Original: Structured Shape-Patterns from a Sketch: A Multi-Scale Approach
Revised: Structured Shape-Patterns from a Sketch: A Multi-Scale Approach

Original: Submission Number:

Original: A BSTRACT
Revised: A BSTRACT

Original: Structured 2D patterns formed by the anisotropic distribution of ar- bitrary shapes are ubiquitous in nature and man-made environments.
Revised: Structured 2D patterns formed by the anisotropic distribution of arbitrary shapes are ubiquitous in nature and man-made environments.

Original: They may include both bounded and unbounded (extended fiber- like) shapes.
Revised: They may include both bounded and unbounded (extended fiber-like) shapes.

Original: In this work, we address the problem of interactively generating such patterns from a single exemplar, sketched by a user.
Revised: In this work, we address the problem of interactively generating such patterns from a single exemplar sketched by a user.

Original: We build our solution on a new data structure, the Support Structure Hierarchy , computed from a multi-resolution analysis of the input exemplar, that encodes the main anisotropy directions at different scales as well as deviations from them.
Revised: We build our solution on a new data structure, the Support Structure Hierarchy , computed from a multi-resolution analysis of the input exemplar, that encodes the main anisotropy directions at different scales as well as deviations from them.

Original: We propose an efficient method based on this structure to synthesize a similar distribution of shapes in an extended 2D domain.
Revised: We propose an efficient method based on this structure to synthesize a similar distribution of shapes in an extended 2D domain.

Original: The user may also choose to hybridize several input exemplars, by combining structural shapes extracted at different scales.
Revised: The user can also choose to hybridize multiple input exemplars by combining structural shapes extracted at different scales.

Original: As shown through a user study, our multi-scale solution generates structured shape-patterns that perceptually compete with state-of-the-art methods, were they from learning-based or not.
Revised: As shown in a user study, our multi-scale solution generates structured shape-patterns that perceptually compete with state-of-the-art methods, whether learning-based or not.

Original: Moreover, our interactive solution, which does not require any precomputation, matches well  the needs of an interactive authoring tool, where the user can not only sketch and extend 2D vector textures but also combine them seamlessly.
Revised: Moreover, our interactive solution, which requires no pre-calculation, fits well with the needs of an interactive authoring tool, where the user can not only sketch and extend 2D vector textures but also combine them seamlessly.

Original: 1 I NTRODUCTION
Revised: 1 I NTRODUCTION

Original: From fibers and cellular organisms at microscopic scales to sea- weeds, schools of fishes, human queues and alignments of trees or buildings at a larger scale anisotropic distributions of shapes are ubiquitous in nature and man-made environments.
Revised: From fibers and cellular organisms at microscopic scales to sea- weeds, schools of fishes, human queues, and alignments of trees or buildings at a larger scale anisotropic distributions of shapes are ubiquitous in nature and man-made environments.

Original: Moreover, such structured shape-patterns have been used extensively in 2D for decorative purposes, from mosaics and wallpapers to the distribution of windows and architectural decorations on building facades.
Revised: Moreover, such structured shape-patterns have been used extensively in 2D for decorative purposes, from mosaics and wallpapers to the distribution of windows and architectural decorations on building facades.

Original: The perceived structure emerges from the anisotropy of these distributions of shapes.
Revised: The perceived structure emerges from the anisotropy of these shape distributions.

Original: In particular, the specific ranges and variances of perceived orientations, both in terms of salient shapes and alignments, convey their unique visual appearance.
Revised: In particular, the specific ranges and variances of perceived orientations, both in terms of salient shapes and alignments, convey their unique visual appearance.

Original: This work explores the synthesis of such structured 2D shape-patterns from a sketch.
Revised: This work explores the synthesis of such structured 2D shape-patterns from a sketch.

Original: Example-based texture synthesis has already been extensively studied.
Revised: Example-based texture synthesis has already been extensively studied.

Original: However, existing methods have mostly focused on point distributions.
Revised: However, existing methods have mostly focused on point distributions.

Original: They have achieved statistical accuracy using noise models, continuous representations of discrete distributions such as pair-correlation or probability-density functions, or neighbor- hood metrics and energy optimization.
Revised: They have achieved statistical accuracy using noise models, continuous representations of discrete distributions such as pair-correlation or probability-density functions, or neighborhood metrics and energy optimization.

Original: The few methods tackling anisotropic distributions of shapes have used multiple point sam- ples or proxy geometries to achieve the analysis and synthesis of structured patterns.
Revised: The few methods that address anisotropic distributions of shapes have used multiple point samples or proxy geometries to perform structured pattern analysis and synthesis.

Original: To the best of our knowledge, none of them have tackled the case of anisotropic shape-patterns that may include both bounded and unbounded (fiber-like) shapes.
Revised: To the best of our knowledge, none of them have addressed the case of anisotropic shape-patterns that can include both bounded and unbounded (fiber-like) shapes.

Original: While the use of deep learning may be a promising alternative, it requires large training databases and long precomputation (learning) times, which has limited its use so far in interactive design scenarios.
Revised: While the use of deep learning may be a promising alternative, it requires large training databases and long precomputation (learning) times, which has limited its use, so far, in interactive design scenarios.

Original: This work tackles the interactive, sketch-based design of anisotropic distributions of shapes in 2D. Given any sketched pattern (a distribution of simple bounded shapes, and/or fiber-like unbounded shapes), our method efficiently synthesizes a perceptually similar, consistent, and non-repetitive distribution of shapes in an extended 2D domain.
Revised: This work tackles the interactive, sketch-based design of anisotropic distributions of shapes in 2D. Given any sketched pattern (a distribution of simple bounded shapes and/or fiber-like unbounded shapes), our method efficiently synthesizes a perceptually similar, consistent, and non-repetitive distribution of shapes in an extended 2D domain.

Original: Note that the input pattern is fully preserved at the synthesis stage.
Revised: Note that the input pattern is fully preserved at the synthesis stage.

Original: Indeed, contrary to previous methods, the input becomes the central part of the extended texture, while being seam- lessly integrated into its larger surrounding.
Revised: Indeed, contrary to previous methods, the input becomes the central part of the extended texture while being seamlessly integrated into its larger surrounding.

Original: Our solution increases user control and also seems to improve the perceived similarity of the results.
Revised: Our solution increases user control and also seems to improve the perceived similarity of the results.

Original: One such result generated by our interactive system is shown in Fig.
Revised: One such result generated by our interactive system is shown in Fig.

Original: 1 with its interface.
Revised: 1 with its interface.

Original: Real-time analysis and synthesis of distributions require an ef- ficient representation, encoding both local and global correlations between shapes.
Revised: Real-time analysis and synthesis of distributions require an ef- ficient representation, encoding both local and global correlations between shapes.

Original: Our first insight is to introduce a compact encoding for anisotropic distributions, called the Support Structure Hierarchy , where individual supporting structures are lead directions of align- ments or line skeletons computed from user strokes, all computed at various scales.
Revised: Our first insight is to introduce a compact encoding for anisotropic distributions, called the Support Structure Hierarchy , where individual supporting structures are lead directions of align- ments or line skeletons computed from user strokes, all computed at various scales.

Original: This representation leads to a particularly simple and efficient multi-scale analysis of the distributions of orientations in the input sketch.
Revised: This representation leads to a particularly simple and efficient multi-scale analysis of the distributions of orientations in the input sketch.

Original: It also enables efficient domain extension.
Revised: It also enables efficient domain extension.

Original: The main challenge at the synthesis stage remains to understand user expectations and the required criteria for perceptual similarity.
Revised: The main challenge at the synthesis stage remains to understand user expectations and the required criteria for perceptual similarity.

Original: The (new) case of fiber-like shapes is particularly challenging, be-
Revised: The (new) case of fiber-like shapes is particularly challenging be-

Original: cause extending fibers that are disjoint in the input exemplar may generate intersections in the extended domain.
Revised: cause extending fibers that are disjoint in the input exemplar may generate intersections in the extended domain.

Original: This could strongly affect our perception of the output as looking different from the input.
Revised: This could strongly affect our perception of the output as looking different from the input.

Original: To support our insights, we formulate a set of perceptual hypotheses to drive our synthesis solution; they were later validated through a user study.
Revised: To support our insights, we formulate a set of perceptual hypotheses to drive our synthesis solution; they were then validated through a user study.

Original: In particular, our solution interprets non-intersecting fiber-like strokes as curves that could slightly bend to prevent intersection in the extended domain.
Revised: In particular, our solution interprets non-intersecting fiber-like strokes as curves that could slightly bend to prevent inter- section in the extended domain.

Original: Thanks to its efficiency, we integrated our solution in an inter- active authoring tool, where users can progressively test and refine their designs.
Revised: Thanks to its efficiency, we integrated our solution into an inter- active authoring tool, where users can progressively test and refine their designs.

Original: They can generate a wider variety of vector textures by interactively hybridizing features extracted from several input exemplars, e.g., combining shapes from an exemplar with larger-scale alignments from another exemplar.
Revised: They can generate a wider variety of vector textures by interactively hybridizing features extracted from several input exemplars, e.g., combining shapes from an exemplar with larger-scale alignments from another exemplar.

Original: In summary the contributions of our work are threefold, as we introduce:
Revised: In summary, the contributions of our work are threefold, as we introduce:

Original: • a fine-to-coarse analysis method that hierarchically clusters user strokes into a Support Structure Hierarchy , based on a new ”perceived distance” between line-segments within a domain, depending on both their position and orientation; • a coarse-to-fine synthesis method that extends the pattern around the input examplar, based on the extracted hierarchy andon a set of perceptual hypotheses validated by a user study; • an interactive authoring tool, enabling both domain extension and hybridization of structured shape-patterns.
Revised: • a fine-to-coarse analysis method that hierarchically clusters user strokes into a Support Structure Hierarchy, based on a new ”perceived distance” between line segments within a domain, depending on both their position and orientation; • a coarse-to-fine synthesis method that extends the pattern around the input sample based on the extracted hierarchy and a set of perceptual hypotheses validated by a user study; • an interactive authoring tool, enabling both domain extension and hybridization of structured shape-patterns.

Original: 2 R ELATED WORK
Revised: 2 R ELATED WORK

Original: This work addresses  2D sketch-based synthesis of anisotropic dis- crete distributions.
Revised: This work addresses the 2D sketch-based synthesis of anisotropic discrete distributions.

Original: It is related to example-based synthesis that aims at generating an output that minimizes some statistical or percep- tual distance with the input, while avoiding artifacts such as salient repetitiveness.
Revised: It is related to example-based synthesis that aims at generating an output that minimizes some statistical or per- ceptual distance from the input while avoiding artifacts such as salient repetitiveness.

Original: We focus below on distributions of 2D shapes, i.e., vector textures formed by arrangements of discrete 2D shapes, and also discuss recent alternatives based on deep learning.
Revised: We focus below on distributions of 2D shapes, i.e., vector textures formed by arrangements of discrete 2D shapes, and also discuss recent alternatives based on deep learning.

Original: We refer the reader to [5,18] for  more general surveys.
Revised: We refer the reader to the surveys by Wei et al. [21] and Gieseke et al. [7] for a more general overview.

Original: Discrete vector textures (or shape-patterns) have been gen- erated by analyzingdistributions of the centroids of individual shapes, and then applying a two-stage synthesis for the new cen- troids, followed by the creation of the associated shapes.
Revised: Discrete vector textures (or shape-patterns) were generated by analyzing the distributions of individual shape centroids, then applying a two-step synthesis for the new centroids, followed by retrieving the associated shapes.

Original: The pioneer work of Barla et al.
Revised: The pioneering work of Barla et al.

Original: [1] aims at synthesizing stroke patterns.
Revised: [3] aims at synthesizing stroke patterns.

Original: Their method computes a Delaunay triangulation from the centroids to retrieve the input distribution connectivity.
Revised: Their method computes a Delaunay triangulation from the centroids to retrieve the input distribution connectivity.

Original: During synthesis, they rely on a Lloyd relaxation and some perturbation to generate a new set of points from which the shapes are recovered.
Revised: During synthesis, they rely on a Lloyd re- laxation and some perturbation to generate a new set of points from which the shapes are recovered.

Original: In the same mindset, Ijiri et al.
Revised: In the same mindset, Ijiri et al.

Original: [7] explore local growth processes before the relaxation process.
Revised: [9] explore local growth processes before the relaxation process.

Original: These two methods arehowever limited respec- tively to quasi-uniform distributions orto 1-ring neighborhoods.
Revised: However, these two methods are limited respectively to quasi-uniform distributions or 1-ring neighborhoods.

Original: To manage more general shape distributions, Hurtut et al.
Revised: To handle more general shape distributions, Hurtut et al.

Original: [6] define the input distribution as a combination of Gibbs point processes from which they generate a new arrangement using Monte-Carlo chains.
Revised: [8] define the input distribution as a com- bination of Gibbs point processes from which they generate a new arrangement using Monte-Carlo chains.

Original: However, allof these methods are unable to analyze and synthesize structured inputs.
Revised: However, all these methods, as well as the patch-based method of AlMeraj et al. [1], are unable to analyze and synthesize structured inputs.

Original: In particular, they cannot tackle anisotropic distri- butions of shapes such as elongated ones, nor analyze correlations between shapes, orientations, and spatial alignments.
Revised: In particular, they cannot handle anisotropic distributions of shapes such as elongated ones nor analyze correlations between shapes, orientations, and spatial alignments.

Original: Instead of using a single centroid point, Ma et al.
Revised: Instead of using a single centroid point, Ma et al.

Original: [12] characterize each input shape by several sample points.
Revised: [15] characterize each input shape by several sample points.

Original: They rely on a neighbor- hood metric and an energy optimization process to insert individual shapes in a predefined output domain.
Revised: They rely on a neighborhood metric and an energy optimization process to insert individual shapes in a predefined output domain.

Original: Their approach has been extended to dynamic textures [11], stroke auto-completion [19], and adapted to other texture workflows [2,8].
Revised: Their approach has been extended to dynamic textures [14], stroke auto-completion [22], and adapted to other texture workflows [4,10].

Original: While the use of multiple sample points has also been extended to  distribution synthesis of arbitrary shapes, these methods only address bounded shapes—as opposed to unbounded shapes—and require post-processing to avoid inter-penetrations at the synthesis stage, which precludes their use in real time.
Revised: While the use of multiple sample points has also been applied to the distribution synthesis of arbitrary shapes, these methods only address bounded shapes—as opposed to unbounded shapes—and require post-processing to avoid inter-penetrations at the synthesis stage, which precludes their use in real-time.

Original: Rather than sampling the input shapes, Landes et al.
Revised: Rather than sampling the input shapes, Landes et al.

Original: [9] propose to simplify them into proxy geometries.
Revised: [11] propose to simplify the input shapes into proxy geometries.

Original: They introduce a spatial relationship measure that takes into account space between pairs of shapes and their relative orientations.
Revised: They intro- duce a spatial relationship measure that takes into account space between pairs of shapes and their relative orientations.

Original: By extending the stochastic models to point distributions [13,20], their synthesis method success- fully maintains distributions of distances and relative orientations of shapes.
Revised: By extending the stochastic models to point distributions [16,23], their synthesis method successfully maintains distributions of distances and relative orientations of shapes.

Original: Although it handles anisotropic distributions, their method does notmeet our goals, as it does not offer real-time performance and is limited to distributions of bounded shapes.
Revised: Although it handles anisotropic distributions, their method does not offer real-time performance and is limited to distributions of bounded shapes.

Revised: However, we follow their proxy geometry’s idea when we first reduce our bounded shapes into support segments (or central points) and bounding boxes.

Original: In contrast, Roveri et al.
Revised: In contrast, Roveri et al.

Original: [15] present the first example-based distribution synthesis method applicable to both bounded and unbounded shapes.
Revised: [18] present the first example-based distri- bution synthesis method applicable to both bounded and unbounded shapes.

Original: Regardless of their dimension, shapes are decomposed into point samples that are encoded in a functional representation.
Revised: Regardless of their dimension, shapes are decomposed into point samples that are encoded in a functional representation.

Original: A similarity measure is defined in the associated functional space to quantify  similarity between input and output.
Revised: A similarity measure is defined in the associated functional space to quantify the similarity between input and output.

Original: Synthesis is achieved in a few minutes through neighborhood matching and energy optimization.
Revised: The synthesis is achieved in a few minutes through neighborhood matching and energy optimization.

Original: As most other neighborhood-based texture synthesis methods, their method requires input patterns with enough repetitions to avoid bad local minima in the optimization, which would distort the synthesized structures.
Revised: Like most other neighborhood-based texture synthesis methods, their method requires input patterns with enough repetitions to avoid bad local minima in the optimization, which would distort the synthesized structures.

Original: Moreover, contrary to our method, the use of a fixed neighborhood size prevents their method from capturing repetitive structures at different scales.
Revised: Moreover, contrary to our method, the use of a fixed neighborhood size prevents their method from capturing repetitive structures at different scales.

Original: Deep learning methods have recently been applied to texture synthesis [4, 10, 14, 16, 17, 21].
Revised: Deep learning methods have recently been applied to texture synthesis [6, 12, 17, 19, 20, 24].

Original: They show promising results for capturing, at least partially, local and global correlations present in an input exemplar.
Revised: They show promising results for capturing, at least partially, local and global correlations present in an input exemplar.

Original: In particular, Fish et al.
Revised: In particular, Fish et al.

Original: [4] enable sketch stylization via the transfer of geometric textural details from different images; this is related to our secondary goal of pattern hybridization.
Revised: [6] allow for sketch stylization via the transfer of geometric textural details from different images; this is related to our secondary goal of pattern hybridization.

Original: However, most of these frameworks are image-based and do not extend well to the discrete distributions of vector shapes, which is the scope of our work.
Revised: However, most of these frameworks are image-based and do not extend well to the discrete distributions of vector shapes, which is the scope of our work.

Original: The method of Tu et al.
Revised: The method of Tu et al.

Original: [17] is closest to our goal of handling vector shape distributions; it characterizes point patterns via a trained VGG network.
Revised: [20] is closest to our goal of handling vector shape distributions; it characterizes point patterns via a trained VGG network.

Original: Our method, based on a simpler but efficient analysis stage, has the advantage of requiring no time-consuming precomputations (which are inherent to deep- learning techniques), while achieving real-time processing of any newly-created input.
Revised: Our method, based on a simpler but efficient analysis stage, has the advantage of requiring tedious precomputation (which is inherent to deep-learning techniques) while achieving real-time processing of any newly-created input.

Original: 3 O VERVIEW 3.1 Hypotheses on Depiction & Perception
Revised: 3 O VERVIEW 3.1 Hypotheses on Depiction & Perception

Original: Extending a sketched pattern in a perceptually similar way requires making some hypotheses about user depiction and perception of the resulting pattern.
Revised: Extending a sketched pattern in a perceptually similar way requires making some hypotheses about the user’s depiction and perception of the resulting pattern.

Original: Our key hypothesis, common to most sketch- based modeling systems, is that users see their input as a general view of the distribution  they want to create.
Revised: Our key assumption, common to most sketch- based modeling systems, is that users see their input as a general view of the distribution 1 they want to create.

Original: Therefore,  the inputis supposed to include all the necessary information, in a perceptually representative way.
Revised: Therefore, we expect the input to include all the necessary information in a perceptually representative way.

Original: This led us to three design hypotheses:
Revised: This led us to three design hypotheses:

Original: H 1 : Groupings and alignments are meaningful: All alignments and groupings are intentional.
Revised: H 1 : Groupings and alignments are meaningful: All alignments and groupings are intentional.

Original: H 2 : Repetitiveness is explicit: All the shapes that a user wants to see repeated in the output, are repeated in the input.
Revised: H 2 : Repetitiveness is explicit: All the shapes that a user wants to see repeated in the output, are repeated in the input.

Original: H 3 : Non-overlapping shapes should remain disjoint: Shapes that do not overlap the input should not overlap in the output.
Revised: H 3 : Non-overlapping shapes should remain disjoint: Shapes that do not overlap the input should not overlap in the output.

Original: These three hypotheses are used as guidelines for our method at the design stage, and then validated by a user study (see Sect. 6).
Revised: These three hypotheses are used as guidelines for our method at the design stage, and then validated by a user study (see Sect. 6).

Original: 3.2 Creation and Preprocessing of an Input Sketch
Revised: 3.2 Creation and Preprocessing of an Input Sketch

Original: During a sketching session, the user successively draws strokes of any color in a square representing our 2D Input Space ( IS ).
Revised: During a sketching session, the user successively draws strokes of any color in a square representing our 2D Input Space ( IS ).

Original: See
Revised: See

Original: Fig. 1, left.
Revised: Fig. 1, left.

Original: Two different pensare provided to denote bounded and unbounded strokes.
Revised: We provide two different pens to denote bounded and unbounded strokes.

Original: The former are limited to the dimensions of IS , while the latter are interpreted as extending beyond the input domain, either in both directions if both extremities reach the border of the IS , or in a single direction (in case an unbounded stroke does not reach any border of IS , we add a segment to connect it to the closest border).
Revised: The first ones are limited to the dimensions of IS , while the second ones are interpreted as extending beyond the input domain, either in both directions if both extremities reach the border of the IS , or in a single direction (in case an unbounded stroke does not reach any border of IS , we add a segment to connect it to the closest border).

Original: The data stored for each stroke are a list of points, a color, a thickness, a type (bounded or unbounded), and a principal direction computed on the fly from the Principal Component Analysis (PCA) on the coordinates for all points of a stroke.
Revised: The data stored for each stroke are a list of points, a color, a thickness, a type (bounded or unbounded), and a principal direction computed on the fly from the Principal Component Analysis (PCA) on the coordinates for all points of a stroke.

Original: The user may sketch the input pattern in any order.
Revised: The user may sketch the input pattern in any order.

Original: As several stokes can be used to represent a shape, we provide an automatic clustering mechanism, presented next, to identify shapes at the beginning of the analysis stage.
Revised: As several stokes can be used to represent a shape, we provide an automatic clustering mechanism, presented next, to identify shapes at the beginning of the analysis stage.

Original: 3.3 Processing Pipeline
Revised: 3.3 Processing Pipeline

Original: Multi-scale analysis: Based on Hypothesis H 1, the  analysis stage consists in iteratively extracting a fine-to-coarse hierarchy of support structures (the Support Structure Hierarchy ) from the input strokes, according to alignments and multi-scale repetitions in the input (see Fig. 2).
Revised: Multi-scale analysis: In the same spirit as StrokeAggrega- tor [13], we rely on our perceptual hypotheses to reduce the input strokes into coarser structures. In particular, our analysis stage con- sists in iteratively extracting a fine-to-coarse hierarchy of support structures (the Support Structure Hierarchy) from the input strokes according to alignments and multi-scale repetitions in the input (see Fig. 2).

Original: We first cluster bounded strokes into shapes composed of one to several strokes, while each unbounded stokeis considered as an individual shape ( Level 0 ).
Revised: We first cluster bounded strokes into shapes composed of one to several strokes, and we consider each unbounded stoke as an individual shape ( Level 0 ).

Original: Note that colors are not used in the clustering, enabling the use of several different colors in a given shape.
Revised: Note that colors are not used in the clus- tering, enabling the use of several different colors in a given shape.

Original: Bounded shapes are then simplified either into a central point or a support segment depending on their degree of anisotropy (Fig. 2c).
Revised: Bounded shapes are then simplified either into a central point or a support segment depending on their degree of anisotropy (Fig. 2c).

Original: Central points and support segments are clus- tered according to both orientation and position to find alignments, and then grouped into fibers (Fig. 2d), forming the Level 1 of the Support Structure Hierarchy .
Revised: Central points and support segments are clustered according to both orientation and position to find alignments and then grouped into fibers (Fig. 2d), forming the Level 1 of the Support Structure Hierar- chy.

Original: Other fibers are directly extracted from the unbounded strokes (Fig. 2c’).
Revised: Other fibers are directly extracted from the unbounded strokes (Fig. 2c’).

Original: To capture repetitions at a larger scale, fibers of similar orientation are clustered into fiber medians ( Level 2 , Fig. 2f), which are finally grouped into lead directions ( Level 3 , Fig. 2g).
Revised: To capture repetitions at a larger scale, fibers of similar orientation are clustered into fiber medians ( Level 2 , Fig. 2f), which are finally grouped into lead directions ( Level 3 , Fig. 2g).

Original: During this hierarchical clustering and simplification process,  the input domain ISis progressively partitioned into a hierarchy of ribbons that express the variability of position of each substructure around its parent structure.
Revised: During this hierarchical clustering and simplification process, we progressively partition the input domain IS into a hierarchy of ribbons that express the variability of position of each substructure around its parent structure.

Original: This partitioningwill be used to allow an adequate degree of variability while avoiding unwanted overlaps at the synthesis stage.
Revised: We use this partitioning to allow an adequate de- gree of variability while avoiding unwanted overlaps at the synthesis stage.

Original: See Section 4 for details.
Revised: See Section 4 for details.

Original: Synthesis stage: Unlike most existing approaches, our method to synthesize distributions consists in directly replicating local and global correlations between the input shapes, encoded by our Sup- port Structure Hierarchy .
Revised: Synthesis stage: Unlike most existing approaches, our method to synthesize distributions consists in directly replicating local and global correlations between the input shapes, encoded by our Sup- port Structure Hierarchy.

Original: To avoid exact repetitions, this is done by instantiating each structure from top to bottom of the hierarchy while perturbing their positions within adequate allowed areas .
Revised: To avoid exact repetitions, this is done by instantiating each structure from top to bottom of the hierarchy while perturbing their positions within adequate allowed areas .

Original: These areasare computed to prevent overlaps between strokes belonging to the same lead direction and at a low cost since no further overlaps detection will be required.
Revised: We compute these areas to prevent overlaps between strokes belonging to the same lead direction and at a low cost since no further overlaps detection will be required.

Original: Structures at the top of the hierarchy are first extended to the user-selected larger 2D domain, defined as a radial extension of ratio k > 1 of IS .
Revised: The structures at the top of the hierarchy are first extended to the user-selected larger 2D domain, defined as a radial extension of ratio k > 1 of IS .

Original: The support hierarchy is then traversed top-down to the individual strokes.
Revised: The Support Hierarchy is then traversed from top to bottom and down to the individual strokes.

Original: At each level, the repetitive structures are repeated within the larger domain, in order to generate the extended structured pattern.
Revised: At each level, the repetitive structures are repeated within the larger domain to generate the extended structured pattern.

Original: This is done in accordance with our design guidelines: a shape thatonly appears  once in the input (such as the vertical see-weed in Fig. 1) will be extended at its extremities in case of an unbounded stroke, but will not be repeated (consistency with H 2).
Revised: This is done following our design guidelines: a shape that appears only once in the input (such as the vertical seaweed in Fig. 1) will be extended at its extremities in case of an unbounded stroke, but will not be repeated (consistency with H 2).

Original: Moreover, at each level of the hierarchy,  allowed areas within ribbons are used to guide the synthesis of substructures while preventing unwanted overlaps (consistency with H 3).
Revised: In addition, at each level of the hierarchy, the allowed areas within ribbons are used to guide the synthesis of substructures while preventing unwanted overlaps (consistency with H 3).

Original: Note that curving some of the supporting structures is necessary to avoid undesired overlaps in the extendedthe domain, as illustrated by the three green waves that do not overlap with the two blue waves in Fig.
Revised: Note that curving some of the supporting structures is necessary to avoid undesired overlaps in the extended domain, as illustrated by the three green waves that do not overlap with the two blue waves in Fig.

Original: 1, right.
Revised: 1, right.

Original: This process, an original step of our solution justified by our perceptual guidelines, will be detailed in Section 5.
Revised: This process, an original step of our solution justified by our perceptual guidelines, will be detailed in Section 5.

Original: Interactivity and hybridization Thanks to its real-time perfor- mance, our method not only allows users to sketch and extend a given shape-pattern, but also to return to the sketching interface to iteratively improve their input.
Revised: Interactivity and hybridization With its real-time performance, our method not only allows users to sketch and extend a given shapepattern, but also to return to the sketching interface to iteratively improve their input.

Original: In our authoring system, all identified shapes are recorded in a shapes database, enabling the user to refine the input by adjusting their position, or to reuse them later for another design.
Revised: In our authoring system, all identified shapes are recorded in a shapes database, allowing the user to refine the input by adjusting their position or to reuse them later for another design.

Original: Hierarchical structures extracted from the analysis stages of different inputs can also be combined to create a different design, a process called hybridization (see Section 6).
Revised: The hierarchical structures extracted from the analysis stages of different inputs can also be combined to create a different design, a process called hybridization (see Section 6).

Original: 4 F INE - TO -C OARSE A NALYSIS 4.1 Level 0: From Strokes to Shapes
Revised: 4 F INE - TO -C OARSE A NALYSIS 4.1 Level 0: From Strokes to Shapes

Original: As illustrated in Fig.
Revised: As illustrated in Fig.

Original: 2b,b’, the bounded and unbounded strokes in the input are analyzed separately, to extract supporting lines that will then be processed in a combined manner.
Revised: 2b,b’, the bounded and unbounded strokes in the input are analyzed separately to extract supporting lines that will then be processed in a combined manner.

Original: We consider the unbounded strokes as individual unbounded shapes.
Revised: We consider the unbounded strokes as individual unbounded shapes.

Original: In contrast, we extract  bounded shapes by clustering the input bounded strokes as follows: we compute the oriented bounding box of each bounded stroke and group these boxes according to their pair-wise distances.
Revised: In contrast, we extract the bounded shapes by grouping the input bounded strokes as follows. We compute the oriented bounding box of each bounded stroke and group these boxes according to their pairwise distances.

Original: We then associate the resulting bounded shapes to a single central point or support segment , according to an anisotropy threshold.
Revised: We then associate the resulting bounded shapes to a single central point or support segment, according to an anisotropy threshold.

Original: The resulting set of support segments and central points is the first simplification of the input, efficiently encoding the principal directions and approximate positions of the bounded shapes.
Revised: The resulting set of support segments and central points is the first simplification of the input, efficiently encoding the principal directions and approximate positions of the bounded shapes.

Original: 4.2 Level 1: From Shapes to Fibers and their Ribbons
Revised: 4.2 Level 1: From Shapes to Fibers and their Ribbons

Original: We approximate each unbounded shape with a line, called fiber , that best matches its principal direction and position.
Revised: We approximate each unbounded shape with a line, called fiber, that best matches its principal direction and position.

Original: This support line, augmented with a perpendicular thickness to cover the whole shape, is called a ribbon .
Revised: This support line, augmented with a perpendicular thickness to cover the whole shape, is called a ribbon.

Original: For bounded shapes, finding such fibers and ribbons requires analyzing anisotropic information such as alignments.
Revised: For bounded shapes, finding such fibers and ribbons requires analyzing anisotropic information such as alignments.

Original: We retrieve the support lines of support segment and cluster them using the Mean Shift algorithm.
Revised: We retrieve the support lines of support segment and cluster them using the Mean Shift algorithm.

Original: We then compute a central fiber within each cluster.
Revised: We then compute a central fiber within each cluster.

Original: Central points are first clustered by position, before using Principal Component Analysis to compute their main directions of alignment.
Revised: The central points are first clustered by position, before using Principal Component Analysis to compute their main directions of alignment.

Original: Representative fibers are defined from the centroid of each cluster and these principal directions.
Revised: Representative fibers are defined from the centroid of each cluster and these principal directions.

Original: Thicknesses are computed for each of these fibers , so that the corresponding ribbon fully covers the shapes associated with the clustered points or segments.
Revised: Thicknesses are computed for each of these fibers, so that the corresponding ribbon fully covers the shapes associated with the clustered points or segments.

Original: 4.3 Level 2: From Fibers to Fiber Medians
Revised: 4.3 Level 2: From Fibers to Fiber Medians

Original: Fibers with similar orientations and close positions are grouped at this stage.
Revised: At this stage, we group the fibers with similar orientations and close positions.

Original: Since we focus on anisotropic distributions, we prioritize  orientations of fibers over their positions.
Revised: Since we focus on anisotropic distributions, we prioritize the orientations of fibers over their positions.

Original: We first compute the histogram of fiber orientations to group those that belong to the same anisotropic distribution.
Revised: We first compute the histogram of fiber orientations to group those belonging to the same anisotropic distribution.

Original: We then refine each cluster using a specific perceived distance , which we define as the minimum distance between each fiber intersection points with the domain contour (see Fig. 3).
Revised: We then refine each cluster using a specific perceived distance , which we define as the minimum distance between each fiber intersection points with the domain contour (see Fig. 3).

Original: This distance takes into account both  position and  orientation of the lines: the more parallel and closer two lines are  in position, the smaller the distance.
Revised: This distance takes into account both the position and the orientation of the lines: the more parallel and closer two lines are to each other in position, the smaller the distance.

Original: Each resulting sub-clusteris stored as a fiber median defined as the mean of parameters both in orientation and in position of the clustered fibers (see Fig. 2f).
Revised: We store each resulting sub-cluster as a fiber median defined as the mean of parameters both in orientation and in position of the clustered fibers (see Fig. 2f).

Original: We also store the circular standard deviation associated to each fiber median for later use at the synthesis stage.
Revised: We also record the circular standard deviation associated to each fiber median for later use at the synthesis stage.

Original: Similarly to the previous hierarchy level, a thickness parameter is associated with each newly created fiber medianin order to define an associated ribbon that fully includes the sub-ribbons of the clustered substructures (see Fig. 4).
Revised: Similar to the previous hierarchy level, a thickness parameter is associated with each newly created fiber median to define an associated ribbon that fully includes the sub-ribbons of the clustered substructures (see Fig. 4).

Original: 4.4 Level 3: From Fiber Medians to Lead Directions
Revised: 4.4 Level 3: From Fiber Medians to Lead Directions

Original: The top level of aggregation in our hierarchical analysis aims at clustering similarly oriented fiber medians.
Revised: The top-level of aggregation in our hierarchical analysis aims to group similarly oriented fiber medians.

Original: We use the same clustering process asat the previous level of the hierarchy.
Revised: We use the same clustering process as the previous level of the hierarchy.

Original: Each clusteris represented by a lead direction , defined using the average of the clustered elements in orientation and position (see Fig.
Revised: We represent each cluster by a lead direction, defined using the average of the clustered elements in orientation and position (see Fig.

Original: 2g).
Revised: 2g).

Original: As at the previous hierarchy level, ribbons are defined by associating a thickness parameter to each lead direction, as to includes ribbons around clustered median fibers.
Revised: As at the previ- ous hierarchy level, ribbons are defined by associating a thickness parameter to each lead direction to include ribbons around clustered median fibers.

Original: As a result, the input space IS is divided into nested ribbon-like structures (see Fig. 4).
Revised: As a result, the input space IS is divided into nested ribbon-like structures (see Fig. 4).

Original: 4.5 Computing the allowed displacement areas
Revised: 4.5 Computing the allowed displacement areas

Original: The last step of the analysis stage consists in computing the avail- able space around each clustered shape or ribbon, within their parent structure in the hierarchy.
Revised: The last step of the analysis stage is to calculate the available space around each clustered shape or ribbon within their parent structure in the hierarchy.

Original: We call this space the allowed displace- ment area , since it will be used at the synthesis stage for adding random displacements to repeated structures, enabling to provide visual diversity while avoiding unwanted overlaps between shapes.
Revised: We call this space the allowed displacement area , as it will be used at the synthesis stage to add random displacements to repeated structures, providing visual diversity while avoiding unwanted overlap between shapes.

Original: Displacement areas for ribbons Starting at the top of the hierarchy, we recursively decompose each ribbon, using splitting linesthat are parallel to its main axis, and evenly split the empty space between neighboring, non-overlapping sub-ribbons (defined as ribbons around one of the clustered sub-structures).
Revised: Displacement areas for ribbons Starting at the top of the hierarchy, we recursively decompose each ribbon, using splitting lines parallel to its main axis, and evenly divide the empty space be- tween neighboring non-overlapping sub-ribbons (defined as ribbons around one of the clustered substructures).

Original: The distance between neighboring sub-ribbons (ie. the minimal distance between their contents), used to position these lines, is computed while con- sidering a toroidal topology for IS .
Revised: The distance between neighboring sub-ribbons (i e., the minimum distance between their contents) used to position these lines is computed while considering a toroidal topology for IS .

Original: Based on this distance, two lines are evenly generated between the neighbouring ribbons, to define the limits of extended regions for each of them, as well  an empty space between them.
Revised: Based on this distance, two lines are evenly generated between the neighbouring ribbons to define the limits of extended regions for each of them, as well as an empty space between them.

Original: This decomposition results into a displacement region around each sub-ribbon, and an given distance, called gap between them.
Revised: This decomposition results in a displacement region around each sub-ribbon, and a given distance, called gap between them.

Original: Note that since  the parent orientationwas used for this decomposi- tion, the sub-ribbons generally have a slightly different orientation.
Revised: Note that since we use the parent orientation for this decomposition, the sub-ribbons usually have a slightly different orientation.

Original: Moreover, they are not necessarily centered in the associated dis- placement region (see Fig. 4) a) and b).
Revised: Moreover, they are not necessarily centered in the associated displacement region (see Fig. 4) a) and b).

Original: Finally, the minimum and maximumgaps values  are stored in the parent structure, together with the set of displacement areas associated to sub-ribbons.
Revised: Finally, the minimum and maximum values of the gaps are stored in the parent structure, together with the set of displacement areas associated with sub-ribbons.

Original: Displacement areas for bounded shapes These rectangular regions, depicted using dashed lines in Fig.
Revised: Displacement areas for bounded shapes These rectangular regions, delimited by dashed lines in Fig.

Original: 4 d), represent the areas within the fiber ribbon of a bounded shape in which its bounding box will be allowed to move during instantiation.
Revised: 4 d), represent the areas within the fiber ribbon of a bounded shape in which its bounding box will be allowed to move during instantiation.

Original: Their two axes ( x , y ) respectively correspond to the direction of the associated fiber median and its orthogonal direction.
Revised: Their two axes ( x , y ) respectively correspond to the direction of the associated fiber median and its orthogonal direction.

Original: The allowed perturbation along x (tangent to the direction) is set to halfof the distance to the next bounding box of a bounded shape, which ensures that overlaps will always be prevented at the synthesis stage.
Revised: The allowed perturbation along x (tangent to the direction) is set to half the distance to the next bounding box of a bounded shape, ensuring no overlap at the synthesis stage.

Original: The allowed perturbation along y is set so that the bounding box can cover the whole associated fiber-median ribbon.
Revised: The allowed perturbation along y is set so that the bounding box can cover the whole associated fiber-median ribbon.

Original: Again, these computations are done while considering a toroidal topology for the input space IS .
Revised: Again, these computations are performed considering a toroidal topology for the input space IS .

Original: Therefore, the computed displacement areas can then expand outside IS (see the orange areas in Fig.
Revised: Therefore, the computed displacement areas can then expand outside IS (see the orange areas in Fig.

Original: 4 d), which is less restrictive when extending the pattern to a larger input domain.
Revised: 4 d), which is less restrictive when extending the pattern to a larger input domain.

Original: 5 S YNTHESIS OF AN EXTENDED SHAPE - PATTERN
Revised: 5 S YNTHESIS OF AN EXTENDED SHAPE - PATTERN

Original: To enablea seamless exploration of a larger 2D domain by simply zooming out after sketching, our objective is to keep the user-drawn strokes within IS while extending and repeating them in a larger output space OS (defined as an expansion of IS by a ratio k > 1).
Revised: To enable seamless exploration of a larger 2D domain by simply zooming out after sketching, our objective is to retain the user-drawn strokes within IS while extending and repeating them in a larger output space OS (defined as an expansion of IS by a ratio k > 1).

Original: This is done through a coarse-to-fine process in which the elements stored in the Support Structure Hierarchy are extended to OS and repeated if necessary.
Revised: This is done by a coarse-to-fine process in which the elements stored in the Support Structure Hierarchy are extended to OS and repeated as necessary.

Original: Extension and repetition of lead ribbons:
Revised: Extension and repetition of lead ribbons:

Original: According to H 2 (see Sect. 3.1), Lead directions consisting of only one fiber median , (such as the vertical lead direction in Fig. 5) should not be repeated.
Revised: According to H 2 (see Sect. 3.1), lead directions consisting of only one fiber median, (such as the vertical lead direction in Fig. 5) should not be repeated.

Original: Therefore, we simply extend them as well as their unbounded child structures to span the whole OS .
Revised: Therefore, we simply extend them as well as their unbounded child structures to span the whole OS .

Original: For the remaining lead directions (corresponding to  repeated substructures in the input), we perform the same extension to the whole OS, but also generate new copies of the structure in the remaining space, through an efficient randomized repetition procedure, as fol- lows.
Revised: For the remaining lead directions (corresponding to the repeated sub-structures in the input), we perform the same extension to the entire OS but also generate new copies of the structure in the remain- ing space through an efficient randomized repetition procedure, as follows.

Original: For each lead ribbon , we start from a displacement area with a single neighbour, randomly generate a new gap using values in the recorded range, and generate the next displacement area as to randomly clone one of the exiting ones (ie. using the same width).
Revised: For each lead ribbon, we start from a displacement area with a single neighbour, randomly generate a new gap using values in the recorded range, and generate the next displacement area as to randomly clone one of the existing ones (i.e., using the same width).

Original: We apply this technique to progressively fill OS .
Revised: We apply this technique to progressively fill OS .

Original: The randomness in the gap values between displacement areas for sub-ribbons generates different lead ribbon configurations, and therefore different outputs from the same input (see Fig. 5).
Revised: The randomness in the gap values between displacement areas for sub-ribbons generates different lead ribbon configurations, and thus different outputs from the same input (see Fig. 5).

Original: Fiber medians ribbons will now be generated within the newly extended and repeated lead ribbons, as presented next, at the cost of slightly bending some of them as well as their child structures, if they happen to overlap when extended to OS .
Revised: Fiber medians ribbons will now be generated within the newly extended and repeated lead ribbons, as presented next, at the cost of slightly bending some of them as well as their child structures if they happen to overlap when extended to OS .

Original: Repetition of fiber medians and ribbons For each newly generated displacement area , we synthesize its fiber median by first copying the parameter values of the original ribbon.
Revised: Repetition of fiber medians and ribbons For each newly generated displacement area, we synthesize its fiber median by first copying the parameter values of the original ribbon.

Original: We then use the circular standard deviation on the medians’ orientations computed during the analysis stage (Sect. 4.3) to perturb its orientation.
Revised: We then use the circular standard deviation on the medians’ orientations computed during the analysis stage (Sect. 4.3) to perturb its orientation.

Original: We also perturb the position of its centroid to place it in the middle of the displacement area .
Revised: We also perturb the position of its centroid to place it in the middle of the displacement area.

Original: While the middle part of generated ribbons are guaranteed to remain within their lead ribbon , this is not necessarily the case when they extend to OS , as illustrated in Fig.
Revised: While the middle part of generated ribbons is guaranteed to remain within their lead ribbon, this is not necessarily the case when they extend to OS , as illustrated in Fig.

Original: 6 (left).
Revised: 6 (left).

Original: When this occurs, we slightly bend a ribbon and its fiber median (see Fig.
Revised: When this occurs, we slightly bend a ribbon and its fiber median (see Fig.

Original: 6 (right))
Revised: 6 (right))

Original: to make it fit entirely inside its allowed displacement area.
Revised: to make it fit entirely inside its allowed displacement area.

Original: Avoiding overlaps by bending structures Inspired by physi- cal properties of (real) fibers, we consider the following assumption: the thinner the ribbon , the more flexible it may be.
Revised: Avoiding overlaps by bending structures Inspired by the physical properties of (real) fibers, we consider the following assumption: the thinner the ribbon, the more flexible it may be.

Original: This can be for- malized through the equation R = τ w , relating the curvature radius R to the ribbon width w and a stiffness parameter τ ∈ R + .
Revised: This can be formalized through the equation R = τ w , relating the curvature radius R to the ribbon width w and a stiffness parameter τ ∈ R + .

Original: In case of overlap, each ribbon will intersect twice with its lead ribbon .For
Revised: In case of overlap, each ribbon will intersect twice with its lead ribbon.

Original:  symmetry reasons, we then bend both sides of the ribbon , even if one of the intersection regions is out of the output domain.
Revised: For symmetry reasons, we then bend both sides of the ribbon, even if one of the intersection regions is out of the output domain.

Original: To preserve continuity between the original borderlines of the ribbon and their curved version, we consider the midpoints ( M 1 and M 2 in green in Fig.
Revised: To preserve continuity between the original borderlines of the ribbon and their curved version, we consider the midpoints ( M 1 and M 2 in green in Fig.

Original: 7) between a projected point and the other intersection point as inflection points.
Revised: 7) between a projected point and the other intersection point as inflection points.

Original: For each of these inflexion points (say M ), the key idea is to find the arc of circle C that passes through M and remains inside the lead ribbon as illustrated in Fig.
Revised: For each of these inflection points (say M ), the key idea is to find the arc of circle C that passes through M and remains inside the lead ribbon as illustrated in Fig.

Original: The details of this computationare provided in the associated Supplementary material.
Revised: We provide the details of this computation in the associated Supplementary material.

Original: The same bending process is applied to the children sub-ribbons , in order tofit them inside their parent curved ribbon .
Revised: The same bending process is applied to the children’s sub ribbons to them inside their parent’s curved ribbon.

Original: Shape distribution synthesis The final step is to synthesize new shapes within each extended or newly created fiber ribbon .
Revised: Shape distribution synthesis The final step is to synthesize new shapes within each extended or newly created fiber ribbon.

Original: a) Unbounded shapes We define four unbounded stroke cat- egories (lines, rays, arcs and curves) that respectively stand for perfectly linear unbounded strokes, half-lines, unbounded strokes with a single curvature extremum in IS , and unbounded strokes with more than one curvature extrema.
Revised: a) Unbounded shapes We define four categories of un- bounded strokes (lines, rays, arcs, and curves) that represent perfectly linear unbounded strokes, half-lines, unbounded strokes with a single curvature extremum in IS , and unbounded strokes with more than one curvature extrema, respectively.

Original: We start by extending these un- bounded shapes to OS along their fiber direction, which is trivial for lines and rays.
Revised: We begin by extending these unbounded shapes to OS along their fiber direction, which is trivial for lines and rays.

Original: Arcs are extended through an alternative mirror duplication that leads to a smooth sinusoidal curve.
Revised: The arcs are extended through an alter- native mirror duplication that leads to a smooth sinusoidal curve.

Original: Curves are first cut at their first and last extrema.
Revised: The curves are first cut at their first and last extrema.

Original: Then, we alternativelyduplicate the mirror  version of the curve-segment to extend it to OS as illustrated in Fig. 
Revised: Then, we alternatively mirror the version of the curve segment to extend it to OS as illustrated in Fig. 8.

Original: These extended strokes are stored in the local frame of their corresponding fiber .
Revised: These extended strokes are stored in the local frame of their corresponding fiber.

Original: They will therefore be automatically repeated and curved if needed through the repetition process of their parent structures in the hierarchy.
Revised: They will therefore be automatically repeated and curved if necessary through the repeti- tion process of their parent structures in the hierarchy.

Original: The resulting curved structures are shown for different sizes of OS in Fig.
Revised: The resulting curved structures are shown for different sizes of OS in Fig.

Original: b) Bounded shapes We process bounded shapes by first, it- eratively repeating their representative support segments or central points along their extended fiber, while using the previously com- puted displacement areas to perturb their positions, and drawing the shapes in the resulting local frames.
Revised: b) Bounded shapes We process bounded shapes by first it- eratively repeating their representative support segments or central points along their extended fiber while using the previously com- puted displacement areas to perturb their positions, and drawing the shapes in the resulting local frames.

Original: We then reuse their local positions with respect to their fiber to repeat them within the parent fiber medians ribbons , but with randomly modified positionswithin in the authorized displacement areas .
Revised: We then reuse their lo- cal positions relative to their fiber to repeat them within the parent fiber medians ribbons, but with randomly modified positions in the allowed displacement areas.

Original: See Figure Fig.
Revised: See Figure Fig.

Original: Avoiding residual overlaps: Given that repetitions in differ- ent lead directions are computed independently, lead ribbons with different orientations may naturally intersect.
Revised: Avoiding residual overlaps: Given that repetitions in differ- ent lead directions are computed independently, lead ribbons with different orientations may naturally intersect.

Original: This may lead to perceptual artifacts if these lead directions both contain initially non-overlapping bounded shapes.
Revised: This may lead to perceptual artifacts if these lead directions both contain initially non-overlapping bounded shapes.

Original: Indeed some undesirable overlaps may occur in the output.
Revised: Indeed some undesirable overlaps may occur in the output.

Original: We use an AABB tree to partition OS and efficiently detect overlaps between  displacement areas of bounded shapes.
Revised: We use an AABB tree to partition OS and efficiently detect overlaps between the displacement areas of bounded shapes.

Original: In  such case, we restrict the corresponding displacement areas .
Revised: In a such case, we restrict the corresponding displacement areas.

Original: If this strategy fails (not enough space to insert a shape), we do not instantiate it (see Fig.
Revised: If this strategy fails (not enough space to insert a shape), we do not instantiate it (see Fig.

Original: 11 (b) for such a challenging example).
Revised: 11 (b) for such a challenging example).

Original: 6 R ESULTS AND D ISCUSSION 6.1 Interactive authoring system
Revised: 6 R ESULTS AND D ISCUSSION 6.1 Interactive authoring system

Original: Our prototype systemis implemented in WebGL.
Revised: We implement our prototype system in WebGL.

Original: Creating and extending highly structured vector patterns is made easy by our method, as shown in Fig.
Revised: Creating and extend- ing highly structured vector patterns is made easy by our method, as shown in Fig.

Original: In addition to the main sketching and texture expansion interface, the user may store and reuse complex shapes, such as the two categories of fishes in Fig.
Revised: In addition to the main sketching and texture expansion interface, the user may store and reuse complex shapes, such as the two categories of fishes in Fig.

Original: The use of our system for creating a complex sketch, inspired from biology, is illustrated in Fig.
Revised: The use of our system for creating a complex sketch, inspired by biology, is illustrated in Fig.

Original: In addition, several input shape-patterns can be interactively com- bined to create a hybrid one, as follows:
Revised: In addition, several input shape-patterns can be interactively com- bined to create a hybrid one, as follows.

Original: Thanks to the Support
Revised: Thanks to the Support

Original: Structure Hierarchy , the user can select the desired level of hier- archy from two different input shape-patterns, and combine them and create an hybrid result.
Revised: Structure Hierarchy, the user can select the desired level of hierarchy from two different input shape-patterns and combine them and create an hybrid result.

Original: We rely on the fact that our Support Structure Hierarchy encodes the input data into structuresthat are defined in the local frames of their upper structure ribbons, them- selves characterized by a main direction anda width.
Revised: We rely on the fact that our Support Structure Hierarchy encodes the input data into structures defined in the local frames of their upper structure ribbons, themselves characterized by their main direction and width.

Original: Therefore, consistent patterns can be generated while the input shapes, fibers, fiber-medians or lead directions are changed.
Revised: Therefore, consistent patterns can be generated while the input shapes, fibers, fiber-medians, or lead directions are changed.

Original: Such an hybridization is shown in Fig.
Revised: Such a hybridization is shown in Fig.

Revised: 13 and its different steps displayed at the end of the accompanying video.

Original: 6.2 Comparison with previous work
Revised: 6.2 Comparison with previous work

Original: We compared our results with both distribution-based and deep- learning-based methods for generating vector textures from examples.
Revised: We compared our results with both distribution-based and deep- learning-based methods for generating vector textures from examples.

Original: Since most classical methods are limited to distributions of bounded shapes, we restricted comparison to this sub-case (see
Revised: Since most classical methods are limited to distributions of bounded shapes, we restricted comparison to this sub-case (see

Original: Fig. 14).
Revised: Fig. 14).

Original: Since our results seemed close to those of the best classi- cal method, Landes et al.
Revised: Since our results seemed close to those of the best classical method, Landes et al.

Original: [9], we selected this method for further comparison in our user study (see below).
Revised: [11], we selected this method for further comparison in our user study (see below).

Original: We also tried our method on examples presented as failure cases in previous papers, such as Fig.
Revised: We also tried our method on examples presented as failure cases in previous papers, such as Fig.

Original: 11 (a failure case of [3]) and Fig.(a failure case of [2]).
Revised: 11 (a failure case of [5]) and Fig. 15 (a failure case of [4]).

Original: In both cases, our solution was robust and managed to maintain the regularity of the structured input for both bounded and unbounded shapes.
Revised: In both cases, our solution was robust and managed to maintain the regularity of the structured input for both bounded and unbounded shapes.

Original: Lastly, we compared our results with those of Tu et al
Revised: Lastly, we compared our results with those of Tu et al.

Original: [17], the only deep-learning method tackling point distributions (see Fig. 16)
Revised: [20], the only deep-learning method tackling point distributions (see Fig. 16).

Original: Although our method is interactive and does not require any precom- putation stage (in contrast to the hours of training of deep learning methods), the quality of our results looks almost as good (we get more artefacts in the first example, while [17] gets more of them in the second one).
Revised: Although our method is interactive and does not require any precom- putation stage (in contrast to the hours of training of deep learning methods), the quality of our results looks comparable to the first example while the method from Tu et al. [20] gets more of them in the second one.

Original: 6.3 User study
Revised: 6.3 User study

Original: We carefully designed an online user study to validate the perceptual hypotheses presented in Sect.
Revised: We carefully designed an online user study to validate the perceptual hypotheses presented in Sect.

Original: 3.1, as well as the perceived quality of the extended textures we generate (See our supplemental document for screenshots and detailed results).
Revised: 3.1, as well as the perceived quality of the extended textures we generate (See our supplemental document for screenshots and detailed results).

Original: The study was conducted by 35 users, from 19 to 61 years old, including 22 males, 9 females and 4 genders unspecified.

Original: 14 had an intermediary or expert experience in digital design and 9 as traditional designers.

Original: It was composed of two parts: an interactive drawing session, and a comparison session.
Revised: This study was composed of two parts: an interactive drawing session and a comparison session.

Revised: We let the reader refer to the supplementary material for an illustration of all the proposed tasks.

Original: In the drawing session,  users were asked to manually draw an extended texture from a given input pattern.
Revised: During the drawing session, the users were asked to manually draw an extended texture from a given input pattern.

Revised: For this interac- tive session, we sorted the tasks in increasing orders of complexity: the first task consists in replicating (H1) the bounded shapes and avoiding overlaps (H3); the second task was about respecting the groupings and alignments (H1) of unbounded shapes while also avoiding overlaps (H3); the third task was mostly about our repetitiveness hypothesis (H2) for unbounded shapes, and the last task was about the respect of groupings and alignments (H1) of bounded shapes while avoiding overlaps (H3).

Original: In the comparison session,  users were asked to select the closest result from a givena 2D input.
Revised: On the other hand, in the com- parison session, the users were asked to select the closest result from a given 2D input.

Revised: The first selection task was to check whether users preferred an output with overlapping unbounded strokes or not (H3).

Revised: The objective of the second task was to verify our repetitiveness hypothesis (H2) for unbounded strokes.

Original: Each experiment lasted around ten minutes, most of which in the drawing session.
Revised: Each experiment lasted around ten minutes, most of which was during the drawing session.

Revised: The study was conducted by 35 users, from 19 to 61 years old, including 22 males, 9 females, and 4 genders unspecified.

Revised: 14 had an intermediary or expert experience in digital design and 9 as traditional designers.

Original: Among the guidelines to validate, H 1 (Groupings and alignments are meaningful) was validated by the drawing session, where 97% of the users preserved the grouping of fiber-like shapes and 76% of the users respected the anisotropy directions of bounded shapes in their drawings.
Revised: Among the guidelines to validate, H 1 (Groupings and alignments are meaningful) was validated by the drawing session, where 97% of the users preserved the grouping of fiber-like shapes and 76% of the users respected the anisotropy directions of bounded shapes in their drawings.

Original: H 2 (repetitiveness is explicit) was validated by most users during the comparison session, and was also observed in the user’s drawings as those of Fig.
Revised: H 2 (repetitiveness is explicit) was validated by most users during the comparison session and was also observed in the user’s drawings as those of Fig.

Original: H 3 (non-overlapping shapes should remain disjoint) was validated as well by the users’ drawings, with 73% of overlapping-free drawings when it was the case in the input.
Revised: H 3 (non-overlapping shapes should remain disjoint) was validated as well by the users’ drawings, with 73% of overlapping-free drawings when it was the case in the input.

Original: As part of the comparison session, user were asked to  choose between our extended textures and the generated ones from Landes et al.
Revised: As in the study of AlMeraj et al. [2], we rely on our user study to let users choose between our extended textures and the generated ones from Landes et al.

Original: [9] (shown in random order, and using the same shape depic- tion), for the ants and the balloons examples of Fig.
Revised: [11] (shown in random order and using the same shape depiction) for the ants and the balloons examples of Fig.

Original: Respectively 86% and 77% of users preferred our results.
Revised: Respectively 86% and 77% of users preferred our results.

Original: We attribute this un- expectedly good results to the fact we keep the exact input pattern at the center of the generated texture, while seamlessly extending it sideways.
Revised: We attribute these unexpectedly good results to the fact we keep the exact input pattern at the center of the generated texture while seamlessly extending it sideways.

Original: 6.4 Performance
Revised: 6.4 Performance

Original: The following table was computed using the Google Chrome run- time performance on an Intel(R) Core(TM) i7-7920HQ CPU at 3.10GHz.
Revised: The following table was computed using the Google Chrome run- time performance on an Intel(R) Core(TM) i7-7920HQ CPU at 3.10GHz.

Original: The second column gives the number of points in the input example, and then the time in milliseconds of respectively the analysis and the synthesis.
Revised: The second column gives the number of points in the input example and then the time in milliseconds of respectively the analysis and the synthesis.

Original: Note that the synthesis has been performed with a ratio of k = 3.
Revised: Note that the synthesis has been performed with a ratio of k = 3.

Original: As can be observed, the overall computation times take less than afraction of second.
Revised: As can be observed, the overall computation times take less than a second.

Original: Example # Points
Revised: Example # Points

Original: Analysis
Revised: Analysis

Original: Synthesis
Revised: Synthesis

Original: Fishes (Fig. 1) 
Revised: Fish & seaweeds (Fig. 1) 73ms 111ms Biology (Fig. 12) [Figure] 86ms Ants (Fig. 14 top)

Original: [Table]
Revised: [Figure]

Revised: 233ms Balloons (Fig. 14 bottom)

Original: [Table] [Table]
Revised: [Figure] [Figure]

Original: 6.5 Discussion and limitations
Revised: 6.5 Discussion and limitations

Original: The specificity of our method compared to previous work is that it does not require any neighborhood matching at the synthesis stage, given that our hierarchical representation already captures correlations.
Revised: The specificity of our method compared to previous work is that it does not require any neighborhood matching at the synthesis stage, given that our hierarchical representation already captures correlations.

Original: This leads to real-time performances, suitable to ourapplication context.
Revised: This leads to real-time performance suitable to our

Original: However, the notion of anisotropy being central in our method,it makes it unsuitable to synthesize isotropic distributions.
Revised: However, the notion of anisotropy, central in our method, makes it unsuitable to synthesize isotropic distributions.

Original: Indeed computing meaningful fiber directions then more difficult, which prevents further extraction  a structural hierarchy.
Revised: Indeed the computation of significant fiber directions becomes more difficult, which prevents further extraction of a structural hierarchy.

Original: While this is the main limitation of our framework. Therefore, in an authoring tool, our solution should be compared with a former method handling isotropic distributions.
Revised: Although this is the main limitation of our framework, in an authoring tool, our solution should be compared with a former method that handles isotropic distributions.

Original: A useful extension would be to give the user the possibility of choosing among different perceptual hypotheses, for instance, regarding explicit repetitiveness, which may notbe always desired, or to enable overlaps between bounded and unbounded shapes.
Revised: A useful extension would be to give the user the ability to choose among different perceptual hypotheses, for example, regarding explicit repetitiveness, which is not always desired or to allow overlaps between bounded and unbounded shapes.

Original: For instance, in the biology illustration of Fig. 12, the cells depicted in pink/red should remain attached to the underlying fibers, which is not the case in our solution.
Revised: For example, in the biology illustration of Fig. 12, the cells depicted in pink/red should remain attached to the underlying fibers, which is not the case in our solution.

Original: Indeed, we never cluster bounded strokes with unbounded strokes, even if they overlap.
Revised: Indeed, we never cluster bounded strokes with unbounded strokes, even if they overlap.

Original: This could easily be added as an option.
Revised: This could easily be added as an option.

Original: Generating waving curves rather than having unbounded  strokes intersect a design choice, which could also be disabled be the user if necessary.
Revised: Generating wavy curves rather than intersecting unbounded [Figure] strokes is a design choice, which could also be disabled by the user if necessary.

Original: Fig. 18 presents a failure case for our solutions, as bounding boxes around strokes overlap, while these strokes should not be grouped.
Revised: Fig.18presents a failure case for our solutions, as bounding boxes around strokes overlap, while these strokes should not be grouped.

Original: To solve this problem, we could allow the choice between bounding-boxes-based distances and centroid-based distances as clustering criteria for bounded strokes.
Revised: To solve this problem, we could allow the choice between bounding-boxes-based distances and centroid-based distances as clustering criteria for bounded strokes.

Original: This would facilitate the processing of any dense distribution.
Revised: This would facilitate the processing of any dense distribution.

Original: As a last limitation,  we do not consider branching curves, which could be another kind of unbounded elements in the input texture pattern. Since it processes the input strokes one by one, our current method would not capture the branching.
Revised: As the last limitation, extracting the main direction for unbounded strokes restrict their shapes to mostly linear ones. In particular, we do not consider branching curves or circular lines.

Original: The input would be split into isolated curves, which would probably intersect when repeated in  the output.
Revised: In the case of branching, individual strokes would be split into isolated curves. This could result in unwanted overlaps during the synthesis.

Original: In addition, even if  grouping was forced, our current way of representing unbounded elements using a single linear ribbon of given width would fail.
Revised: In addition, even if the grouping was forced, our current way of representing unbounded elements using a single linear ribbon of given width would fail.

Revised: In the case of circular lines, we would need to cut them into linear pieces and treat each piece individually, which would result in handling branching.

Original: Therefore, seamlessly extending patterns that  such branching structures remains  open problem
Revised: Therefore, seamlessly extending patterns that include such complex structures remains an open problem.

Original: 7 C ONCLUSION
Revised: 7 C ONCLUSION

Original: Motivated by the interactive design of vector textures, we presented a multi-scale method to efficiently extract anisotropic properties from an input pattern, and seamlessly extend it to a larger 2D domain.
Revised: Motivated by the interactive design of vector textures, we presented a multi-scale method to efficiently extract anisotropic properties from an input pattern, and seamlessly extend it to a larger 2D domain.

Original: While our method runs in real time, visual quality of results compare well with those of state-of-the-art vector texture generation methods, including those requiring either higher computational time and/or training data to learn from. M
Revised: Although our method runs in real-time, the visual quality of results is comparable to that of state-of-the-art vector texture generation methods, including those that require higher computational time and/or training data.

Original: The new Support Structure Hierarchy we introduced is crucial to our method.
Revised: The new Support Structure Hierarchy we introduced is crucial for our method.

Original: Extracted at the analysis stage based on a new perceived distance between the salient anisotropic structures withinin the input domain, it allows  to  capture and reproduce the multi-scale structuresin an efficient way, while maintaining a good level of visual diversity in the synthesized distribution of shapes.
Revised: Extracted at the analysis stage based on a new perceived distance between the salient anisotropic structures within the input domain, it allows us to efficiently capture and reproduce the multiscale structures while maintaining a good level of visual diversity in the synthesized distribution of shapes.

Original: In terms of the interface, our system can be used to quickly design new vector textures by interactively creating new patterns or combining existing ones.
Revised: In terms of the interface, our system can be used to quickly design new vector textures by interactively creating new patterns or combining existing ones.

Original: Future work While our solution is well suited to most structured shape-patterns our use of linear ribbon-like shapes to capture multiscale anisotropy prevents us from handling more complex, branching structures.
Revised: Future work Although our solution is well suited for most struc- tured shape-patterns, our use of linear ribbon-like shapes to capture multi-scale anisotropy prevents us from handling more complex and branched structures.

Original: Addressing this specific case would be an interesting avenue for future work.
Revised: Addressing this specific case would be an interesting avenue for future work.

Original: In addition, a challenging open  problem would be to generate a 3D texture from the 2D exemplar interactively sketched by the user.
Revised: In addition, an open and challenging problem would be to generate a 3D texture from the 2D exemplar interactively sketched by the user.

Original: In cases such as biological illustrations, this would enable users to navigate in a 3D structure created from the sketch, leading to a better understanding  the depicted environment. 
Revised: In cases such as biological illustrations, this would allow users to navigate in a 3D structure created from the sketch, leading to a better understanding of the depicted environment. A CKNOWLEDGEMENT

Revised: We would like to thank Pierre Poulin for his feedback that helped us improve this paper.