc space during te parsing process. A ig precision and recall in te identification of symbol ypoteses and relations is tus desirable to efficiently constrain te searc space Label list pruning To define te labels and respective likeliood scores of symbol and relation ypoteses, we could use te confidence scores returned by te respective classifiers. However, to manage complexity, only class labels tat present ig confidence scores sould be kept. Selecting te labels to be kept based on a fixed global confidence tresold value is not adequate since label distributions vary greatly among symbols and relations. An effective metod to select te most likely labels for eac ypotesis is described next. Let {(l i, s i ), i = 1,..., n } be te pairs of labels and respective scores initially attributed to, sorted in descending order according to te likeliood scores s i. Ten, given a distribution tresold tr (between 0 an 1), we define te minimum number of k top ranked labels wose confidences sum up to at least tr: k = arg min x x s i > tr (1) i=1 Hypotesis is rejected if it presents igest score for te junk class label and if tat score is above te tresold tr. Oterwise, we set L() = {(l i, s i ) : i = 1,..., k}. We define label pruning tresolds t symb for symbols and t rel for relations. 3.4 Grap parsing Te goal of te parsing process is to build a parsing tree tat explains te set of input strokes S input, according to a grammar. Since tere migt be more tan one interpretation, multiple trees migt be generated, possibly saring subtrees eac oter. Tus, tey will be stored in a parse forest. Figure 8 sows a parse forest calculated from te ypoteses grap of Figure 7, using te grap grammar of Figure 4. As can be seen, te root node (top of te figure) corresponds to te starting non-terminal M E. Two brances are generated from rules associated to ME. Te left branc is generated by applying rule 2 and te rigt branc by applying rule 1. Note tat, for eac rule, any of te resulting partition of te strokes induces a grap tat is isomorpic to te RHS grap of te respective rule. Te same principle olds for te remaining of te nodes. Te parsing process follows a top-down approac. To understand te parsing process, a key step is to understand ow a stroke set is partitioned wen a rule is applied. More specifically, given a set of strokes S and a non-terminal NT, for eac rule A := B associated to NT, we must find every partition of S tat is a valid matcing to B. A partition of S is a matcing to B if its number of parts is equal to te number of vertices of B, so tat eac part can be assigned to one vertex in B. A matcing is valid if te following two conditions old: (1) te partition of S induces a grap tat is isomorpic to B, and (2) eac subset of strokes assigned to a vertex of B must be parsable according to te grammar. Supposing te number of vertices in B is k and te number of strokes in S is n, witout any constraint, te total number of possible stroke partitions to be examined to generate te valid matcings would be O(k n ). Exaustively examining eac of tese partitions is not computationally practical. A main strategy of our metod is to constrain te number of partitions to be examined wit te aid of te ypoteses grap. We assume tat all meaningful interpretations are present in te ypoteses grap as a subgrap. Tus, before

7 SUBMITTED ON SEPTEMBER Algoritm 1 : parsegrapic(s,nt ) Parses a set of strokes S from a non-terminal NT Input: (S, NT ) Output: parsedg = {(G 1, r 1 ),..., (G q, r q )} 1: parsedg 2: if parsed[(s, NT )] ten 3: parsedg T BL[(S, NT )] 4: else 5: for all rule in ruleswitlhs(nt ) do 6: if rule is A b ten 7: if l(b) L(S) ten 8: G buildgrap(s, l(b)) 9: parsedg parsedg {(G, rule)} 10: end if 11: else 12: for all G in validmatcinginstances(s, B = RHS(rule)) do 13: if v V G, parsegrapic(stk(v), l(v)) ten 14: parsedg parsedg {(G, rule)} 15: end if 16: end for 17: end if 18: end for 19: T BL[(S, NT )] parsedg 20: parsed[(s, NT )] T rue 21: end if 22: return parsedg Fig. 8. A parse forest representing multiple interpretations of a matematical expression. Labels on arrows refer to grammar rules of Figure 4. Red arrows represent a parse tree tat corresponds to te interpretation P b 4. starting te parsing process, we build te set of all stroke groups, denoted ereafter as ST K, underlying any valid connected subgrap of H. Note tat tese stroke groups must not contain repeated strokes, i.e., a valid subgrap is one in wic a same stroke is not present twice. Furtermore, not all stroke groups will be necessarily parsable. Te relation between two stroke groups is also recorded in STK as being te same between te corresponding subgraps. Hence, during parsing, te searc space of valid matcings will be restricted to tose present in ST K. Once a valid matcing is found, an instance of B, wic we call instantiated grap, will be recursively parsed and will become a parsed grap wen eac of its vertices is successfully parsed. Te complete algoritm is described next. For te sake of simplification, we will assume tat te input grammar contains only two types of rules: terminal and non-terminal. Terminal rules are productions of te form A := b, were te RHS grap b is a single vertex grap, wit labels in te terminal set, suc as rules from r-7 to r-73 of te grammar of Figure 4. Non-terminal rules are productions of te form A := B, were B is a grap containing one or more vertices, eac of tem wit non-terminal labels, suc as rules r-1 to r-6 of te grammar of Figure 4. Tus, Algoritm 1 considers only tese two types of rules. Its extension to treat rules tat contain bot terminals and non-terminals in its rigt-and side is a straigtforward combination of te previous two cases. Algoritm 1 receives as inputs a stroke set S = {stk 1,..., stk n } and a non-terminal NT. Initially, te set of strokes is te wole input set S input and te non-terminal is te starting node I. Ten, it applies eac of te production rules tat ave NT as te LHS grap and returns a set (parsedg) containing all parsed graps, togeter wit te respective rules tat generated tem. To avoid recomputation, a global table T BL indexed by pairs (S = {stk 1,..., stk n }, NT ) is used. An entry in TBL is of te form T BL[(S, NT )] = {(G 1, r 1 ),..., (G q, r q )} were G i is a parsed grap and r i is te rule tat generated G i. At te end of te algoritm, if te pair (S, NT ) is not parsable, te corresponding entry in T BL is empty. Lines 2-3 verify if te pair (S, NT ) as already been processed. If so, results are retrieved from T BL and returned. Oterwise, lines 5-18 iterate over te rules tat ave NT in its LHS grap. If te rule is of terminal type (lines 6-10), it suffices to ceck if te RHS vertex label, l(b), is contained in te set of labels L(S) attributed to te underlying stroke set. Tis verification is done by cecking if te stroke set S corresponds to a vertex in te ypoteses grap and if te label set of te corresponding 1 vertex includes l(b). Ten a single vertex grap is built and stored togeter wit te rule in parsedg. If te rule is of non-terminal type (lines 11-17), for eac valid matcing between S and B (line 12) we verify if te instantiated grap is parsable. Te parsing result, eiter a list of parsed graps, or an empty list (in case of parsing failure), is added to T BL. As already mentioned, table T BL is used to avoid parsing recomputation of pairs (S, NT ). At te end of te parsing process, te parse forest can be extracted from T BL by traversing it starting from index (S input, I) Pruning strategies Besides constraining te partitions to be examined to only tose formed by stroke groups tat underlie a subgrap of H, tere are oter strategies tat can be used to speed up computation. For example, determining te maximum and minimum size of non-terminal nodes is a strategy tat as been previously used in text parsing [29]. Te sizes, in terms of grapic symbols or strokes, can be computed directly from te grammar. Based on tese numbers, during parsing any stroke subsets tat are out of te min-max ranges do not need to be evaluated. Tis information can be calculated wen building ST K. Moreover, to find valid

8 SUBMITTED ON SEPTEMBER matcing partitions, te minimum and maximum sizes of te stroke subsets already matced to some vertices can be used to determine te minimum and maximum size of te stroke groups tat still can be matced to te rest of te nodes. Anoter useful information is to explore te knowledge tat a non-terminal can generate only a specific subgroup of te terminals. For instance, in te grammar of Figure 4, non-terminal OP can generate only symbols +,, <, or >. Tus, stroke subsets tat do not contain any ypotesis wit one of suc labels as terminals are not evaluated during te parsing process. Analogously, stroke groups tat correspond to symbol and relation ypoteses wit ig mean junk score can be disregarded. Specifically, stroke subsets wit a certain number (five, for example) symbol ypoteses, aving mean junk score, including bot symbol and relation labels, above a given junk tresold t junk will not be considered. Tis pruning is mainly useful wen te symbol and relation ypoteses ave a large number of labels. Hig mean junk score indicates tat it is unlikely tat te underlying group of strokes is parsable. 3.5 Optimal parse tree extraction Once a parse forest is built, te final step consists in traversing it to extract te best tree (interpretation). To caracterize wat is an optimal tree (best interpretation), we first define a cost function for trees. Rougly stating, an interpretation will be considered of low cost if its corresponding parse tree includes substructures wit ig confidence scores. We introduce a few notations tat will be elpful. Let x denote a node in te parse forest. Let G x = (V x, E x ) be te grap instantiated at node x. Eac vertex v V x as an underlying set of strokes, stk(v). For eac terminal vertex v V x tere will be a pair (label(v), score(v)) SL [0, 1] and for eac edge e E x will be a pair (label(e), score(e)) RL [0, 1]. Te cost of a tree can be computed bottom-up. We first define individual costs relative to symbols and relations, and ten define ow to combine te two to determine te cost of a tree. Let t be a parse tree and let x be a node in t. Let cild(x) denote te cild nodes of x. Te subtree in t wit root at x is denoted t x. We first assign to a node x a symbol cost J s (x): J s (x) = log score(v), y cild(x) and a relation cost J r (x): J r (x) = J s (y) e E x log score(e) + Ten, te cost of t x is defined as if x is terminal, wit V x = {v}, if x is non-terminal, y cild(x) (2) J r (y) (3) J(t x ) = α n s J s (x) + 1 α n r J r (x) (4) were n s and n r are, respectively, te number of symbols and relations under t x. Parameter α weigts bot types of costs, and could be adjusted to give more relevance to one or to te oter. An example of a tree is sown in Figure 9. Its root node is x 1 and tus te tree is denoted t x1. Te cost of tree t x1 is given in Eq. 5. x 3 x 4 x 5 v 5 TRM v 7 v 8 P e 3(sp) x 6 x 2 v 6 v 9 b v 2 T RM x 1 e 1() v 1 ME v 3 OP e 2() v 10 x 7 x < 8 v 4 T RM v 11 v 12 x 9 1 Fig. 9. Parse tree of expression P b < 1, extracted from te parse forest of Figure 8. Nodes are indexed as x i, i = 1,..., 9. Similarly, vertices and edges of te instantiated graps are respectively indexed as v j, for j = 1,..., 12, and e k, for k = 1,..., 3. Nodes wit terminal symbols are depicted wit double line borders. J(t x1 ) = α ( ) J s (v 8 ) + J s (v 9 ) + J s (v 10 ) + J s (v 12 ) + ( 4 ) 1 α ( J r (e 1 ) + J r (e 2 ) + J r (e 3 )) (5) 3 In order to extract te best tree, te cost of eac tree in te parse forest must be computed. Since te trees in te parse forest sare subtrees, tis fact can be explored to avoid computing te cost of a sared subtree repeatedly. In addition, from an application point of view, being able to efficiently retrieve a number of best parse trees rater tan just te best one is often desirable. We borrow ideas from te tree extraction tecnique, in te context of string grammars, proposed by Boullier et al. [30]. Given a parse forest, tey proposed a metod tat builds a new parse forest wit a fixed number of n-best trees, using a bottomup approac. Te resulting parse forest can be furter processed to improve te recognition result, for example, by doing a re-ranking of te trees, a processing tat could be too expensive to be done in te original parse forest. Note tat tere migt be multiple subtrees wit root at a node x in te parse forest. For instance, in te parse forest of Fig 8, te vertex in te bottom left non-terminal node grap as two possible derivations ( P or p ). Wenever tere are multiple derivations from a non-terminal vertex, only one of tem will be present in a parse tree. Tus, given a node x in te parse forest, let us denote by t (i) x, i I x, te spanned trees from x. Te number of possible trees in te forest is combinatorial wit respect to te multiple subtrees spanned from te nodes in a pat from te root node to a leaf node.

9 SUBMITTED ON SEPTEMBER We use a bottom-up approac to compute, for eac node x in te forest, a list of subtrees spanned from it. Tis information is kept as a table in te node, and eac row of te table stores information to recover one of te spanned trees (specifically, it stores te partition of te stroke set resulting from te corresponding derivation). After te bottom-up process finises, individual trees can be extracted by performing a top-down traversal, starting from eac row of te table at te root node of te forest. Te best tree, according to te specified cost, is te one recovered by starting te traversal from te first row of te table. However, since tere migt be a large number of parse trees in te forest, a naive application of te metod described above may be computationally proibitive. To overcome tis problem, a pruning strategy can be applied during te bottom-up step to keep table sizes manageable: for eac table, spanned trees tat ave a cost muc iger tan te best tree are discarded. To compute relative differences of cost, let minj(x) be te minimum cost tree spanned from x. Ten, given t pr [0, 1], a spanned tree t (i) x is kept if J(t (i) x ) minj(x) < t pr minj(x). (6) Tis strategy resembles te one proposed in [30], but it differs in te sense tat wile tey keep a fixed number of best trees, we keep only te relatively likely ones. Te more ambiguous te input, te more parse trees are kept. Te pruning tresold t pr can be empirically estimated. 4 APPLICATIONS Te application of te framework requires te definition of some key elements. First, a grap grammar tat models te family of grapics to be recognized must be defined. A set of labels for te relations (RL) and for te symbols (SL), including junk, must be defined. Second, a ypoteses grap generated from te set of input strokes, wit symbol labels in SL and relation labels in RL, must be provided. Terminal nodes of te grammar are named using te labels in SL, wile edges in te graps of te grammar are labeled using labels in RL. For parsing, an embedding metod must be defined for eac grammar rule. In tis section, we detail ow tese elements as well as important parameter values ave been defined for te recognition of matematical expressions and flowcarts. Results and discussions are presented in te next section. Te grammars in xml format are available at frank.aguilar/grammars/. Before applying te recognition metod itself, we applied to te set of strokes te smooting and resampling metods described in [31]. Smooting removes abrupt trajectory canges in te strokes and resampling makes point distribution uniform equally spaced along te strokes. In te evaluating datasets, eac stroke belongs to only one symbol; tus no additional preprocessing was needed. 4.1 Recognition of matematical expressions Dataset and Grammar We use te CROHME-2014 dataset [32]. It consists of andwritten expressions divided into training and test sets, wit 9, 507 and 986 expressions, respectively. Te expressions include 101 symbol classes, and six relation classes (orizontal as in ab, above as in x, below as in, superscript as in a b, subscript as in a b, and inside as in x ). CROHME-2014 dataset provides a string grammar for te corresponding L A TEX expressions. Based on tat string grammar, we defined a grap grammar wit 205 production rules, including te rules to generate te 101 symbol labels (terminals). To define te embeddings, we use te concept of baseline. A baseline in a grap is defined as a maximal pat wose connecting edges ave only te orizontal () label (tis definition can be seen as a grap version of te baseline definition of [20]). A baseline is considered nested to a vertex v if it is connected to v by an edge (v, v ), were v is te first vertex of te baseline. A baseline tat is nested to no vertex is called dominant baseline. Note tat a baseline may consist of a single vertex. Ten, te embedding is defined as follows. Let r : G l := G r be a rule and let v be te leftmost and v be te rigtmost vertices of te dominant baseline of G r. Let also G be a grap wit an occurrence of G l, identified as a vertex u V G. Te embedding associated to te application of rule r on G replaces u wit G r, generating an updated grap G, suc tat V G = V G \ {u} V Gr and E G = [E G \ ({(u, u) : u V G } {(u, u ) : u V G })] ɛ were ɛ = {(u, v ) : (u, u) E G } {(v, u ) : (u, u ) E G }. (7) In oter words, all edges tat were incident on u will be made incident to v and all edges tat were originated from u will be made originating from v Hypoteses grap building To generate te ypoteses grap, we used te symbol segmentation and classification metods described in [33], [34], along wit te spatial relation classification metods described in [35]. Tey are based on multilayer neural networks wit sape context descriptor [36], and images created from symbols and relations, including neigboring strokes to be used as contextual information. Te networks use a softmax output wic is ten converted to a cost measure (applying te negative logaritm to te output) in order to be used in te cost function defined in Eq. 4. An important parameter to build te ypoteses grap is te symbol and relation label pruning tresolds, t symb and t rel (see Eq. 1). Tese tresold values determine ow many and wic labels will be attaced to eac vertex and edge. Since during te parsing process te partitions of te stroke set and labels are constrained by te ypoteses grap, te acievable maximum recognition rates are bounded by possibilities encoded in te ypoteses grap. From te training set, we randomly selected 950 expressions (about 10%) to serve as a validation set and used te rest for training. Using te trained symbol and relation classifiers, we evaluated te effect of varying values of t symb and t rel on te validation set. For eac tresold value we computed te symbol, relation and complete expression recalls, tat is, ow many of eac of tese components were present in te ypoteses grap. Figure 10 sows te results relative to tis evaluation, over t symb in te range [0.4 1] (for values less tan 0.4, x

10 SUBMITTED ON SEPTEMBER te performance was similar to te case of 0.4) and t rel in te range [0.1 1]. Note tat tis evaluation is concerned wit verifying ow many of te elements of interest are, in fact, present in te ypoteses grap; it is not related wit parsing. Recall (%) Symbol (below) and rela>on (above) classifier tresolds Symbol Rela>on Expression Fig. 10. Symbol, relation and expression level recall of te ypoteses grap generation step. For eac symbol classification tresold t symb in te range [ ], relation classification tresold t rel is varied in te range [ ]. We can see in Figure 10 tat even for te lowest tresold values te recall of symbols and relation is about 90%. For complete expressions (i.e. all symbols and relations of te expressions are in te ypotesis grap), owever, te recall for te lowest tresold values is 40%. If symbol classification tresold is set to 1, 99, 75% of te symbols are correctly included. Since in tis case no stroke group is rejected, 99, 75% is also te percentage of symbols identified by te stroke grouping metod. If, in addition, we also set te relation classification tresold to 1, almost all relations and expressions are included (99, 45% and 98.11%, respectively) Grap parsing and tree extraction We also analyzed te effect of different values of t symb and t rel on te recall after parsing. We set te maximum value for t symb to 8 and for t rel to 0.85, as parsing large expressions wit tresolds larger tan tose takes muc time to be considered in a real application. In tis evaluation, for optimal tree extraction we set α = (same weigt for te symbol and relation costs, see Eq. 4) and t pr = 0.1 (tree pruning tresold, see Eq. 6). Figure 11 sows te expression recall obtained by te parsing metod and te corresponding recall obtained by te ypoteses grap generation step (note tat te second indicates te maximum acievable recall). Altoug for values above t symb = and t rel = 0.8 no considerable improvements are observed in te parsing recall, te gap between ypoteses grap recall and parsing recall increases up to about 40%. Tus, we cose t symb = 8 and t rel = 0.85, as tese values allow to keep more ypoteses and can be useful during parsing of unseen expressions (better generalization). Using t symb = 8 and t rel = 0.85, we ave also evaluated te effect of different values of t pr (tree pruning tresold) and α (weigting in te cost function) on tree extraction on validation set. Troug tis evaluation, we set t pr = 0.1 and α = 0.4 (tis coice was based on te best expression recall). Expresion recall (%) Symbol (below) and relabon (above) classifier tresolds Grap Parsing Hypoteses grap Fig. 11. Expression recall obtained at grap parsing and ypoteses grap generation steps, for different symbol and relation tresolds. 4.2 Recognition of flowcarts Dataset and grammar We use te flowcart dataset described in [37]. Te dataset includes 7 symbol classes (arrow, connection, data, decision, process, terminator, and text), and tree relation classes (Src, Targ, and AssTxt). An example was presented in Section 2 (Fig. 2), wit strokes colored according to te symbol type tey belong to. In tis dataset, relations in eac flowcart are establised between adjacent symbols. For instance, in te flowcart of Figure 2, Src and Targ relations are defined between te top arrow and a terminal and data, respectively. In te same way, an AssTxt relation is defined between te top terminal and te text inside it. Te flowcarts ave been written by 36 people, and te dataset is divided into a train set wit 248 and a test set wit 171 flowcarts. Te total number of symbols is about 9, 000. As described in Section 2, text symbols ave different caracteristics tan oter flowcart symbols, and tey are usually recognized troug specific metods. Since flowcart recognition is addressed in tis work wit te aim of illustrating te application of te proposed framework, we are not specially concerned wit recognition performance. Tus, we ave opted on removing strokes corresponding to text symbols, as well as te respective relations (AssTxt) from te flowcarts. Symbol class text and relation class AssTxt were not considered. We note, owever, tat it would be equally possible to parse te integral flowcart witout any canges in te parsing and tree extraction steps once adequate symbol and relation classifiers are developed for texts. In contrast to te CROHME-2014 dataset, we found no grammar defined for te flowcart dataset. Tus, we defined a grammar wit 16 production rules, were six of tem generate te terminal symbols. Te embedding is defined in a similar way to te one defined for matematical expressions, except for te set of edges to be added. Let u denote te vertex to be replaced in G and v i V Gr te vertices in te replacing grap. Te edges to be added are defined by: ɛ ={(u, v) (u, u) E G and v = arg min cost r (u, v i )} v i {(v, u ) (u, u ) E G and v = arg min cost r (v i, u )} v i were cost r (u, v) is te minimum relation cost among relations between a symbol ypotesis under u and a symbol ypotesis under v

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