Sophie-Caroline Campana

DDS Thibault Canceill
email: thibault.canceill@sfr.fr.

DDS Rémi Esclassan

Thibault

Oral care of a patient with a SAPHO syndrome and a nickel allergy

Keywords: nickel allergy, prosthesis, SAPHO syndrome ORCID

SAPHO is an acronym for Synovitis, Acne, Pustulosis, Hyperostosis and Osteitis. The syndrome is difficult to diagnose because it may present many different manifestations in adults and children. Its origin is still unknown, although some infectious, genetic and immune hypotheses have been put forward. We report the case of a 49-year-old woman with SAPHO syndrome, who developed a serious cutaneous allergy following the insertion of a removable partial denture (RPD). The oral care and treatment of this patient are described.

INTRODUCTION

SAPHO syndrome (SAPHO is an acronym for Synovitis, Acne, Pustulosis, Hyperostosis, Osteitis) is a poorly known disease, classified as rare with a prevalence close to 1/10 000. [START_REF] Aljuhani | The SAPHO syndrome: a single-center study of 41 adult patients[END_REF][START_REF] Gharsallah | SAPHO syndrome][END_REF][START_REF] Nguyen | The SAPHO syndrome[END_REF] It was long considered as a spondylarthrosis because of the spinal manifestations but the association with other symptoms led to a distinction being made between the two diseases. [START_REF] Carranco-Medina | Thrombotic manifestations in SAPHO syndrome. review of the literature[END_REF] It was first reported in the late 1980s by Chamot et al who defined some diagnosis criteria for this syndrome at the time. [START_REF] Benhamou | Synovitis-acne-pustulosis hyperostosis-osteomyelitis syndrome (SAPHO). A new syndrome among the spondyloarthropathies?[END_REF][START_REF] Chamot | Acne-pustulosis-hyperostosis-osteitis syndrome. Results of a national survey. 85 cases[END_REF][START_REF] Kahn | The SAPHO syndrome[END_REF] The syndrome brings together inflammatory osteitis, osteoarticular features, hyperostosis, and skin manifestations such as severe acne or palmoplantar pustulosis. [START_REF] Aljuhani | The SAPHO syndrome: a single-center study of 41 adult patients[END_REF][START_REF] Carneiro | SAPHO syndrome[END_REF] Peripheral arthritis and synovitis are also noted in a third of patients [START_REF] Carneiro | SAPHO syndrome[END_REF] and, in others, cutaneous reactions can resemble. It is often noted that skin features occur before osseous ones but there are strong variations among patients and skin manifestations can also appear later in the evolution of the syndrome. [START_REF] Cianci | Clinical heterogeneity of SAPHO syndrome: challenging diagnose and treatment[END_REF] Back pain and anterior chest wall (sternum, clavicle, sternocostoclavicular) involvement [START_REF] Benhamou | Synovitis-acne-pustulosis hyperostosis-osteomyelitis syndrome (SAPHO). A new syndrome among the spondyloarthropathies?[END_REF][START_REF] Carneiro | SAPHO syndrome[END_REF][START_REF] Okuno | Clinical features and radiological findings of 67 patients with SAPHO syndrome[END_REF] are typical early presentations [START_REF] Aljuhani | The SAPHO syndrome: a single-center study of 41 adult patients[END_REF] and considered main diagnostic criteria and distinguished from other types of pathologies (osteitis, spondyloarthritis, or osteoslerotic metastases of breast cancers). [START_REF] Cianci | Clinical heterogeneity of SAPHO syndrome: challenging diagnose and treatment[END_REF][START_REF] Okuno | Clinical features and radiological findings of 67 patients with SAPHO syndrome[END_REF][START_REF] Hagiwara | SAPHO syndrome: synovitis, acne, pustulosis, hyperostosis, and osteitis[END_REF] Differential diagnosis is complicated by the multiplicity of symptoms and the strong variations in their occurrence. [START_REF] Aljuhani | The SAPHO syndrome: a single-center study of 41 adult patients[END_REF][START_REF] Carneiro | SAPHO syndrome[END_REF][START_REF] Firinu | SAPHO syndrome: current developments and approaches to clinical treatment[END_REF] All of these symptoms of chronic intense pain can affect the patient's quality of life [START_REF] Witt | Disease burden, disease manifestations and current treatment regimen of the SAPHO syndrome in Germany: results from a nationwide patient survey[END_REF] and are associated with a 7.5% risk of depressive episodes. [START_REF] Lu | Depression in patients with SAPHO syndrome and its relationship with brain activity and connectivity[END_REF] Osseous manifestations explain why scintillography and magnetic resonance imaging are commonly used to establish or confirm the diagnosis. [START_REF] Canella | SAPHO syndrome[END_REF][START_REF] Kundu | Diagnosing the SAPHO syndrome: a report of three cases and review of literature[END_REF][START_REF] Schaub | Imaging for synovitis, acne, pustulosis, hyperostosis, and osteitis (SAPHO) syndrome[END_REF] The general management of SAPHO syndrome is empirical and the long-term prognosis has not been established. [START_REF] Lu | Depression in patients with SAPHO syndrome and its relationship with brain activity and connectivity[END_REF] Bisphosphonates are commonly used against osteomyelitis and osteitis 18,19 but they are not efficient in every case and many reports have been published on other treatments attempted in the absence of response to bisphosphonates, such as anti-IL1, Janus kinase inhibitor or, recently, TNF blockers. [START_REF] Aljuhani | The SAPHO syndrome: a single-center study of 41 adult patients[END_REF][START_REF] Firinu | SAPHO syndrome: current developments and approaches to clinical treatment[END_REF][START_REF] Daoussis | Biologics in SAPHO syndrome: a systematic review[END_REF][START_REF] Yang | Case report: successful treatment of refractory SAPHO syndrome with the JAK inhibitor tofacitinib[END_REF] However, no therapy has demonstrated an effect in all patients and remission may be only partial, with disappearance of skin symptoms but persistence of bone lesions. [START_REF] Yang | Case report: successful treatment of refractory SAPHO syndrome with the JAK inhibitor tofacitinib[END_REF] Oral manifestations of SAPHO syndrome predominantly concern mandibular osteomyelitis [START_REF] Baba | SAPHO syndrome with mandibular manifestation[END_REF][START_REF] Hatano | A case of SAPHO syndrome with diffuse sclerosing osteomyelitis of the mandible treated successfully with prednisolone and bisphosphonate[END_REF][START_REF] Marí | Diffuse sclerosing osteomyelitis (DSO) of the mandible in SAPHO syndrome: a novel approach with anti-TNF therapy. Systematic review[END_REF][START_REF] Pottecher | SAPHO syndrome revealed by sclerosing mandibular osteomyelitis[END_REF][START_REF] Suei | Diagnostic points and possible origin of osteomyelitis in synovitis, acne, pustulosis, hyperostosis and osteitis (SAPHO) syndrome: a radiographic study of 77 mandibular osteomyelitis cases[END_REF][START_REF] Zemann | SAPHO syndrome with affection of the mandible: diagnosis, treatment, and review of literature[END_REF] but some necrosis may also occur due to dental surgery in concomitance with the administration of bisphosphonate treatments. However, the mandible is affected by osseous manifestations in only 10% F I G U R E 1 Panoramic radiograph performed during patient's first visit to the dental unit of cases, [START_REF] Carneiro | SAPHO syndrome[END_REF][START_REF] Baba | SAPHO syndrome with mandibular manifestation[END_REF] that is, less often than the anterior chest wall or spine. [START_REF] Aljuhani | The SAPHO syndrome: a single-center study of 41 adult patients[END_REF] In this context, the aim of this case report is to describe the oral care of a SAPHO patient who suffered a sudden allergy after the insertion of a metal removable partial denture (RPD).

CASE REPORT

A 49-year-old woman with SAPHO syndrome came to the dental unit of Toulouse Hospital because of acute dental pain and for aesthetic considerations. Because of her long dental history, she "wanted "to have all her teeth extracted." At this time, she had never had a RPD and never reported any allergy.

Her SAPHO syndrome had been discovered some years previously after a work-related accident when she was working as a nurse. She experienced severe pain on attempting to lift a patient and therefore underwent a medical examination including radiography. The diagnosis was based on the full body and long bone radiographs and the concomitant severe back and articular pain. During the first consultation in the dental unit, a medical check-up revealed osteoarticular degeneration, peripheral arthritis and massive cutaneous involvements. The patient was not treated with bisphosphonates but only with anti-inflammatory (Ketoprofen), antalgic (Tramadol, Gabapentin), and psychotropic medicines (Hydroxyzine, Zopiclone). Ketoprofen is commonly used for the symptomatic long-term treatment of chronic inflammatory rheumatism, and Gabapentin may be prescribed not only for its analgesic properties but also its propensity to reduce epileptic and spasmodic seizures. During an oral examination, generalized chronic periodontitis was observed with mobilities and bleeding on probing. A panoramic radiograph confirmed the diminution of the alveolar bone level and the presence of foci of infection, especially on teeth 47 and 48 (Figure 1). The radiograph showed no evidence of osteoarticular degeneration in the jaws and is consistent with the low incidence in the reported in the literature. [START_REF] Baba | SAPHO syndrome with mandibular manifestation[END_REF] All molars exhibited severe periodontitis and the decision was rendered to extract them at this time, owing in part, to the absence of bisphosphonate intake. General anaesthesia was F I G U R E 2 Perioral eczema observed a few weeks after hypersensitive reaction to nickel contained in prosthesis clasps preferred because her general osteoarticular pain, joint pain in the temporomandibular articulation and mouth opening limitation made it difficult for the woman to lie down and to keep her mouth open for a long time. Her daily treatment was not modified for the surgery but antibiotics (amoxicillin and clavulanic acid, 7 days) and mouthwashes (chlorhexidine digluconate 0.12%, 10 days) were also prescribed. The patient was happy to brush her teeth with a postsurgical toothbrush and preferred fluid and soft food for a few days. To accustom her to wearing the prosthesis, it was decided not to remove all the teeth at first but to keep teeth 11, 12, 13, 22, 23, 31, 32, 33, 41, 42, and 43. Therefore, two RPDs were made, which would be transformed into complete removable dentures later.

Unfortunately, 3 days after the placing of the first elements, the patient presented severe cutaneous reactions of the face and throat and mucogingival erythema on the palatal and vestibular gingiva under the devices (Figure 2). The dermatology clinician's report (Toulouse Rangueil Hospital, Toulouse, France) described "a very oedematous perioral eczema with throat diffusion." An allergist examined the patient and concluded on a "possible hypersensitivity reaction." The patch test performed revealed an exacerbated nickel-sulphate response at 72 hours (Figure 3). The allergist established a link between the cutaneous reaction and the materials used to make the clasps of the prostheses (nickel, iron, chromium, and molybdenum). Also, the laboratory's prosthesis identification card confirmed that the metal was a quaternary alloy (Figure 4). The patient's daily treatment was modified by the addition of sterile water and dermal corticoids to apply to the cutaneous reaction. Then, the decision was taken in accordance with the patient to cut all clasps and to speed up the care procedures, which meant extracting the remaining teeth without giving the patient the possibility of a testing period with the RPD. A second general anaesthesia was not performed because the woman's general health was less degraded at the time of the second surgery and also because all teeth to be extracted were mobile and located in the anterior sector, so it was not necessary for her to open her mouth wide. Mouthwashes and advice on food were again given to the patient. In this patient, general osteoarticular pain complicated dental care through the difficulties she experienced in staying in a decubitus position and opening her mouth for a long time. She also presented an articular pain crisis during a dental appointment, which made it necessary to interrupt tooth extractions and wait for a few minutes. In addition, sterilized petroleum jelly body lotion (Vaseline Stérilisée, Cooper, France) was applied to skin irritations all around the oral cavity before each dental appointment, to protect them from dehydration and reduce care-related irritation (Figure 5).

Forced to stop work several years earlier and fully aware of her state of health, the patient demonstrated extreme motivation and cooperation for care. Despite a lack of healing of the perioral and throat eczema due to her syndrome, she was extremely patient and not emotional. The patient never missed F I G U R E 6 Two complete removable prostheses were inserted after tooth extractions an appointment and she was satisfied with the aesthetic result, fulfilling her initial motivation to seek care (Figures 6 and7). In addition, the patient began transition into a more normal diet.

DISCUSSION

Although it could be expected that SAPHO is infrequently encountered in dental practice, [START_REF] Cotti | A case of SAPHO syndrome with endodontic implications and treatment with biologic drugs[END_REF] the attending clinician should be familiar with management strategies for oral and perioral allergic reactions.

The patient's demographic information in this case report was consistent with some other epidemiological studies revealing a slight female predominance for SAPHO syndrome [START_REF] Aljuhani | The SAPHO syndrome: a single-center study of 41 adult patients[END_REF][START_REF] Nguyen | The SAPHO syndrome[END_REF][START_REF] Li | Synovitis, acne, pustulosis, hyperostosis and osteitis syndrome: a single centre study of a cohort of 164 patients[END_REF] and for nickel allergy. [START_REF] Magnusson | Nickel allergy and nickel-containing dental alloys[END_REF] The patient developed a nickel allergy, which is found in 13% of the general population. [START_REF] Bakula | Contact allergy in the mouth: diversity of clinical presentations and diagnosis of common allergens relevant to dental practice[END_REF] The literature on SAPHO syndrome includes some reports concerning children 2 but the syndrome is essentially found in adults. Its aetiology and its physiopathological mechanism are still unknown. [START_REF] Lu | Depression in patients with SAPHO syndrome and its relationship with brain activity and connectivity[END_REF][START_REF] Carr | The "hidden" SAPHO syndrome[END_REF][START_REF] Rukavina | SAPHO syndrome: a review[END_REF] A possible association with HLA-B27 was initially suspected [START_REF] Kahn | The SAPHO syndrome[END_REF] (as in spondylarthrosis) but recent studies have dismissed this hypothesis [START_REF] Aljuhani | The SAPHO syndrome: a single-center study of 41 adult patients[END_REF][START_REF] Canella | SAPHO syndrome[END_REF][START_REF] Hayem | SAPHO syndrome: a long-term follow-up study of 120 cases[END_REF] F I G U R E 7 Recall appointment 2 months after fitting of prosthesis. Peri-oral and throat eczema still present and has not healed yet due to skin healing difficulties in patients with SAPHO syndrome given that only a minority of patients present the antigen. Thus SAPHO syndrome may have a genetic or immune origin or occur after a Propionibacterium Acnes infection. [START_REF] Cianci | Clinical heterogeneity of SAPHO syndrome: challenging diagnose and treatment[END_REF] An antibiotic treatment targeting this bacterium has already shown good results on patients' health [START_REF] Aljuhani | The SAPHO syndrome: a single-center study of 41 adult patients[END_REF][START_REF] Carneiro | SAPHO syndrome[END_REF] but disease progression starts again when the treatment is stopped and not all the patients respond well to the antibiotic. [START_REF] Aljuhani | The SAPHO syndrome: a single-center study of 41 adult patients[END_REF] Thus, the etiopathogenesis of SAPHO syndrome may be multifactorial and not solely regarded as an infectious process.

Nickel is one of the most common allergens used in dental prostheses [START_REF] Bakula | Contact allergy in the mouth: diversity of clinical presentations and diagnosis of common allergens relevant to dental practice[END_REF] and hypersensitivity reactions may develop 24-72 hours after contact with saliva. Manifestations include perioral dermatitis, burning mouth syndrome, lichenoid reactions, cheilitis, gingivitis, and orofacial granulomastosis. [START_REF] Bakula | Contact allergy in the mouth: diversity of clinical presentations and diagnosis of common allergens relevant to dental practice[END_REF][START_REF] Lygre | Prosthodontic biomaterials and adverse reactions: a critical review of the clinical and research literature[END_REF][START_REF] Alnazzawi | Oral diseases associated with fixed prosthodontic restorations[END_REF] In the particular case of patients with SAPHO syndrome, the main complications occur in presence of mandibular osteomyelitis and bisphosphonate treatments but all other symptoms of this syndrome must be considered before starting any dental therapy. Because of the low incidence of this syndrome, there are no clear recommendations for its oral management. Nevertheless, it is possible to identify some important principles applicable to dental care in general. When articular pain occurs, modification of dental care may be necessary, such as reducing the length of the patient's dental appointment and their time maintaining an open mouth.

The complexity of these cases makes it essential to coordinate medical and dental care. A multidisciplinary team approach may be warranted within a hospital environment and perhaps augmented with the services of a physiotherapist when an affected patient experiences acute exacerbations. [START_REF] Skamagki | A systematic review on workplace interventions to manage chronic musculoskeletal conditions[END_REF] The onset of oral and perioral allergic reactions, such as nickel (as seen in the featured case report) may further complicate dental therapy. Moreover, the dentist may play integral role in the diagnosis of SAPHO syndrome. In a recent case report, mandibular pain, the first symptom reported, led to SAPHO discovery. [START_REF] Kikuchi | Mandibular osteitis leading to the diagnosis of SAPHO syndrome[END_REF] Elimination of any other dental cause of pain is essential and, in the case in question, CBCT was performed, revealing a sclerotic change and a periosteal reaction on the affected side of the mandible. [START_REF] Kikuchi | Mandibular osteitis leading to the diagnosis of SAPHO syndrome[END_REF]

F I G U R E 3

 3 Cutaneous test report revealing the hypersensitivity reaction to nickel sulfate at 72 hours F I G U R E 4 Extract from the identification card of the prosthesis, confirming the presence of nickel in the composition F I G U R E 5 Application of sterilized petroleum jelly body lotion to skin irritations during dental care

ACKNOWLEDGMENT

The authors thank Alice Cros for her help in oral surgery.

CONFLICTS OF INTEREST

The authors declare no conflict of interest.


Axel Paris

Adrien Peytavie

Eric Guérin

Jean-Michel Dischler

Eric Galin

• A Peytavie

• E Guérin

J-M Dischler

• E Galin

Modeling Rocky Scenery using Implicit Blocks

Keywords: Implicit surfaces, Terrain Synthesis, Procedural Modeling

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

Three-dimensional and vertical landforms such as cliffs, steep-walled canyons, crags, promontories, or overhangs are fundamental visual elements of scenic terrains. Despite the wide application of artificial terrains in the entertainment industry as well as simulation, and extensive research in this area, modeling truly 3D landforms with a high level of detail such as eroded karst tunnel networks, rocky cliff overhangs and arches with bare rock strata remains an unsolved problem. The vast majority of existing techniques addresses only 2 12 D

Fig. 1 Given an low resolution input terrain T , our method generates fractured mesoscale blocks featuring small-scale details inside a cubic tile, and replicates them realistically in the scene according to the geology of the different strata.

heightfield terrains which do not allow for an accurate representation of vertical landforms. Even though recent advanced techniques for modeling truly 3D terrains have been proposed [START_REF] Peytavie | Arches: A framework for modeling complex terrains[END_REF][START_REF] Becher | Feature-based volumetric terrain generation and decoration[END_REF][START_REF] Paris | Terrain amplification with implicit 3D features[END_REF], most of them have a limited resolution and only address the generation of large-scale features. Hyper-textures [START_REF] Perlin | Hypertexture[END_REF] can synthesize fractal mesoscale and microscale details over the surface of a coarse terrain, however, the self-similar appearance of the resulting bedrock, often corresponding to sandstone, lacks structure. Generating the mesoscale block structures and small-scale patterns of bedrock that appear on bare rocky terrain and the vertical walls of canyons, steep-walled cliffs or promontories, has received little attention. The challenge stems from the fact that the geometry of rocky surfaces results from different physical processes (including fracturing, percolation, and erosion), depends on the materials involved (such as limestone, dolomite, sandstone or basalt), and shows in a variety of forms (from regular hexagonal prisms to seemingly chaotic polyhedral shapes) and scales (from a few decimeters to meters).

In this paper, we propose an amplification method that enhances an otherwise smooth input terrain by au-tomatically carving geomorphologically consistent volumetric details over cliffs and overhangs. Our method consists in reproducing the fractures that exist in the bedrock to generate blocks that will be visible on the vertical walls of cliffs (Figure 1).

More precisely, the main contributions of our work are as follows: 1) we present an original geologicallybased fracturing process to generate realistic blocks of rock separated by fractures tiling space; 2) we introduce a novel gradient-based warping operator for adding surface details to implicit primitives which allows us to carve small-scale rock patterns from synthetic or real images over blocks; 3) we define a field function node compatible with any hierarchical implicit surface modeling framework for replicating the field functions characterizing the blocks, therefore implicitly replicating them over a volumetric terrain or a heightfield. Altogether, we provide a unified implicit framework for the representation of mesoscale (≈ 1 m) and small-scale (≈ 1 cm) bedrock details allowing to reproduce complex bedrock patterns and shapes. Moreover, our method provides multiple levels of control, allowing the user to author blocks and tune the placement rules.

Related work

Our method relates both to 3D terrain modeling and to hyper-textures or amplification techniques that enhance an existing coarse terrain representation with smaller-scale details. For a more complete coverage of terrain modeling techniques, the reader is referred to the review in [START_REF] Galin | A review of digital terrain modeling[END_REF].

Voxel representations were first employed for modeling volumetric rocky landforms such as cliffs [START_REF] Ito | Modeling rocky scenery taking into account joints[END_REF] by simulating fracturing between neighboring cells along rock joints. Stability analysis is then applied to remove disconnected cells. Besides memory and computational costs, the method does not scale to large terrains and does not address the distribution of joints crucial for synthesizing patterns observed in nature. Voxels were also used for simulating spheroidal and cavernous erosion producing volumetric structures such as hoodoos and goblins [START_REF] Beardall | Goblins by spheroidal weathering[END_REF][START_REF] Jones | Directable weathering of concave rock using curvature estimation[END_REF]. Limiting the range to small features allows to reproduce small-scale details on such specific landforms, but does not lend itself to simulating erosion on large terrains.

Implicit surfaces provide an alternative functionoriented mathematical framework for modeling 3D terrains. Peytavie et al. extend the concept of stacked layers, first introduced for erosion simulation, by incorporating water and air layers and thereby enabling the construction of caves and overhangs [START_REF] Peytavie | Arches: A framework for modeling complex terrains[END_REF]. The final smooth surface is defined as a convolution of these layers. Although more compact than voxels, material stacks need to be stored explicitly and are limited in terms of precision. Another approach consists in combining voxels and feature curves [START_REF] Becher | Feature-based volumetric terrain generation and decoration[END_REF] for modeling arches and overhangs. Again the method explicitly stores the voxels of the terrain making it memory demanding. Recently, a hierarchical implicit construction tree combining feature primitives [START_REF] Paris | Terrain amplification with implicit 3D features[END_REF] was proposed for modeling a vast variety of landforms including arches, caves, karsts, sea cliffs, Goblins by simulating erosion processes by an invasion percolation algorithm.

While effective for modeling general landforms, most existing terrain modeling techniques successfully generate smooth large-scale landforms but suffer from a lack of precision and fail at representing mesoscale and small-scale details. While procedural sum of scales-noise functions [START_REF] Ebert | Texturing and Modeling: A Procedural Approach[END_REF] can in theory compute an infinite amount of details, the self-similar geometries resulting from the fractal process do not capture the characteristic structures and patterns observed on real terrains. In contrast, our method generates an implicit tiling function combined with small scale gradient-based warping for representing the mesoscale structure and small-scale details conforming to observations in geomorphology.

A common approach often used in the entertainment industry consists in decorating a heightfield by distributing 3D meshes taken from an atlas of characteristic landforms over the vertical parts on the terrain. A specific technique for generating rock piles was proposed in [START_REF] Peytavie | Procedural Generation of Rock Piles Using Aperiodic Tiling[END_REF]. It consists in creating aperiodic rock tiles to avoid computationally intensive physical simulation for stabilizing rocks. Rock shapes are generated by computing the Voronoi cells of a set of seed points using a parametrized anisotropic distance. While this approach can reproduce rock piles found at the bottom of cliffs, it does neither accounts for geological correctness nor follows a joint model, and therefore fails at reproducing certain types of blocks. Ghost Tiles introduced in [START_REF] Guérin | Efficient modeling of entangled details for natural scenes[END_REF] use a densely populated tile with pre-computed intersections to synthesize the multitude of entangled objects fallen to the ground such as rocks, branches or leaves. Those methods do not address the synthesis of geologically consistent fractures and do not create realistic block structures.

Finally, textures without uv-maps can be used to add details terrains by displacement mapping [START_REF] Yuksel | Rethinking texture mapping[END_REF], but only few texturing methods can be applied to introduce real 3D features. Hyper-textures [START_REF] Perlin | Hypertexture[END_REF][START_REF] Ebert | Texturing and Modeling: A Procedural Approach[END_REF] often based on noise [START_REF] Perlin | An image synthesizer[END_REF] allow the synthesis of a theoretically infinite amount of volumetric details. Hyper-textures perturb an initial smooth volumetric model by adding a fractal density function or warping space. Although this method generates fractal surfaces with self-similar fea-tures such as powdery rocks or eroded limestone, it fails at reproducing block structures such as fractured sandstone or columnar basaltic shapes. Cellular textures [START_REF] Worley | A cellular texture basis function[END_REF] can be easily adapted to generalized Voronoi diagrams and synthesize volumetric blocks patterns. Discrete element textures [START_REF] Ma | Discrete element textures[END_REF] synthesize distributions of objects using a small input examplar, however, there is no guarantee to obtain geologically correct block structures. Texel mapping enhances the surface of an object with details stored in voxel grids and replicated around the surface of the input object [START_REF] Neyret | Modeling animating and rendering complex scenes using volumetric textures[END_REF]. This method produces 3D features on a thick layer only, requires an explicit parametrization of the surface, but fails at generating geomorphologically coherent mesoscale block structures, which is the focus of our work. An important aspect of our method is that it is compatible with these texturing techniques, which can be invoked to synthesize small and micro-scale details.

Contrary to previous approaches, our method is capable of reproducing geologically correct blocks enhanced with small-scale details and implicitly places and combines them over the original terrain to produce a detailed representation featuring the complex patterns observed in real landscapes.

Overview

Real terrains often feature complex rock formations in areas such as cliffs or caves. These landforms are the result of complex, interconnected physical processes that include glacial erosion, catastrophic rock collapses due to instability, or aeolian erosion. To avoid computationally intensive simulations, we propose a procedural approach to create block formations based on classifications used in geomorphology.

Polyhedral

Equidimensional Prismatic Tabular Rhombohedral Columnar Fig. 2 Categories of blocks identified in geomorphology corresponding to various fracture distributions.

Our work comes from the observation that fractures in bedrock form blocks exhibiting a variety of shapes of different sizes. Archetype structures such as prismatic, equidimensional or rhombohedral (Figure 2) have been identified in geomorphology [START_REF] Palmstrom | Measurement and characterization of rock mass jointing[END_REF][START_REF] Dearman | Engineering geological mapping. Butterworths advanced series in geotechnical engineering[END_REF]. A fracture is a break of continuity in the body of rock whose origin is natural. It most frequently occurs as fracture sets that are defined as a family of parallel, evenly spaced broken fractures that can be identified by analyzing their orientations, spacing, and physical properties. Regular and periodic distributions of fractures result in regular columnar blocks, whereas distributions with three major orthogonal directions result in equidimensional blocks; randomly distributed fractures produce polyhedral type. Fracturing the entire input terrain T would be computationally and memory intensive. Therefore, we propose a procedural tiling method. The goal is to compute cubic tiles containing geologically correct blocks that will be placed over the vertical parts of bare bedrock.

The overall terrain amplification process is composed of two steps (Figure 3). We first generate a set of blocks B i organized into a cubic tile denoted as C using a procedural fracturing approach based on a geomorphological classification (Section 4). These blocks B i are implicitly defined by scalar functions b i : R 3 → R, which allows to obtain a 3D volumetric representation of the mesoscale patterns and small-scale details using a new gradient-based warping operator.

The second step consists in amplifying a smooth input terrain T by replicating blocks over the bare vertical parts of the bedrock (Section 5). The input terrain can be any kind of 2 1 2 D heightfield (resulting from edit-ing, simulation, procedural generation or synthesized from examples, we refer to [START_REF] Galin | A review of digital terrain modeling[END_REF]), or a 3D terrain model (see Section 2). The new 3D detailed terrain model T is defined as an implicit surface whose scalar field function f is a construction tree combining the field function of the initial terrain f and the scalar functions of the blocks b i .

The cubic tile C is virtually tiling R 3 , and only the blocks B i that straddle the vertical parts of the terrain are replicated. To avoid explicit instantiation, we propose an original selective field function replication method inspired from [START_REF] Stolte | Infinite implicit replication: Case study for voxelizing and representing cyclical parametric surfaces implicitly[END_REF] that allows to virtually replicate the field functions b i of the blocks only over the vertical parts of the input terrain.

Block tile generation

The key process in the formation of blocks is fracturing. In geomorphology, fractures are represented and simulated using 3D discs that define the formation of blocks by breaking the continuity of the bedrock.

From this observation, we propose a procedural and controllable method to simulate the different types of block formations as found in nature. We address the generation of blocks tiling space. In the following section, we consider periodic tiling out of clarity, the generation of aperiodic tiling using Wang Cubes [START_REF] Culk | An aperiodic set of 13 wang tiles[END_REF] or Corner Cubes [START_REF] Lagae | An alternative for Wang tiles: Colored edges versus colored corners[END_REF][START_REF] Peytavie | Procedural Generation of Rock Piles Using Aperiodic Tiling[END_REF] is a direct technical generalization of this work. Therefore, we address the generation of blocks in a cubic tile C of size s (in our implementation we use a cube with size s ≈ 20 m). The algorithm proceeds in two steps as depicted in Figure 4.

Fracturing Block generation

Clusters G 4 Blocks generation pipeline inside a cubic tile C. We first compute a nearest neighbor graph G over a set of sample points P inside the tile. Fracturing discs D remove edges crossing fractures. We extract clusters G i as the set of disconnected sub-graphs, and generate implicit primitives B i for each cluster. For the sake of clarity, the method is depicted in 2D and edges of G are not represented.

Fractures D i Blocks B i Cubic tile C Points P k B i Field function b (p) i G i p Fig.
Fracturing. Starting from an initial set of points P = {p k } sampling the cubic tile, we compute the geometric nearest neighbor graph G over this set. A set of fractures D = {D i } where D i are discs is then generated using procedural rules. These fractures cut edges from the graph, thus creating connected sub-graphs called clusters and denoted as G i that will finally form the blocks.

Implicit primitive generation. The coarse geometry of the blocks is defined as the convex hull of the point sets for every cluster G i . For every block, we define a scalar function b i that computes a signed distance bound to the surface of the block. Its corresponding construction tree combines the planes of the convex hull using a smooth intersection operator. The convex blocks are responsible for the mesoscale details of the bare rock. We finally apply a new gradient-based warping operator for generating the small-scale volumetric details.

Fracturing

The fracturing process starts by generating a set of sample points p k inside the cube, using a Poisson sphere distribution. Experiments demonstrated that a regular sampling leads to 3D aliasing with unnatural axisaligned fractured shapes.

The Poisson radius r influences the size and shape of the blocks. Higher radius values yield tiles with fewer points inside, which in turn leads to blocks with coarser convex shapes and fewer polygonal faces after the fracturing process. For a cubic tile size of s ≈ 20 m, we generate points with a Poisson radius r ≈ 10 cm which leads to ≈ 14 000 points. After fracturing, each block contains between 80 and 150 sample points (see Table 1). We then compute the nearest neighbor graph G connecting the set of points P using r as the neighboring distance threshold, i.e. an edge between points p i and p j exists if p i -p j < r.

Fractures D i (c i , r i , n i ) are defined as discs in space characterized by their centers c i , radii r i and normal orientation n i . Geomorphological types are distinguished by the way fractures are distributed in the cube, i.e. by the following parameters: number of fractures, average radius size, distribution of disc centers, and relative orientations. Without loss of generality, discs centers c i are randomly generated in the cube using a Poisson sphere sampling with an average radius of ≈ 2 m. Both r i and n i distributions are tuned according to the block type (see Figure 5).

The user may control the fracturing process either by tuning the disc distribution parameters, or by placing specific fractures in the cubic tile, or by directly placing several authored blocks in the tile (see Figure 6), and the system will adapt. It is also possible to increase the overall amount of fracture by decreasing the Poisson radius of the fracture centers distribu-tion. Other disc sampling strategies could be used, however we found Poisson sampling practical for obtaining regular-pattern-free distributions with a practical control over the disc centers spacing.

The following paragraphs explain the different fracture distributions, as described in [START_REF] Palmstrom | Measurement and characterization of rock mass jointing[END_REF] and their control parameters.

Polyhedral Tabular

Equidimensional Rhombohedral Fig. 5 Different types of blocks generated with different disc distributions.

Equidimensional block type has three dominant sets of fractures, approximately orthogonal, with occasional irregular fractures, giving almost cubic-shaped blocks. Prismatic blocks are similar in shape but are formed under slightly more irregular fracture distributions. For these types, normals n i are determined using a random axis-aligned direction. Radii r i are randomly chosen in an interval given by a user parameter. In our experiments, we set the radii to ≈ 5m.

Polyhedral type shows irregular small blocks without explicit arrangement into distinct sets. This type requires more fractures due to the completely stochastic essence of the distribution, therefore we set the radius of the fracture centers distribution to ≈ 1 m and create more fractures.

Rhombohedral blocks have three (or more) dominant mutually oblique sets of fractures, giving oblique-shaped blocks. The parameters are the same as for equidimensional blocks, but with tilted axis-aligned directions to obtain diagonal orientations.

Tabular block type has one dominant set of parallel fractures orthogonal to a dominant axis direction, for example bedding planes, with other non-persistent fractures; thickness of blocks is much lower than length or width. Discs orthogonal to the dominant axis orthogonal have a large radius equal to the size s of the cubic tile C, whereas the other discs parallel to the two orthogonal axes occur less frequently, and have with small radii r i ≈ s/10.

Columnar type is composed of several (usually more than three) sets of continuous, parallel fractures. The length is much greater than other dimensions. The next step consists in removing edges cut by a disc D i and then create clusters G i of connected nodes p k . This is performed by using a greedy algorithm: starting from a random point, aggregation propagates through the graph until no more points can be connected. Formally, an edge p j p k is kept if it does not intersect any fracture:

∀ D i ∈ D, D i ∩ p j p k = ∅
Finally, we check that all the points in a given cluster are visible to each other, i.e. that fractures do not cut an edge connecting any pair of points in the cluster. This guarantees that clusters should not spread around fracture discs, which would create blocks that do respect the constraints. The complexity of the fracturing step depends on the number of fractures #D: a highly fractured cube will require more intersection tests between edges of the graph and the discs. Table 1 reports some statistics for the different types illustrated in Figure 5.

Rhombohedral Tabular

Implicit block generation

Recall that we aim at generating an amplified terrain model T as an implicit surface. Implicit surfaces allow us to create a unified volumetric representation for both the base terrain and the mesoscale and small-scale details of the bedrock. We create the scalar field b i from the previously computed clusters G i as a construction tree defined as the smooth-intersection of half-spaces forming a base convex shape, and then deformed using a gradient-based warping operator (Figure 7). We first compute the convex hull of each cluster and extract a corresponding polygonal mesh M i . We compute the planes H k = (o k , u k ) associated to every polygon of M i such that their normal u k should be oriented towards the exterior of M i . The corresponding scalar function of the convex base c i is defined as the smooth intersection of the half-spaces functions h k associated to planes H k . Every half-space is defined by the signed distance to the plane h k (p) = (p -o k ) • u k . The final implicit convex blocks B i are finally defined as:

b i (p) = c i • ω -1 (p)
where c i (p) denotes the function associated to the block B i , and ω -1 denotes a gradient-based warping operator modeling small-scale details. Should the points of the cluster G i be coplanar, or contain less than 3 points, we define the base geometry as a thick polygon, line segment or point primitive.

Smooth convex base Using a traditional intersection operator [START_REF] Wyvill | Extending the CSG tree -Warping, Blending and Boolean Operations in an Implicit Surface Modeling System[END_REF] defined as f A∩B = max(f A , f B ) creates gradient discontinuities (Figure 8) which prevents the use of gradient-based warping as the resulting field function would no longer be continuous. Therefore, we use the smooth intersection operator introduced in [START_REF] Barthe | Combining implicit surfaces with soft blending in a CSG tree[END_REF] over the n planes of the block B i . Without loss of generality, we consider the intersection in the case of two half-spaces A and B. The field function of the smooth 8 Smooth intersection generates a convex block shape with rounded edges and preserves the continuity of the gradient ∇c i , therefore allowing a correct gradient-based warping.

Continuous gradient

Smooth intersection Intersection

f A ∩ B Gradient discontinuities f A ∩ B Planes H k Fig.
intersection A ∩B is parameterized by a radius R and defined as:

f A ∩B = max(f A , f B ) -R k(f A , f B ) 3 /6 k(f A , f B ) = max(1 -|f A -f B |/R, 0)
Higher values for R lead to smoother shapes, whereas smaller values preserve sharp features. In our method, we set R = 25 cm. Since the smooth intersection operator is not associative, the order in which intersections are performed may influence the final scalar field. In practice, the impact of the ordering is limited and appears not to have any visual impact over the block shape. Different operators [START_REF] Gourmel | A gradient-based implicit blend[END_REF] could also be used, as long as they do not introduce gradient discontinuities. Surface details Adding surface details to implicit surfaces is a challenging problem as implicit surfaces do not provide an explicit parameterization. Existing methods rely either on interactive authoring [START_REF] Sugihara | Warpcurves: A tool for explicit manipulation of implicit surfaces[END_REF] or require an explicit parameterization of the implicit surface [START_REF] Zanni | Geometric Details on Skeleton-based Implicit Surfaces[END_REF], and do not lend themselves for generating the surface details of blocks. We introduce a new gradientbased warping operator for adding details to any implicit surfaces, taking inspiration from tri-planar projection [START_REF] Geiss | Generating complex procedural terrains using the GPU[END_REF]. This warping, applied for every block, does not require an explicit parameterization of the surface, and introduces small-scale volumetric details as displacements that enhance the model. Figure 9 shows the effect of warping with different relief functions d encoded as images. The operator is defined as:

ω -1 (p) = p + δ(p)
Let g(p) = ∇b(p)/ ∇b(p) denote the normalized gradient. Let γ i (p) : R 3 → R 2 denote the projection of p on the i-th plane, namely xy, xz and yz. The function δ(p) : R 3 → R 3 computes the 3D warping of p according to a 2D displacement function d : R 2 → R:

δ(p) = g(p) 3 i=0 α i • g(p)) • d • γ i (p)
The weighting function α i : R 3 → R weights the contributions of the three displacements d • γ i (p) according to the scalar product between the normalized gradient and the unit axis-aligned vectors u i : The function d : R 2 → R computes the displacement distance and can be effectively defined either as a procedurally defined turbulence, or from real displacement images. It is computed for every projection of the point p, thus three times. Figure 10 shows the effect of gradient warping for the tabular type.

α i (p) = |g(p) • u i |.

Terrain amplification

The amplification process aims at generating a modified terrain T = T ∪ R defined as the union of T and the replication R of some of the blocks B i in C over the bare vertical walls of the cliffs. We propose a new replication operator transforming the initial scalar field f into an amplified model f .

Infinite replication of a scalar field b i over the entire space, first addressed in [START_REF] Stolte | Infinite implicit replication: Case study for voxelizing and representing cyclical parametric surfaces implicitly[END_REF], can be obtained by directly computing b i with the modulo between the argument point p and the size s of the tiling cube: b i (p mod s) with p mod s = p -s p/s . In our case, the challenge consists of computing the scalar fields b i only for the blocks B i that straddle the terrain at certain positions, therefore according to the field function representing the terrain f . We employ a presence function e : R 3 → {0, 1} that evaluates whether a block B i should be replicated or not.

For every block B i , we define a corresponding anchor point a i that will be used to evaluate its presence.

Here we present the concept with only one anchor point per block out of clarity, but the method can be easily generalized for a set of anchor points. Recall that p/s denote the integer coordinates of the virtual cell containing p. Blocks are selectively replicated by computing the presence e function at the virtual anchor point in world space a i + s p/s . The replication operator is a function t which defines its field function as the union of all block field function b i times their presence function e i .

t(p) = max i b i (p mod s) e i (a i + s p/s )
The final function f of the terrain is defined as f = max(f, t).

More precisely, the evaluation of the replication operator at a point p is performed as follows. We evaluate in which cell of the grid lies p, using a modulo operation on the floating point coordinates of p. We compute the contribution of each block function b i , virtually translated in the cell for the point p.

If the distance from p to a border of a cell is less than a given threshold we compute the contribution in neighboring cells to account for blocks straddling C. Therefore, we compute the total contribution as the union between blocks in the current cell and 2, 4 or 8 neighboring cells.

In the case of a set A = {a k } of several anchor points, the presence function e i computes the percentage of anchor points a i of B i which satisfy geometric criteria. In our implementation, we replicate a block if more than half, i.e. #A/2, of the anchor points satisfy e(a k ) = 1.

The presence function can be any combination of a variety of criteria; here we briefly review some important ones that allow for the automatic placement of blocks over the vertical walls, and that provide control by prescribing a geological strata definition as presented in [START_REF] Paris | Terrain amplification with implicit 3D features[END_REF].

We first define criteria based on the distance to the surface: in our model, a block can be replicated only if enough anchor points a k are in a given distance range [v a , v b ] to the surface, i.e f (a k ) ∈ [v a , v b ]. To only replicate blocks on steep slopes and vertical walls, we compare the direction of the gradient with the up direction u z ; recall that g denotes the normalized gradient of the terrain, we set e(a k ) = 1 if |g(p) • u z | < ε, with ε the maximum slope parameter. Finally, we extend on the implicit geological function γ defined in [START_REF] Paris | Terrain amplification with implicit 3D features[END_REF] and define different bedrock material at different location in the scene. Therefore, an anchor point a k satisfies the repli-

Tabular Equidimensional

Fig. 11 A canyon amplified with equidimensional blocks located at the bottom of the canyon, and tabular blocks placed on the higher parts of the cliff walls.

cation criteria if the material γ(a k ) corresponds to the material of the underlying block B i . Note that the presence function is generic and can be easily extended to account for other criteria such as the presence of a water level, trees and obstacles.

Results

We implemented our method in C++ and all the scenes were generated on a desktop computer equipped with Intel R Core i7, clocked at 4 GHz with 16GB of RAM, and an NVidia GTX 970 graphics card. The implicit surface representing the amplified terrain was polygonized [START_REF] Wyvill | Data structure for soft objects[END_REF] and the resulting mesh directly streamed into Vue Xstream R to produce the final images (Figures 5,[START_REF] Ebert | Texturing and Modeling: A Procedural Approach[END_REF][START_REF] Geiss | Generating complex procedural terrains using the GPU[END_REF][START_REF] Gourmel | A gradient-based implicit blend[END_REF][START_REF] Guérin | Efficient modeling of entangled details for natural scenes[END_REF][START_REF] Ito | Modeling rocky scenery taking into account joints[END_REF][START_REF] Jones | Directable weathering of concave rock using curvature estimation[END_REF][START_REF] Lagae | An alternative for Wang tiles: Colored edges versus colored corners[END_REF]. Table 2 reports some statistics about the different scenes. Contrary to noise-based hyper-textures that only require a few dozens of parameters, our method needs to store the hierarchical implicit models of the different blocks; the required amount of memory remains small (less than a megabyte).

Control

Figure 5 a variety of types of blocks: different fractures distributions were prescribed (Section 4) and lead to different shapes such as polyhedral, rhombohedral or tabular blocks. The user can tune the parameters of different fractures within a tile, and our method automatically computes block shapes that adapt to these constraints; it is also possible to sculpt specific blocks and let the system automatically adapt and generate the remaining ones, as demonstrated by Figure 6, where the user placed specific tabular or rhombohedral blocks at the desired locations in the tiles.

Figure 11 illustrates user-control over the block placement: an equidimensional bedrock strata was defined by the user at the bottom of the cliff resulting in regular cliff walls. A second strata with tabular type was prescribed above the previous one and tabular blocks were automatically added to account for the specified material. In practice, it takes a few iterations for the user to tune the placement rules of the different block tiles for a specific scene; the main limitation remains the computationally intensive visualization of the final implicit surface (see Section 6.4).

Figure 11 also demonstrates the effectiveness of control based on the computation of the gradient ∇f for detecting the vertical parts of the terrain: no blocks were generated on the top of the plateau, thus keeping the ground flat where trees appear. Figure 14 shows different styles of blocks applied to a sea cliff scene, completely changing the overall mesoscale geometry of the final landscape.

Finally, micro-scale details are defined by using the gradient-based warping operator combined with different displacement maps as showcased in Figure 9. The relief function d can be painted by the user; procedurally defined by combining noise functions in construction tree of scalar primitives or scanned from real rock surfaces.

Comparison with other methods

While the sum of scaled-noise functions [START_REF] Ebert | Texturing and Modeling: A Procedural Approach[END_REF] can, in theory, produce an infinite amount of details, the selfsimilar geometries resulting from the fractal process do not capture the characteristic structures and patterns observed on real terrains. Figure 12 shows a comparison of different methods used to add details onto a sphere. Hyper-textures [START_REF] Perlin | Hypertexture[END_REF][START_REF] Ebert | Texturing and Modeling: A Procedural Approach[END_REF] may generate holes and disconnected parts, star-shaped primitives with a radial turbulence [START_REF] Paris | Terrain amplification with implicit 3D features[END_REF] avoids artifacts but both methods lack geological structure. In contrast, fracturing generates geomorphologically consistent blocks, and gradient-based warping allows for the generation of small-scale details captured from real rocks. Hyper-textures [START_REF] Perlin | Hypertexture[END_REF] could also be used to add small-scale details, however gradientbased warping allows for better control over the displacement by using reliefs from real rocks. In terms of mesoscale details, our method improves existing volumetric terrain models. Global warping operators [START_REF] Gamito | Procedural landscapes with overhangs[END_REF] do not provide sufficient accuracy to reproduce the complex geometry of the vertical parts of the bedrock. Voxels and features curves-based approaches [START_REF] Becher | Feature-based volumetric terrain generation and decoration[END_REF] are limited by the grid resolution and generate smooth large-scale terrains.

While the system described in [START_REF] Paris | Terrain amplification with implicit 3D features[END_REF] can generate large scale landforms such as arches, overhangs, and simulate large scale erosion effects, the vertical surface of cliffs lacks geometrical patterns and bedrock details. Although primitive-based implicit surfaces can theoretically reproduce small-scale details, their modeling comes at the price of defining a huge number of small primitives in the construction tree, a memory-intensive process. In contrast, our method defines a memoryefficient tiling and replication function that implicitly tiles space with a union of blocks to amplify and add details to cliffs.

In spirit, generating and replicating instances resembles the Ghost Tiling approach [START_REF] Guérin | Efficient modeling of entangled details for natural scenes[END_REF], and the rock pile generation based on aperiodic tiling [START_REF] Peytavie | Procedural Generation of Rock Piles Using Aperiodic Tiling[END_REF] that instantiate rock meshes. Our approach differs in the sense that blocks are defined as implicit primitives and virtually replicated by using a replication operator combined with a presence function, which allows us to combine them to produce a consistent volumetric model. More-over, these methods would need to be improved to account for fracture distributions to guarantee the replicability of certain block types -which is one primary focus of our work.

Compatibility with other techniques

An important aspect of this work is that it is compatible with other terrain modeling techniques. Our method can benefit to implicit surface-based terrain representations such as the hybrid layer-stack convolution-surface framework described in [START_REF] Peytavie | Arches: A framework for modeling complex terrains[END_REF] or the primitive-based system [START_REF] Paris | Terrain amplification with implicit 3D features[END_REF]. Those models, designed for modeling large scale terrains, can be amplified with mesoscale details as demonstrated in Figure 13. Moreover, our method lends itself to amplifying heightmaps or procedurally defined elevations since a three-dimensional scalar field f : R 3 → R can be directly derived from the elevation function h : R 2 → R by defining f (p) = p z -h(p xy ).

Arches

Our method Finally, although we used implicit surfaces for modeling the complex mesoscale 3D features of terrains, the block generation algorithm (Section 4) can also produce a mesh representation M k for every block B k in the tile. In this context, our method can be streamed in production environments using mesh representations and is compatible with standard instantiation techniques such as the ones described in [START_REF] Peytavie | Procedural Generation of Rock Piles Using Aperiodic Tiling[END_REF] or [START_REF] Guérin | Efficient modeling of entangled details for natural scenes[END_REF] to distribute block meshes over the vertical parts of the input terrain.

Limitations

The block tile generation process (Section 4) can take up to 30s because of the many intersection tests between segments (linking two points in the tile) and fractures discs. However, time was not spent on optimizing Tabular Equidimensional Rhombohedral this step of our pipeline, since it is done once in pre processing. A more limiting property of our method is the fact that our block primitives can only model convex shapes. This could be resolved by developing primitives more suited for non-convex shapes, but it is beyond the scope of this paper and is left as future work.

Although implicit surfaces provide a powerful, sparse, and compact framework for generating complex volumetric structures, their visualization remains computationally intensive. In particular, the computation of the field function b i for one single block B i requires many half space distance computations, combined with a warping operator: blocks could be optimized with a limited number of primitives to reduce the overall number of field function queries.

Using a limited set of block tiles sometimes yields visible repetitions when applied to flat cliff walls parallel to the tiling directions. Aperiodic tiling using Wang cubes [START_REF] Culk | An aperiodic set of 13 wang tiles[END_REF] or Corner Cubes [START_REF] Lagae | An alternative for Wang tiles: Colored edges versus colored corners[END_REF][START_REF] Peytavie | Arches: A framework for modeling complex terrains[END_REF] would reduce visible artifacts, yet at the price of a larger number of precomputed tiles and, therefore, a more memory demanding implementation.

Finally, the procedural fracturing tool only provides an indirect control over the distribution of fractures, and thus the shape of the blocks: a better interactive material editing tool would allow the user to author a wider range of shapes conforming to his intent.

Conclusion

We have introduced an implicit representation of blocks for automatically generating mesoscale and microscale details on the vertical walls of rocky cliffs, crags, or promontories. Our method relies on the decomposition of a bedrock tile into blocks using a fracturing approach based on classifications in geomorphology. The implicit surface description allows generating realistic shapes enhanced with details organized into patterns and structures as observed in geomorphology. Our method is compatible with other terrain modeling techniques: the block generation algorithm can additionally produce triangle meshes, which in turn can be instantiated to enhance heightfields. We also introduced a warping operator that can be applied to any implicit surface model to add small-scale details.

A direct extension of this work would consist of generalizing with different types of blocks, in particular prismatic and columnar configurations that require specific fractures distributions. Another avenue of research worth investigating would be to generate largescale fractures spreading over hundreds of meters and shaping the vertical walls of cliffs by taking into account the underlying strata of the bedrock.

Fig. 3

 3 Fig.3Overview of the algorithm: during a pre-processing step, we generate cubic tiles containing 3D blocks of distinct types according to different fracture distributions, then given an initial coarse heightfield, strata definition and control regions, we automatically generate a 3D implicit model defined as a field function replicating the blocks for creating mesoscale and small-scale details.

Fig. 6

 6 Fig. 6 Example of user-control during the fracturing step: the user authored specific tabular blocks (left), and rhombohedral blocks (right) inside an equidimensional block tile.

Fig. 7

 7 Fig. 7 Simplified hierarchical representation of blocks B i : the base convex shape is defined as the smooth intersection of a set of plane primitives H, and procedurally warped.

Fig. 9

 9 Fig. 9 Examples of gradient-based warping with different reliefs applied to a sphere primitive.

Fig. 10

 10 Fig. 10 Tabular block tile without warping (left) and after gradient-based warping (right): this new operator allows to create highly detailed blocks without holes.

Fig. 12

 12 Fig. 12 Comparison of different methods used to add details onto an implicit sphere, with an ambient shading (top) and a high resolution texture (bottom).

Fig. 13

 13 Fig. 13 Side by side comparison showing a canyon with smooth overhangs produced by the layer-stack model, and the rocky amplified vertical-wall generated by blocks.

Fig. 14

 14 Fig. 14 Different styles of blocks generated on a cliff and arches. From left to right, tabular block style, equidimensional blocks and finally rhombohedral block style.

Table 1

 1

	Type	#P #D #B #P i Time
	Equidimensional 14 524	15 125	110	6.9
	Rhombohedral	14 589	24	78	161	12.0
	Polyhedral	14 532	44 100	140	25.4
	Tabular	14 554	30	82	143	16.3

Statistics for the generation tiles: number of points #P, number of fracture discs #D, number of blocks #B, average number of points #P i inside a block B i , and generation time (in seconds).

Table 2

 2 Statistics for the scenes: size (in meters), number of replicated blocks #R, and amount of memory (in megabytes) needed to store the field functions b i representing the blocks B i in the cubic tiles C.

	Scene	Figure	Size	#R Memory
	Sea cliff	1, 14	100 × 100	959	0.57
	Canyon	11	100 × 50 1041	0.54
	Sea spire	3	150 × 150 1680	0.53

CNRS, Université de Lyon, LIRIS, France

ICube, Université de Strasbourg, CNRS, FranceTabular

Acknowledgments

This work was part of the project HDW ANR-16-CE33-0001, supported by Agence Nationale de la Recherche.

Conflict of Interest

The authors declare that they have no conflict of interest.