Mihnea Tufis
email: mihnea.tufis@lip6.fr

Jean-Gabriel Ganascia
email: jean-gabriel.ganascia@lip6.fr

Adding semantics to comics using a crowdsourcing approach

With over 85 million print units sold last year only for the top 300 comic book titles, the comics industry is reaching a new high for the first time since 2007 (before the economical crisis). And this excludes the increasingly popular graphic novels or the increasingly more accessible digital comics. Digital comics however find themselves at a point where the difficulty of producing high quality content is directly related to the lack of semantic information linked to it. In this short paper, we propose a crowdsourcing approach for annotating comic books for the purpose of producing XML-based encodings to assist publishers, researchers or collection curators.

INTRODUCTION

Most current comic book formats don't go beyond the simple packaging of resources and metadata in comic book archive files (essentially a regular archive file; e.g., CBZ is a ZIP archive, CBR is a RAR archive) that different reading systems un-package and commonly render on a page-by-page basis. Others (e.g., comiXology, Aquafadas) have taken a step forward in meeting the increasing demands of consumption of comic books on mobile screen devices (smartphones, tablets, phablets) and have introduced a panel-to-panel type of navigation. Critics of the latter are complaining though about the inaccuracy of the rendering, the loss of the contextual information that a classic comic book reader has by examining the proximity of the current panel and even about the fact that the original artistic intention of the creator is basically violated.

In this paper we discuss the use of crowdsourcing for gathering annotations for comic books content. We believe that the fan-base comics have and the increased consumption of digital comics will enable us to tap into a large, passionate crowd with deep and diverse knowledge about the genre. As a result, we are generating a corpus of structured metadata as Comic Books Markup Language (ComicsML) encodings for each annotated page. We discuss their potential use in the publishing industry (for digital comics), in digital humanities research as well as in curating comics collections and databases.

COMICS AND DIGITAL FORMATS

A synthetic comparison of e-book formats1 reveals the multitude of digital publication formats available today and enumerates their capabilities with respect to what can be considered the key factors in terms of rendering digital comics: support for image, sound, interactivity, embedded annotations, word-wrap support or digital rights management. The linked table shows us that the most compelling case seems to be made by the EPUB format. In addition to this, EPUB seems to be the digital publication format supported by most mobile platforms (with Amazon's Kindle being the only one not integrating it).

The ever-growing popularity of EPUB as a publishing open standard for e-books has brought a series of developments concerning image based publications (i.e. magazines, atlases, art albums etc.) with a focus on comics. The latest come from the EPUB Advanced and Hybrid Layouts working group (AHL) in the form of one of their newest specifications -Region Based Navigation [START_REF] Conboy | [END_REF]. This is essentially generalizing the guided navigation by describing a way to break the page of a comic book into different regions of interest (ROI) as well as their rendering order. In addition to this, a Metadata Structural Vocabulary for Comics [START_REF] Ichikawa | EPUB Region Based Navigation Markup Guide 1.0[END_REF] is describing a minimal set of terms for annotating regions of interest: panel, panel-group, balloon, text-area and sound-area. The primary goal is assisting creators of digital content in using the combined metadata and reading system capabilities for an enhanced rendering of the content.

While tremendously helpful, the Region Based Navigation specification is taking care exclusively of the presentation layer (i.e. rendering a publication on a screen device). We believe that the artistic nature of classic comics as well as the great potential digital comics have already showcased allow us to go beyond simple content presentation. Comics have always engaged their readers and have always drawn passionate people not only into reading them. Comic books aficionados are proud of their collections, they establish an intimate link with their charactersthey want to learn their history (e.g., when did Batman appear for the first time, in which issue, which cartoonist drew him?), they follow their lives (e.g., how many times did Wolverine die, did Superman and Spiderman ever cross paths, where are the Watchmen meeting?). For all these cases and numerous others there is a clear need for richer descriptive metadata, which go beyond the presentation scope for which EPUB's Structural Vocabulary for Comics has been created.

CROWDSOURCING ANNOTATIONS FOR COMICS

We rely on the digital comic book readers as participants to our crowdsourcing experiment. The tasks we propose to them are organized around a set of questions regarding a series of problems related to comics on which computer algorithms are not performing well enough: page structure, identification of narration elements (characters, places, events), stylistic elements (bubble shapes, onomatopoeia, movement lines).

We aggregate the answers taking into account the reliability of a user in a given context (task difficulty, user experience with the task, type of question) and the agreement between the annotators.

A quality score is thus generated for each annotation with the best of them being selected as solutions.

We subsequently are able to generate the ComicsML [START_REF] Walsh | Comic Book Markup Language: an Introduction and Rationale[END_REF] encodings. This XML derived format is particularly useful since it's based on the already successful Text Encoding Initiative (TEI), allowing for declarations for page structure and composition, panels, characters, text (in all the varieties hosted by the comic medium: different types of balloons, diegetic text, onomatopoeia), events and even panel-to-panel transitions First, it could enable comics publishers to generate better content: the ComicsML content is aggregated to the EPUB layer, functioning as a semantic complementary layer to the (mainly) presentation layer offered by EPUB. In addition to that, the collected annotations can be directly used within the EPUB Rendition Navigation document [START_REF] Conboy | [END_REF], according to the recommendations from the EPUB Structural Vocabulary for Comics [START_REF] Ichikawa | EPUB Region Based Navigation Markup Guide 1.0[END_REF].

Digital Humanities scholars could use such encodings to optimize their research on comics: structured ComicsML encodings of comics or graphic novels could be used to explore the content in a multitude of ways. This applies not only to digital comics, but also to ComicsML encodings of physical comics.

Comics curators and collectors (professional or amateur) of physical or online collections could be provided with a structured content which could be more easily integrated within their collections or databases. This may assist them into enlarging public or private databases of characters or comics series and enable the creation of artefacts such as comic books dictionaries, indices, and dictionaries of onomatopoeia. A certain number of projects are already in place and could greatly benefit from the creation of comic books annotations. We mention here the Grand Comics Database (an online database of printed comics), Comic Book Database, Digital Comics Museum (a collection of scanned public domain comics from the Golden Age) or the Catalogue of the Cite Internationale de la Bande Dessinee et de l'Image from Angouleme.

The crowd and the platform

With several hundred different crowdsourcing platforms already up and running, we turned our attention to the Citizen Science Data Factory comprehensive technical report [START_REF] Azavea | Citizen Science Data Factory -Part 1: Data Collection Platform Evaluation[END_REF] and their recommendations for platform design.

Then, a project that became a great source of inspiration was the tagger, from the Public Catalogue Foundation (PCF). Set to annotate the entire collection of paintings from UK museums, PCF's tagger is a great application of crowdsourcing in the field of arts. With their concept in mind, we wanted to take a step in a different direction -producing annotated content for publishers, researchers or curators to use.

When designing a crowdsourcing project or an entire platform there are several factors that absolutely need to be taken into account [START_REF] Sharma | Crowdsourcing Critical Success Factor Model: Strategies to harness the collective intelligence of the crowd[END_REF]:

• What crowd are we targeting?

• Subsequently, how do we motivate it? What incentives will we use?

• How will tasks be designed?

Concerning motivation we use exactly the same incentives as PCF's tagger -offering the possibility to passionate comic readers to offer their (potentially undiscovered) expertise while doing something they love. Our hypothesis is that comic book readers are so passionate about their comics that they would pay a great deal of attention to details; giving them the opportunity to showcase their "skills" should be a strong enough incentive. In addition to that, offering step-by-step instructions, examples and tying them to the comic book theory might actually get nonfrequent readers more involved into the comic books world. For all we know, it's a virus that catches on fast! Our platform needed to be simple, yet offer us the possibility of creating a high quality user experience. The state of the art research pointed us to Crowdcrafting -a web platform dedicated to creating, managing and running crowdsourcing projects. Developed over PyBossa, a BOSSA API for crowdsourcing engines, the platform has a series of advantages: i) easy authentication (also available through 3rd party sites: Google, Facebook) -great feature, meant to get everyone started fast and prevent potential users from getting bored before even diving into the tasks [START_REF]CIA: Operation Ajax[END_REF], ii) the possibility to configure the user interface using open web technologies -the UI has been entirely developed in HTML and JavaScript (the use of Bootstrap is encouraged!), iii) the tasks are described using JSON objects or CSV files, and iv) the answers are stored as JSON objects. Besides the actual annotation, PyBossa enables us to collect useful information about the users working on each task and the time they are spending on it. This in turn allows us to develop our quality evaluation algorithm, which aggregates the annotations based on contextual information about both the task and the users.

General task design

Before designing our tasks we needed to elicit a set of requirements, based on the needs of the potential stakeholders (readers, publishers, researchers). Thus, the following aspects were taken into account. First, we looked at the limits of digital formats for comics (notably EPUB), which we discussed in Section 1. Second, we considered the ComicsML schema as a strong reference for some of the metadata we'd like to collect. Finally, principles of the comic book theory emerging from the work of masters of the genre such as Will Eisner's instructional books [START_REF] Eisner | Comics and Sequential Art: Principles and Practices From the Legendary Cartoonist[END_REF] and Scott McCloud's scholarly work in "Understanding Comics" [START_REF] Nowak | How reliable are annotations via crowdsourcing? A study about inter-annotator agreement for multi-label image annotation[END_REF] provided us with a greater understanding of the sequential art in general. We have designed our tasks with attention to concepts such as page composition, page structure, reading behaviour, ellipsis and transitions or the interplay between text and visual.

Structurally each task asks the annotator to answer a series of sequential questions about a comic book page. The questions are centred on 4 key themes (Figure 2).

Figure 1. The 4 key annotation themes

Considering the theory related to the unity of the panels of a comic book page [START_REF] Nowak | How reliable are annotations via crowdsourcing? A study about inter-annotator agreement for multi-label image annotation[END_REF] and to prevent annotators to spend too much time on a task [START_REF]CIA: Operation Ajax[END_REF], we decided to centre our tasks on the entire page, as opposed to presenting them with each panel. Also in order to keep the process fast and simple as in the case of the PCF tagger, the annotators are not allowed to return to a previous step.

Task description

Step 1 -Page structure

The annotators are presented with a simple interface in which they will have to make a choice between a set of suggested grid layouts. These layouts are the output of applying the automatic frame extraction algorithm developed by Rigaud et al. [START_REF] Sharma | Crowdsourcing Critical Success Factor Model: Strategies to harness the collective intelligence of the crowd[END_REF].

In addition, our mobile annotation app (see Figure 2) also allows the annotators to draw the grid layout as they wish, by tapping over the presented page.

Step 2 -Panel reading order

Sometimes the reading order of the panels of a page is clear. This happens in older comics in which a regular (e.g., the classic 3x3) grid is fairly obvious to read. As the comics evolved as a genre, so did the artistic means become more and more sophisticated. It is not rare to encounter difficult to read grids, as a way for the artist to challenge and engage the reader. At this stage we apply the previously identified grid over the presented page and we ask the annotator to tap / click over each panel in the reading order.

Figure 3. Panels reading order

This step is particularly important since we use the difficulty of interpreting a page as a proxy for the difficulty of the task. We compare it with the time the contributor is spending on the task and with the aggregate answers of the other annotators on the same task; thus we establish the quality of the current annotator in the given context and the quality of his answer.

Finally, considering the answers of steps 1 and 2 we can already initialise the ComicsML encoding of the current page with the tags related to the page structure and composition (i.e. insert the cbml:panel elements in the given order).

Step 3 -Character identification

At this step, we ask the crowd to simply enumerate all the characters they can identify in the current page. Characters can be identified by reading their names in the text, recognising them from experience or simply giving a general statement about the character (e.g., "masked man" may be referring to Batman). The difficulty here is that the page is taken out of the context of the whole comic book that might make it difficult for an annotator to name a secondary or referential character whose name has been mentioned on a different page. At this point we will be using state of the art symbolic learning algorithms to fusion generic with specific information (e.g. if Batman and "masked man" both have high quality scores, they will both denote the same concept and will be considered as valid annotations). In this case, upon detection of a potential fusion, our algorithm will suggest the more specific term, leaving the final word to the human annotator.

Step 4 -Places identification

At this step the annotators are asked to simply enumerate all the places they can recognise on the current page. We are particularly looking for named places (e.g., Gotham City, NY, planet Mars), but will also ask the annotator to signal any generic place that he might deem as important for the scenes in the page (e.g., "the interior of a bank" [in case of a robery], "inside a space ship" [in case of a space battle]). These are exactly the kind of very specific annotation tasks for which state of the art image recognition algorithms are expected to fail.

Step 5 -Events identification

This is yet another highly specific recognition task. Annotators are asked to describe the most important actions occurring in the page, or if they are willing the actions taking place in each of the panels. At this stage we expect a heterogeneous corpus of annotations and we have yet to figure out what techniques will we use to validate and finally aggregate them. This is a pivotal step, since the solutions generated here, together with the annotations obtained in steps 3 and 4 will be used to further build the ComicsML encoding of the page (i.e. inserting the characters attribute of the cbml:panel element, the events element, the places element).

Scott McCloud stresses the role of ellipsis ("the blood in the gutters" -the space between two panels, where the reader fills in the gaps between the events depicted in two adjacent panels) as an artistic mean for authors to engage their readers and describes a typology of these transitions [START_REF] Nowak | How reliable are annotations via crowdsourcing? A study about inter-annotator agreement for multi-label image annotation[END_REF]. It turns out ComicsML allows us to declare such transitions through the #ana attribute of the cbml:panel element. Our ambition is to use the events descriptions for each panel and try to automatically infer the type of transition between successive panels. This might prove to be of utmost importance to digital comics publishers (they exploit the ellipsis more and more, by adding different transition effects, animations and transformations between panels and pages) as well as comics scholars working in digital humanities (e.g. it is well known that traditional comics from different geographic areas -Japanese manga, American comics, Franco-Belgian bande dessinnee -have different frequency distributions for each type of such transitions).

Step 6 -Non-visual cues

Comics are a special medium, making use of the visual to depict all other non-visual senses, by making use of different drawing tricks:

• Smoke coming out of a cigarette may engage the reader's smelling sense

• Onomatopoeia form a particular language of their own; comics and especially manga authors have proven to be extremely creative when it comes to expressing different sounds via stylised text (e.g., "POW!" can be a punch, "BAM!" can be a gun being fired, the Japanese manga has developed a complex vocabulary of its own)

• Horizontal lines around a car can suggest the car is moving at high speed, while around a ball they express the simple movement of the ball At this step, we ask the users to identify precisely these 3 types of non-visual sensations and the way they are expressed: smell, sound and movement.

Digital comics publishers could largely benefit from this kind of annotations, by adding visual and sound effects to their content [START_REF]CIA: Operation Ajax[END_REF]. On the other hand, researchers could study, for instance, the drawing style of an author and his use of non-visual cues, and go as far as creating onomatopoeia dictionaries for comics (to our knowledge, such dictionaries already exist for manga, but not for American or European comics; curiously enough the complexity of the meaning of the onomatopoeia in Japanese manga far exceeds that of its American and European counterparts).

Step 7 -Putting everything together

At this step, annotators are presented with all the previously identified narrative elements (characters, places, non-visual cues) and are asked to simply drag and drop them over the panels in which they appear.

At the end of this stage, we should be able to generate a reasonably complete ComicsML encoding of the current page (see Figure 3).

A note on the quality of annotations

In this section we briefly discuss our method for assessing the quality of crowdsourced annotations, prior to their aggregation.

To this respect, recent body of work in the field of crowdsourcing focuses on methodologies, factors and measures for crowdsourced annotations. The problem seems to be that there isn't a one size fits all approach. Not for developing, nor for assessing the success of a project. Each project, depending on the crowdsourced tasks requires tinkering with at least two parameters: the size of the crowd and the time required for the project to converge to satisfactory results.

Alonso and Mizzaro [START_REF] Alonso | Using Crowdsourcing for TREC assessment[END_REF] use K-statistics to measure interannotator agreement (IAA) and compare experts to non-experts on binary relevance tasks, Snow et al. [START_REF] Snow | Cheap and Fast -But is it Good? Evaluating Non-Expert Annotations for Natural Language Tasks[END_REF] obtain good results with the same kind of measures for 4 different types of tasks (affect recognition, word similarity, textual entailment, word sense disambiguation), but do encourage to model the bias of individual annotators and correct it. Feng et al. [START_REF] Feng | Acquiring High Quality Non-Expert Knowledge from On-demand Workforce[END_REF] as well as Kazai and Milic-Frayling [START_REF] Mccloud | Understanding Comics -The Invisible Art[END_REF] work on relevance assessment tasks and use inter-annotator agreement measures, but while the former suggests the removal of outlier annotators, the latter propose an approach for modelling trust for annotators as a function of the IAA, their practical involvement in offering comments to tasks and the degree to which they accumulate knowledge while performing tasks.

Sheng et al. [START_REF] Sheng | Get Another Label? Improving Data Quality and Data Mining Using Multiple, Noisy Labelers[END_REF] use the majority voting system and try to model the uncertainty associated to each label to determine whether further labelling is required or not; however, in response to their strict assumptions about their crowd, Donmez et al. [START_REF] Donmez | Efficiently Learning the Accuracy of Labeling Sources for Selective Sampling[END_REF] suggest to work with unknown label accuracies, and in turn estimate the accuracy of the labellers.

Considering crowdsourcing image annotations, Nowak and Lukashevich [START_REF] Rigaud | Robust text and frame extraction from comic books[END_REF] use predictors in order to place a new annotation in a ontology of concepts and then measure the ontology distance between the annotation and a gold standard corresponding concept. Later, Nowak and Ruger [START_REF] Nowak | Multilabel Classification Evaluation using Ontology Information[END_REF] combine different evaluation metrics (ontology score, equal error rate, area under the curve, ranking correlation) with classic inter-annotator agreement and conclude that while generally reliable the IAA results are not consistent over the different evaluation metrics. According to them, finding a more robust quality measure for measuring agreement in image annotation tasks is still an open research question.

We base our approach on the idea that each annotator performs differently on different tasks according to his motivation, level of expertise and task difficulty. Moreover, we expect annotators to become better in time -we expect an increase in the level of expertise for the different types of tasks.

The quality of an annotator a i on task t is a function of the difficulty of task t and expertise of annotator a i on the task.

Q annotator (a i , t) = f (difficulty(t), expertise(a i , t))

The quality of the annotations corresponding to task t is given by the inter-annotator agreement between annotators from A, working on task t.

Q annotation (t) = IAA (A, t),

where A = {aj, Q annotator (a j , t) > s} is the set of annotators for which the annotator quality on task t is above a certain threshold s.

CONCLUSIONS AND FUTURE WORK

The practical motivations behind our work can first be found in the increased market share of comics in the publishing industry (and the constant increase of digital comics there within). Second, taking into account the increased interest from the part of digital humanities scholars with respect to comics and graphic novels as well as the special interest collection curators (digital or physical, amateurs or professionals) take into collecting, classifying or searching comic books corpora, we have stressed the lack of, and therefore necessity for a standardised, digital encoding format for comic books, graphic novels, manga or bandes dessinnees.

It is difficult however to develop automatic information retrieval solutions to populate such ComicsML encodings. We have thus shown how crowdsourcing the resulting tasks over the web, we can involve a potentially large number of a-priori non-expert annotators, working on their passion and expertise with reading and experiencing comics.

We have presented the general architecture of our system, as well as details of how we have designed our tasks, having in mind three main aspects: the limits of current digital comic book formats, the specifications behind the ComicsML metadata schema and theoretical principles of the comics as a genre. For each step we have discussed the challenges we have faced or will be facing as well as the way the collected results will merge into the final ComicsML encoding.

Finally, we have brought to our readers' attention the sensitive topic of crowdsourced data quality. Thus, based on state of the art research we have introduced the general idea behind our approach to annotation quality: estimating context-dependent annotator quality, which in turn is used to determine the quality score for a given annotation.

Future Work

At the time this paper was submitted, we have finished the prototype development for both the mobile app and the web application. At the same time, we are closing in on refining our data quality algorithm that should be ready in the following month.

Considering the corpus of comic book pages, we are currently negotiating the use of copyright-free comics from the online Digital Comics Museum, as well as licensed digital comics from one of our commercial partners.

Based on the approach of Feng et al. [START_REF] Feng | Acquiring High Quality Non-Expert Knowledge from On-demand Workforce[END_REF] our methodology for collecting data will comprise of two steps: a calibration step with the purpose of establishing tasks typologies, the number of annotators required for each tasks as well as enough testing data to calibrate our quality algorithm should be released before June 2015; the second, large-scale release is to be launched before September 2015. However, the first ComicsML prototype encodings and hopefully the first hybrid EPUB+ComicsML digital comic book should be ready as early as the summer of 2015.

Figure 2 .

 2 Figure 2. UI for Step 1 -Page Structure

Figure 4 .

 4 Figure 4. A fragment of the ComicsML encoding for the page presented above

http://en.wikipedia.org/wiki/Comparison_of_e-book_formats#Comparison_tables

Columns on Last Page Should Be Made As Close As

Possible to Equal Length


Raluca Vlad-Debusschere

Nathalie Guyader

Anne Guérin-Dugué

A bio-inspired model of central and peripheral vision for scene categorization

La vision périphérique a été souvent négligée dans les modèles, qui simulent pour une grande partie la vision centrale, et dans les expériences comportementales, qui étudient majoritairement les capacités de la vision centrale. A partir d'un modèle de vision centrale déjà existant, nous testons si l'ajout d'un filtrage spatialement variant modélisant la diminution d'acuité visuelle en périphérie nous permet de répliquer des performances humaines de catégorisation de scènes présentées à différentes excentricités. Les résultats obtenus correspondent aux résultats comportementaux pour une sous-base d'images sans être généralisables à la totalité de la base. Nous expliquons ces résultats par le fait que notre modèle ne prend pas en compte l'information de couleur et utilise uniquement l'information de luminance pour décrire les images.

Abstract -Peripheral vision has been left aside both in modeling approaches and in psychological experiments, with the majority of existing studies mainly dedicated to our central vision exclusively. Using an existing model of central vision, we investigate whether adding a spatially variant filter simulating the decrease of visual acuity toward periphery allows us to reproduce human performances in a task of categorization of scenes positioned at different eccentricities in the visual field. The results we obtain fit human performances for a subset of images of natural scenes, however, for the moment these results cannot be extended to the entire database. We explain this result by the fact that the human performances were measured using colored scenes and the proposed model only processed the luminance information.

Introduction 1.Context

Most models of the human visual system (HVS) only simulate the central vision. However, a better understanding and quantification of the capacities of our peripheral vision is crucial, for example, for people suffering from age-related macular degeneration (AMD), who are subject to partial or complete loss of their central vision and use their peripheral vision in everyday life. Thus, understanding the ability of performing various tasks in function of the eccentricity of the visual stimulus could give comprehensive insights on how visually impaired persons perform their everyday life activities and, more interestingly, on how new devices could be developed for facilitating these activities.

In this paper, we present a simplified bio-inspired model of the HVS, adapted to both central and peripheral vision. We illustrate the results obtained with this model compared to behavioral data on a visual task of scene categorization at various eccentricities.

Existing models

The decomposition of visual content with cortical-like filters using a bank of Gabor filters has already been exten-sively used for reproducing the performances of the HVS on various tasks, from texture segmentation [START_REF] Guérin-Dugué | Texture segmentation using pyramidal gabor functions and self-organising feature maps[END_REF], to more complex image categorization [START_REF] Oliva | Modeling the shape of the scene: A holistic representation of the spatial envelope[END_REF], [START_REF] Guyader | Image phase or amplitude? rapid scene categorization is an amplitude-based process[END_REF]. Other models have gone further on the modeling of the HVS by adding the pre-processing performed by retinal cells [START_REF] Marat | Modelling spatio-temporal saliency to predict gaze direction for short videos[END_REF], [START_REF] Benoit | Using human visual system modeling for bio-inspired low level image processing[END_REF]. However, these models have only focused on the central vision and are thus only able to reproduce the performances of the HVS for stimuli visualized in the center of the visual field.

In the present article, we propose to extend these previous results, by taking into account the fact that our central vision has a high acuity whereas our peripheral vision has a lower visual acuity. Note that in this model we only investigate the processing of static stimulus without studying the peripheral vision particularly efficient to detect moving objects. Hence we modeled the visual processing made by our HVS for a stimulus appearing anywhere in the visual field.

The proposed model

The proposed model is organized in two main layers, the retina and the cortex. Before these two layers, we include a spatially variant (SV) processing layer to model the loss of visual acuity with the increasing eccentricity of the visual stimulus. This pre-processing step takes into ac-count the position of the visual stimulus in the visual field and consequently establishes the degree of precision with witch the stimulus is further processed. The structure of the proposed model is schematized in Figure 1.

Input

The input of the model has to be an image in a graylevel format, since for the moment we only focus on the luminance information, without taking into account the chrominance. Along with the input image, its visualization distance must be provided, since this parameter determines directly its size in the visual field and therefore some of the parameters of the subsequent processing.

SV processing

The SV processing layer not only models the non-uniform density of retinal photoreceptors but also the ratio of retina output cells for input cells that dramatically decreases with eccentricity. The aim is to take into account the loss in visual acuity from the central retina towards the periphery of the visual field [START_REF] Sere | Nonhomogeneous resolution of images of natural scenes[END_REF].

It has already been shown that the retinal and cortical cells distribution is reflected by experimental data on the sensitivity to contrast of the HVS [START_REF] Banks | Peripheral spatial vision: limits imposed by optics, photoreceptors, and receptor pooling[END_REF]. Thus, the SV processing consists of applying the contrast sensitivity curve of [START_REF] Geisler | A real-time foveated multiresolution system for low-bandwidth video communication[END_REF], which was obtained by modeling the data in [START_REF] Robson | Probability summation and regional variation in contrast sensitivity across the visual field[END_REF] and [START_REF] Banks | Peripheral spatial vision: limits imposed by optics, photoreceptors, and receptor pooling[END_REF]. More precisely, we use this curve as it was reproduced in [START_REF] Ho-Phuoc | Développement et mise en oeuvre de modèles d'attention visuelle[END_REF]:

contrast threshold (e) = A α α + e , ( 1
)
where e is the eccentricity in degrees of visual angle at which the image is visualized. The A constant related to the visualization distance is set to 1 and the α parameter controlling the decrease in contrast is set to 2.3 • . This law thus explains the contrast to be perceived when visualizing an image at a given eccentricity. Its graphical illustration is given in Figure 2. By using this contrast sensitivity curve and by considering the parameter giving the position of the input image in the visual field, we are able to filter this image accordingly, leading to a visual content similar to what an observer would see if the image was displayed in his or her field of view.

Retina

The retina layer reproduces the functioning of the principal types of cells in the human retina. Hence, the retina processing step corresponds to a series of digital filters, as explained in [START_REF] Hérault | Vision: Images, Signals and Neural Networks Models of Neural Processing in Visual Perception[END_REF]. Such processing leads to local contrast enhancement and spectral whitening as illustrated by Figure 3.

Cortex

The cortex layer models the functioning of some cortical cells. These cells have been found to preferentially respond to precise orientations and spatial frequencies of the visual stimulus [START_REF] Hubel | Receptive fields, binocular interaction and functional architecture in the cat's visual cortex[END_REF]. Moreover, it has been shown that the response of these cells can been successfully modeled by Gabor filters [START_REF] Daugman | Uncertainty relation for resolution in space, spatial frequency, and orientation optimized by twodimensional visual cortical filters[END_REF]. A specific type of such filters, the log-Gabor filters, which have the profile of Gabor filters viewed on a logarithmic scale, was found to be particularly adequate for processing natural images.

Therefore, in order to simulate a column of cortical cells sensitive to multiple orientations and spatial frequencies, we used a set of log-Gabor filters with predefined parameters. The frequency response of each of these filters is defined as:

G(u, v) = exp - log(u /f 0 ) 2 2σ 2 u + v 2 2σ 2 v , (2)
where f 0 is its central frequency, (u , v ) are obtained by rotating the filter axes, (u, v), with an angle θ, and σ u and σ v are the standard deviation values of the Gaussian in the u and v directions, determined as:

σ u = log 2 B ξ b 4α and σ y = f 0 tan Ω 2 2α , (3)
with B ξ b , the radial band in octave, set to 1.

The filter bank configuration we use is that of 20 log-Gabor filters, capturing 4 different orientations and 5 different spatial frequencies. The maximum central frequency is 0.25 and the central frequencies of the following k th filters are computed using f k = 0.25/2 k . This configuration is illustrated in Figure 4.

Output

To summarize, for any input image, the model produces the output of the layers of filters presented. Thus, as a last step, by applying the series of 20 cortex-like filters to the retina-processed image, a decomposition into 20 cortical features is obtained. As a consequence, for an input image, the model allows the computation of a 20-element numeric descriptor, where each value represents the total energy of a cortical feature.

Scene categorization

Scene descriptors as tho ones provided by the proposed model might be efficient to predict the category of a scene or at least the gist of a scene [START_REF] Oliva | Modeling the shape of the scene: A holistic representation of the spatial envelope[END_REF], [START_REF] Guyader | Image phase or amplitude? rapid scene categorization is an amplitude-based process[END_REF].

The aim of our model is to test whether such results might be extended to scenes presented in the periphery. To validate our model we used behavioral results on large eccentricity scene categorization described below.

Experimental data

The experimental data that we consider as ground-truth reference for our computational model is the data collected during Experiment 1 in [START_REF] Boucart | Scene categorization at large visual eccentricities[END_REF]. This data corresponds precisely to a natural vs. urban categorization task. In this experiment, observers were attributed a target category, i.e. natural or urban, then watched random natural-urban or urban-natural pairs of images, with one image displayed to the left and the other to the right of the central fixation point that observers were supposed to fixate, at the same eccentricity. For each pair of images, they voted left or right, in function of the side where they saw the target category scene. The images were displayed at 10 • , 30 • , 50 • , and 70 • of horizontal eccentricity. The results obtained in this study consist of mean percent correct responses obtained in function of the horizontal eccentricity at which the visual stimuli have been displayed. They are plotted in Figure 5.

Test data set

For comparing the performances of our model to those obtained experimentally, we use the same image data set as in [START_REF] Boucart | Scene categorization at large visual eccentricities[END_REF]. Hence, we consider 200 images visualized at a distance of 2.1 m and covering 20 • × 20 • of eccentricity in the visual field. Half of the images are natural and half are urban.

Method

We perform the scene categorization using a machine learning approach with a comparison of distances between 20-element vectors (our image descriptors). We perform the training procedure on the scenes processed as visualized at 0 • of eccentricity and the testing procedure on the same images, but processed as visualized at the eccentricities used in the original experiment: 10 During the training procedure we computed two mean descriptors, one for the natural images and one for the urban images, using the 20-element descriptors obtained on the training images of each of these two classes.

Then, during the testing procedure, we test the same trial configurations as those used for the 12 participants of the study. We thus consider the target category attributed to each participant and, for each natural-urban pair of images visualized by this participant, we apply the decision rule relative to that target category. This rule decides which among two images belongs to the target category by choosing the shortest euclidean distance between the mean descriptor of the target category and each of the two image descriptors.

Results

The results are shown in Figure 5.

Figure 5 --the experimental results in [START_REF] Boucart | Scene categorization at large visual eccentricities[END_REF]; and * -the computational results obtained with our model on the 200 images and 20 images data sets, respectively. Despite the fact that the mean percent correct responses obtained with our model are inferior to the experimental ones, the decrease in performance with eccentricity approaches the slope of the experimental results. The major factor influencing the low percentages of correct responses obtained seems to be the large variability of the images in the data set under study, visible in the large variability of the spectral features inside of each category. The data set from [START_REF] Boucart | Scene categorization at large visual eccentricities[END_REF] contains, especially for the urban category, images of various contents, which are to be argued whether representative for the category they represent. To illustrate this point, we also show in Figure 5 the results obtained with our model, by following a similar procedure, on a selection of only 20 images from the initial data set. This small test data set was selected so that the 10 images of each category would unambiguously represent the two categories. The results of this test illustrate how, for images that are characteristic of the two categories, the bio-inspired model of the HVS that we propose reaches very good categorization performances.

Conclusion

The study presented in this article is work in progress. At present, we could show that the decrease in performance of the HVS with eccentricity can be reproduced with the bio-inspired model proposed, however more insight is necessary on the influence of the data set content on the results.

An interesting perspective envisaged is the extension of the current model by taking into account the chromatic information as well and a more in-depth analysis of the luminance approach vs. the chrominance approach.

Figure 1 -

 1 Figure 1 -The structure of the proposed model.

Figure 2 -

 2 Figure 2 -The contrast sensitivity curve of [8] that explains the decrease in visual acuity in peripheral vision. The 0 • eccentricity point corresponds to the center of the retina and subsequently to the fixation point.

Figure 3 -

 3 Figure 3 -An example of photographed image (left) and the profile of its amplitude spectrum (right) before (top) and after (bottom) the retina processing.

Figure 4 -

 4 Figure 4 -The contours of the filter banks at half-height and 99% height of the amplitude spectrum (left) and their profile along the horizontal spatial frequency axis (right).


Olivier Brunet
email: olibrunet@free.fr

An Intrisic Topology for Orthomodular Lattices

We present a general way to define a topology on orthomodular lattices. We show that in the case of a Hilbert lattice, this topology is equivalent to that induced by the metrics of the corresponding Hilbert space. Moreover, we show that in the case of a boolean algebra, the obtained topology is the discrete one. Thus, our construction provides a general tool for studying orthomodular lattices but also a way to distinguish classical and quantum logics.

Introduction

The study of quantum structures (also called quantum logics [START_REF] Dalla Chiara | Quantum logic[END_REF]) relates to algebraic order-theoretic structures (such as orthomodular lattices, orthomodular posets, orthoalgebras [START_REF] Foulis | Filters and supports on orthoalgebras[END_REF], etc.) abstracted from Hilbert lattices, that is the collection of closed subspaces (or equivalently, of projection operators) of a Hilbert space. But Hilbert lattices are also much more than just order-theoretic objects, since they possesses a rich topological structure. It is thus reasonable and tempting to study the possibility of defining topologies on more general quantum structures.

There already exists some litterature concerning topology and quantum structures. One can mention study of orthomodular lattices which are also topological lattices [START_REF] Greechie | Representations of locally compact orthomodular lattices[END_REF][START_REF] Greechie | Profinite orthomodular lattices[END_REF][START_REF] Pulmannová | Block-finite orthomodular lattices[END_REF] or of topological orthoalgebras [START_REF] Wilce | Compact orthoalgebras[END_REF][START_REF] Wilce | Symmetry and topology in quantum logic[END_REF][START_REF] Wilce | Topological test spaces[END_REF]. However, there does not exists general results concerning the possibility of defining a topology on orthomodular lattices. This remark motivates the present article.

In the next section, we present a geometric lemma which relates the metrics of the projective sphere of a Hilbert lattice to lattice-theoretical considerations. We then show how this result can be used to define a topology, first on some particular atomic orthomodular lattices and then on any orthomodular lattice. We also show that the obtained general construction permits to recover the metric-induced topology in the case of a Hilbert lattice. Finally, we prove that in the case of a boolean algebra, this construction leads to the discrete topology. As a result, our topological construction provides, together with a general tool for studying orthomodular lattices, a way to discriminate classical boolean structures from some interesting quantum structures.

A Mathematical Preliminary

We first show how it is possible to recover the metrics of the projective sphere of a Hilbert lattice in a purely lattice-theoretical way. This result is based on a geometric lemma first mentionned in [START_REF] Brunet | A priori knowledge and the kochen-specker theorem[END_REF].

A Geometric Lemma

Let us consider the Hilbert space R 3 , and let u and v denote two non-nul vectors such that u • v > 0. In an appropriate orthonormal basis e = {e 1 , e 2 , e 3 }, one can write:

u =   1 0 0   v =   cos θ sin θ 0   with θ ∈ ]0, π 2 [.
In the same basis, given a real ϕ, let us introduce

w ϕ =   0 cos ϕ sin ϕ   .
Let E ϕ be the plan spanned by u and w ϕ : E ϕ = span {u, w ϕ } and let π ϕ (v) denote the orthogonal projection of v on E ϕ . Finally, let v ϕ denote this projection after normalization:

v ϕ = π ϕ (v) ||π ϕ (v)||
Simple calculations show that:

v ϕ = 1 cos 2 θ + sin 2 θ cos 2 ϕ   cos θ sin θ cos 2 ϕ sin θ cos ϕ sin ϕ  
As a consequence, one has:

v ϕ • v ψ = cos 2 θ + sin 2 θ cos 2 ϕ cos 2 ψ + cos ϕ cos ψ sin ϕ sin ψ cos 2 θ + sin 2 θ cos 2 ϕ cos 2 θ + sin 2 θ cos 2 ψ
The study of this equality leads to:

Proposition 1 (Geometric Lemma) With the previous notations, if 0 < θ < π 2 , one has: { v ϕ • v ψ | ϕ, ψ ∈ [0; 2π]} = 3 cos θ -1 cos θ + 1 ; 1 Proof It is clear that the function ϕ, ψ → v ϕ • v ψ is a C ∞ function,
so that the set on the left-hand side of the equality has to be an interval. In order to find the extremal values, we need to solve the system:

       ∂(w ϕ • w ψ ) ∂ϕ = 0 ∂(w ϕ • w ψ ) ∂ψ = 0
Simple calculations show that:

       ∂(w ϕ • w ψ ) ∂ϕ = 0 ∂(w ϕ • w ψ ) ∂ψ = 0 ⇔ 2 cos 3 ϕ cos ψ sin 2 θ sin(ϕ -ψ) + cos 2 θ sin 2(ϕ -ψ) = 0 2 cos ϕ cos 3 ψ sin 2 θ sin(ϕ -ψ) + cos 2 θ sin 2(ϕ -ψ) = 0 ⇔ 2 cos 3 ϕ cos ψ sin 2 θ sin(ϕ -ψ) + cos 2 θ sin 2(ϕ -ψ) = 0 cos ϕ cos ψ sin(ϕ -ψ)(cos 2 ϕ -cos 2 ψ) = 0
Now, considering the second equality, one has:

cos ϕ cos ψ sin(ϕ -ψ)(cos 2 ϕ -cos 2 ψ) = 0 ⇔ cos ϕ = 0 or cos ψ = 0 or sin(ϕ -ψ) = 0 or cos 2 ϕ = cos 2 ψ ⇔ ϕ ≡ π 2 [π] or ψ ≡ π 2 [π] or ϕ ≡ ψ [π] or ϕ ≡ -ψ [π] If ϕ ≡ ψ [π], then v ϕ • v ψ = 1. If ϕ ≡ π 2 [π] or ψ ≡ π 2 [π]
, the first equality yields:

2 cos 3 ϕ cos ψ sin 2 θ sin(ϕ -ψ) + cos 2 θ sin 2(ϕ -ψ) = 0 ⇔ sin 2(ϕ -ψ) = 0 ⇔ ϕ ≡ ψ [ π 2 ]
In that case, v ϕ • v ψ equals either 1 or cos θ. Finally, if ϕ ≡ -ψ [π], then:

2 cos 3 ϕ cos ψ sin 2 θ sin(ϕ -ψ) + cos 2 θ sin 2(ϕ -ψ) = 0 ⇔ 2 cos 4 ϕ sin 2 θ sin 2ϕ + cos 2 θ sin 4ϕ = 0 ⇔ sin ϕ cos ϕ sin 2 θ cos 4 ϕ + 2 cos 2 θ cos 2 ϕ -cos 2 θ = 0 ⇔ sin ϕ = 0 or cos ϕ = 0 or sin 2 θ cos 4 ϕ + 2 cos 2 θ cos 2 ϕ -cos 2 θ = 0
In the third case, the equation sin 2 θX 2 + 2 cos 2 θXcos 2 θ = 0 has a single positive solution:

X = -2 cos 2 θ + 2 cos θ 2 sin 2 θ = cos θ 1 + cos θ . Thus, if ϕ ≡ -ψ [π]
, the first equality becomes:

2 cos 3 ϕ cos ψ sin 2 θ sin(ϕψ) + cos 2 θ sin 2(ϕψ) = 0

⇔ sin ϕ = 0 or cos ϕ = 0 or cos 2 ϕ = cos θ 1 + cos θ And consequently, v ϕ •v ψ equals either 1 (if cos ϕ = 0 or sin ϕ = 0) or 3 cos θ -1 cos θ + 1

(if cos 2 ϕ = cos θ 1 + cos θ ).
To sum things up, the values of v ϕ • v ψ obtained when ∂(w ϕ • w ψ ) ∂ϕ = 0 and ∂(w ϕ • w ψ ) ∂ψ = 0 are 1, cos θ and 3 cos θ -1 cos θ + 1 . We conclude by remarking that

3 cos θ -1 cos θ + 1 ≤ cos θ ≤ 1. Let us define f on [0, 1] by f (x) = 3x-1 x+1 . It is easy to verify that f (0) = -1, f ( 1 3 ) = 0, f (1)
= 1, f is strictly increasing and that f (x) < x. Moreover, let us define a sequence (c n ) by:

   c 0 = 0 c n+1 = f -1 (c n ) = 1 + c n 3 -c n
From its definition, it appears that (c n ) is an homographic sequence and one can express c n as a function of n:

∀ n ∈ N, c n = n n + 2 .
Finally, we define (θ n ) by:

∀ n ∈ N, θ n = arccos (c n ) = arccos n n + 2 .
Clearly, θ 0 = π 2 and lim n→∞ θ n = 0.

The Topology of P H from a Lattice Point of View

It is a common result that given a Hilbert space, its projective space P H is a metric space, hence it is a topological space. Its metrics is given by:

∀ A, B ∈ P H , d(A, B) = arccos |u • v| ||u|| ||v||
where u and v are vectors of H such that A = span(u) and B = span(v).

Thus, the metrics (and the topology) of P H can be directly derivated from the metrics of H.

The previous lemma suggests another way to define the topology of P H out of the structure of the Hilbert lattice L H of all closed subspaces of H. In order to present this, let us first introduce a few definitions.

Given a vector v ∈ H and a closed subspace E ∈ L H , let Π E (v) denote the orthogonal projection of v on E. Now, given two closed subspaces E and F , let E & F denote the closed subspace obtained as the union of the projection on F of all vectors of E:

E & F = { Π F (v) | v ∈ E} Lemma 2
Let H be a Hilbert space of dimension at least 3. Let A and B be in P H and n an integer such that d(A, B) ≥ θ n . Then there exists two closed subspaces A 1 and A 2 greater than A, such that

d(B & A 1 , B & A 2 ) = θ n-1 .
Proof Let u and u be two normalized vectors of H such that:

A = span{u} B = span{v} u • v = cos d(A, B)
Using previous notations, it is possible to define an orthonormal basis e = {e 1 ; e 2 ; e 3 ; . . .} of H such that:

u = e 1 v = cos d(A, B) e 1 + sin d(A, B) e 2
Now, for every ϕ ∈ R, let us define as previously:

w ϕ = cos ϕ e 2 + sin ϕ e 3 E ϕ = span{u, w ϕ } v ϕ = Π Eϕ (v) ||Π Eϕ (v)|| Then, span{v ϕ } = span{Π Eϕ (v)} = span{v} & E ϕ = B & E ϕ , with A ⊆ E ϕ .
From the hypothesis that d(A, B) ≥ θ n (or equivalently that cos d(A, B) ≤ cos(θ n )) and using the monotony of f , one has:

3 cos d(A, B) -1 cos d(A, B) + 1 = f cos d(A, B) ≤ f cos θ n ) = cos θ n-1
Using our Geometric Lemma, this implies that there exists two real numbers α and

β such that v α • v β = cos θ n-1 . Finally, with A 1 = E α and A 2 = E β , we get the expected result, as A ⊆ A 1 , A ⊆ A 2 and d(B & A 1 , B & A 2 ) = d span(v α ), span(v β ) = θ n-1 .
Proposition 3 Let H be a Hilbert space of dimension at least 3. For A and B in P H and n an integer, one has:

d(A, B) ≥ θ n ⇔ ∃ A 0 , B 0 , . . . , A n , B n ∈ P H :    A n = A and B n = B A k , B k ∈ & # (A k+1 , B k+1 ) A 0 ⊆ B ⊥ 0 where & # (A, B) = { B & A ′ | A ′ ∈ L H and A ⊆ A ′ }.
Proof This follows from an induction on n.

If n = 0, one has d(A, B) = π 2 ⇔ A ⊆ B ⊥ . For n ≥ 1, using lemma 2, if d(A, B) ≥ θ n , then there exists A n-1 and B n-1 in & # (A n , B n ) such that d(A n-1 , B n-1 ) = θ n-1 .
Proposition 3 shows that in dimension at least 3, for all A ∈ P H , the open ball B(A, θ n ) can be defined without any reference to the metric space of P H and thus to H. Instead, it can be defined using solely the structure of L H . Moreover, since lim

n→∞ θ n = 0, the collection {B(A, θ n )} n∈N is a neighbourhood base of x.
As the consequence, the usual metric-based topology of P H is equivalent to the topology generated by these neighbourhood bases.

This discussion suggests strongly that a similar construction can be defined for generalizations of Hilbert lattice, namely orthomodular lattices, since in particular the binary operation & used to express orthogonal projection is actually a usual orthomodular lattice operation called the Sasaki projection and defined as x & y = y ∧ (x ∨ y ⊥ ) [START_REF] Brunet | Representation systems and quantum structures[END_REF][START_REF] Dalla Chiara | Quantum logic[END_REF]. In complement to its definition, we give two important properties that the Sasaki projection verifies: Proposition 4 Given an orthomodular lattice L and three elements a, b, c ∈ L, one has:

1. a ≤ b ⇒ a & c ≤ b & c 2. a & b ≤ c ⇒ c ⊥ & b ≤ a ⊥ 3 A First Atom-Based Topology
In this section, unless stated otherwise, L is an atomic orthomodular lattice which verifies the following property, where At(L) denotes the set of atoms of L and ⊥ is the least element of L:

∀ a, b ∈ At(L) × L, a & b ∈ At(L) ∪ {⊥} Atom Projection
This property states that the Sasaki projection of a atom on any element of L is either ⊥ or an atom. It is easy to verify that if H is a Hilbert space, then L H is atomic and verifies this property.

A Topology on Atoms

Let us first define the following sequence of subsets of At(L) 2 :

R At 0 = a, b ∈ At(L) 2 a ≤ b ⊥ R At n+1 = a, b ∈ At(L) 2 ∃ a ′ , b ′ ≥ a, b : a & b ′ , b & a ′ ∈ R At n Intuitively, a, b is in R At n if
and only if two orthogonal atoms can be reached in n steps similar to that of proposition 3. And for a Hilbert space, this corresponds to d(A, B) ≥ θ n , as we will soon show.

Proposition 5 Here are some easy facts about R At n , with n ∈ N and a, b ∈ At(L):

R At n ⊆ R At n+1 a, a ∈ R At n a, b ∈ R At n ⇔ b, a ∈ R At n
Proof First, for all x, one has x ≤ ⊤ and thus

x & ⊤ = x, so that if a, b is in R At n , then a, b = a & ⊤, b & ⊤ ∈ R At n+1 .
To show the second point, for n ≥ 1, if a, a ∈ R At n then it is easy to show that a, a ∈ R At n-1 and, by induction, that a, a ∈ R At 0 which is not possible since a ≤ a ⊥ . The third point follows directly from the fact that it holds for R At 0 and from the symmetry of the definition.

The collection R At n specifies how much outspread atoms are. Furthermore, we define:

R At ∞ = ∞ n=0 R At n ∀ a ∈ At(L), ∀ n ∈ N, B At n (a) = b ∈ At(L) a, b ∈ R At ∞ \ R At n
These subsets B At n (a) will play a role similar to "open balls" in our topology. Here are a few facts following directly from the definitions and from proposition 5:

Proposition 6 If n 1 ≤ n 2 then B At n2 (a) ⊆ B At n1 (a). Moreover, one has b ∈ B At n (a) ⇔ a ∈ B At n (b)
and n B At n (a) = ∅. We now have the elements needed to define our first, atom-based topology: Definition 1 (Atom-Based Topology, version 1) Let T At (L) be defined as:

T At (L) = O ⊆ At(L) ∀ a ∈ O, ∃ n ∈ N : B At n (a) ⊆ O Proposition 7
The set T At (L) is a topology on At(L).

Proof Clearly, both At(L) and ∅ are in T At (L), and T At (L) is stable by arbitrary union. Now, let O 1 and O 2 be two elements of T At (L), and let a be an element of O 1 ∩ O 2 . By definition of T At (L), there exists two integers n 1 and n 2 such that

B At n1 (a) ⊆ O 1 and B At n2 (a) ⊆ O 2 . With n = max(n 1 , n 2 ), B At n (x) = B At n1 (x) ∩ B At n2 (x) so that B At n (x) ⊆ O 1 ∩ O 2 . This shows that O 1 ∩ O 2 is in T At (L).
Let us now turn to the case where L is a Hilbert lattice L H with dim H ≥ 3. We first give two technical results before showing that T At (L H ) and the metricsinduced topology on P H are equivalent.

& (a ∨ c) ≤ c ⇔ a & (b ∨ c) ≤ c Proof One has b & (a ∨ c) ≤ c ⇔ c ⊥ & (a ∨ c) ≤ b ⊥ . Now, since c ⊥ and c ∨ a are compatible, c ⊥ & (a ∨ c) = c ⊥ ∧ (a ∨ c) = a & c ⊥ . This leads to: b & (a ∨ c) ≤ c ⇔ a & c ⊥ ≤ b ⊥ ⇔ b & c ⊥ ≤ a ⊥ ⇔ a & (b ∨ c) ≤ c

Extending the Topology to the Entire Lattice

In order to generalize the definition of our topology to the whole lattice, we introduce the following collection of "balls":

∀ a ∈ L, B L n (a) = b ∈ L ∀ x ∈ At(a), ∃ y ∈ At(b) : y ∈ B At n (x) and ∀ y ∈ At(b), ∃ x ∈ At(a) : x ∈ B At n (y)
Here, given an element x of L, At(x) denotes the set of atoms below x. The sets B L n (a) enable us to define a topology on L the following way.

Definition 2 (Atom-Based Topology, version 2) Let T L (L) be defined as:

T L (L) = O ⊆ L ∀ a ∈ O, ∃ n ∈ N : B L n (a) ⊆ O Proposition 11 The set T L (L) is a topology on L.
Proof The proof is similar to that of proposition 7.

The next proposition shows that the first topology, defined on atoms, can be obtained from the topology on the whole lattice:

Proposition 12 One has T At (L) = { O ∩ At(L) | O ∈ T L (L)}.
Proof This follows from the definition of B L n (a) and the fact that if a is an atom, then At(a) = {a}.

We now turn once again to the case where the considered orthomodular lattice is actually a Hilbert lattice L H . The metrics induced on P H by the metrics of H can be generalized on L H by defining: 2 ). Thus, we have:

d L (a, b) = max max
B L n (a) = { b ∈ At(L) | 0 < d(a, b) < θ n } = { b ∈ L | 0 < d L (a, b) < θ n }
2. If a is not an atom, suppose that b is in B L n (a). From the discussion for the previous case, b cannot be an atom. And following the definition of B L n (a), one has:

∀ x ∈ At(a), ∃ y ∈ At(b) : 0 < d(x, y) < θ n and ∀ y ∈ At(b), ∃ x ∈ At(a) : 0 < d(x, y) < θ n
For x ∈ At(a), two subcases have to be considered. If x ∈ At(b), then we have:

(∃ y ∈ At(b) : 0 < d(x, y) < θ n ) ⇔ 0 < d π (x, b) < θ n
And if x ∈ At(b), then the condition ∃ y ∈ At(b) : 0 < d(x, y) < θ n is always verified since dim b ≥ 2. This implies that if a is not an atom, then:

B L n (a) = b ∈ L ∀ x ∈ At(a) \ At(b), 0 < d π (x, b) < θ n and ∀ y ∈ At(b) \ At(a), 0 < d π (x, a) < θ n = b ∈ L ∀ x ∈ At(a), 0 ≤ d π (x, b) < θ n and ∀ y ∈ At(b), 0 ≤ d π (x, a) < θ n = { b ∈ L | 0 ≤ d L (a, b) < θ n }
Thus, for a subset S ⊆ L and an element a ∈ S, then the following two statements

∃ θ > 0 : ∀ b, (d L (a, b) < θ ⇒ b ∈ S) and ∃ n : B n (a) ⊆ S are equivalent.
As a conclusion, the two topologies are equivalent.

The General Construction

In order to extend our topology to the general case, we first need to adapt the needed definitions in order to avoid any references to atoms. This is done using the tools described in the next section.

Operations on Lower Sets

We recall that given poset P , a subset I ⊆ P is called a lower set if and only if it verifies:

∀ x ∈ I, ∀ y ≤ x, y ∈ I
Let ℘ ↓ (P ) denote the lower sets of a poset P . In the following, we introduce a complementation and a closure operation on lower sets which will enable us to avoid direct reference to atoms.

Definition 3 (Complement of a Lower Set) For I ∈ ℘ ↓ (P ), we define its complement ¬I as:

¬I = { x ∈ P | ∀ y ≤ x, (y ∈ I ⇒ y = ⊥)}
We first present some easy fact about this operation:

Proposition 14 For I ∈ ℘ ↓ (P ), one has:

¬I ∈ ℘ ↓ (P ) I ⊆ ¬¬I ¬¬¬I = ¬I I ∩ ¬I = {⊥} Corollary 14.1 The operation I → ¬¬I is a closure operator on ℘ ↓ (P ).
This closure operator is not new and appears for instance in the domains of Kripke's possible worlds semantics of modal and intuitionnistic logic. In intuitionnistic logic, if a proposition p is interpreted by an ideal I, then its doublenegation ¬¬p is interpreted by ¬¬I [START_REF] Van Dalen | Intuitionnistic logic[END_REF]. Similarly, in S4 modal logic, ¬¬I is the interpretation of ♦p [START_REF] Chellas | Modal Logic, an Introduction[END_REF][START_REF] Hughes | A New Introduction to Modal Logic[END_REF]. In both case, the intuitive idea is the same:

an element x is in the closure ¬¬I if and only if every path starting from x downwards ends in I. This appears clearly in the following expression of ¬¬I:

¬¬I = { x | ∀ y ≤ x, ((∃ z ≤ y : z ∈ I \ {⊥}) ∨ y = ⊥)} ,
Proposition 15 Given an atomic poset P , a ideal I ∈ ℘ ↓ (P ) and an element x ∈ P , one has:

x ∈ ¬¬I ⇔ ∀ y ∈ At(x), y ∈ I Proof First, suppose that x is in ¬¬I and let y be in At(x). Since y ≤ x and y = ⊥, there exists a z such that z ≤ y and z ∈ I \ {⊥}. But since y is an atom, one has z = y and thus y ∈ I. We have shown that ∀ y ∈ At(x), y ∈ I.

Conversely, let x be in P such that x = ⊥ and ∀ z ∈ At(x), z ∈ I and let y be such that ⊥ < y ≤ x. For all z be in At(y), we have z ∈ At(x) and thus z ∈ I. As a consequence, we have shown that ∀ y ≤ x, y = ⊥ ⇒ ∃ z ≤ y : z ∈ I \ {⊥}.

Corollary 15.1 Let I be an ideal of an atomic poset P . If At(P ) denotes the atoms of P , then one has:

¬¬I ∩ At(P ) = I ∩ At(P )
This proposition and its corollary show that the complement operation and the derived closure operation provide an interesting basis for generalizing the definition of our topology.

Proposition 16 Given a poset P , a lower set I ∈ ℘ ↓ (P ) and an atom x of P , one has:

x ∈ ¬I ⇔ x ∈ I

Definition of the Topology in the General Case

Before turning to the definition of our topology, we introduce the smashedproduct of a poset which will avoid excessive trouble when dealing with ⊥.

Definition 4 (Smashed-product) Given a poset P with a least element ⊥, we define the ⊥-smashed-product P #2 as the poset with elements a, b ∈ P 2 a = ⊥ and b = ⊥ ∪ { ⊥, ⊥ } equipped with the point-wise partial order induced by P .

Obviously, if P is atomic, then so is P #2 , in which case At(P #2 ) = At(P ) × At(P ). In the following, we will consider the ⊥-smashed-product L #2 of an orthomodular lattice L. Of course, L #2 is a poset (and even a lattice) but not an orthomodular lattice.

We now have the tools to adapt our topology-related definitions to the general case.

Definition 5 (Topology) Given an orthomodular lattice L, we first define a collection {R n } telling how "far apart" elements of L are:

R 0 = ¬¬ a, b ∈ L #2 a ≤ b ⊥ R n+1 = ¬¬ a, b ∈ L #2 ∃ a ′ , b ′ ≥ a, b : a & b ′ , b & a ′ ∈ R n
Then, we introduce subsets B n (a) specifying which elements are close enough from a:

R ∞ = ∞ k=0 R k B n (a) = b ∈ L ∀ a ′ ≤ ⊥ a, ∃ a ′′ ≤ ⊥ a ′ : ∃ b ′ ≤ ⊥ b : a ′′ , b ′ ∈ R ∞ ∩ ¬R n and ∀ b ′ ≤ ⊥ b, ∃ b ′′ ≤ ⊥ b ′ : ∃ a ′ ≤ ⊥ a : a ′ , b ′′ ∈ R ∞ ∩ ¬R n
In this definition, x ≤ ⊥ y means ⊥ < x ≤ y. Finally, we define the topology T (L):

T (L) = { O ⊆ L | ∀ a ∈ O, ∃ n N : B n (a) ⊆ O}
Some remarks have to be done concerning the definition of

{R n }. First, it is clear that a, b ∈ L #2 a ≤ b ⊥ is a lower set of L #2
so that the definition of R 0 makes sense. Similarly, for I ∈ ℘ ↓ (L #2 ), the set J defined as:

J = a, b ∈ L 2 ∃ a ′ , b ′ ≥ a, b : a & b ′ , b & a ′ ∈ I
is also a lower set. To show this, let a, b be in J and α, β be such that α, β ≤ a, b . Since a, b ∈ J, there exists a

′ , b ′ ≥ a, b such that a & b ′ , b & a ′ is in I. But then, a ′ , b ′ ≥ α, β and α & b ′ , β & a ′ ≤ a & b ′ , b & a ′ . As a consequence, α & b ′ , β & a ′
is in I and thus α, β is in J. This shows that J is an ideal, and that the definition of {R n } by induction makes sense.

Of course, we have: Proposition 17 Given an orthomodular lattice L, T (L) is a topology on L.

Proof Once again, the proof is the same as that of proposition 7.

Equivalence of Topologies

Proposition 18 For all n ∈ N, one has

R n ∩ At(L #2 ) = R At n .
Proof We show this by induction. It is clearly true for n = 0. Now, suppose that it is true for n and let a and b be two atoms. As a consequence of corollary 15.1, one has:

a, b ∈ R n+1 ⇔ ∃ a ′ , b ′ ≥ a, b : a & b ′ , b & a ′ ∈ R n
As a consequence, two possibilities have to be considered:

1. If a ≤ b ⊥ , then a, b is in both R 0 ∩ At(L #2
) and R At 0 and thus, it is in both R n+1 ∩ At(L #2 ) and R At n+1 . 2. Otherwise, for all b ′ ≥ b and a ′ ≥ a, neither a & b ′ nor b & a ′ are equal to ⊥, so that they are both atoms due to the atom projection property, and by induction:

a & b ′ , b & a ′ ∈ R n ∩ At(L #2 ) ⇔ a & b ′ , b & a ′ ∈ R At n
One then has:

a, b ∈ R n+1 ∩ At(L #2 ) ⇔ ∃ a ′ , b ′ ≥ a, b : a & b ′ , b & a ′ ∈ R n ∩ At(L #2 ) ⇔ ∃ a ′ , b ′ ≥ a, b : a & b ′ , b & a ′ ∈ R At n ⇔ a, b ∈ R At n+1
As a result, the equality R n ∩ At(L #2 ) = R At n holds for all n ∈ N.

Proposition 19 If L is atomic and verifies the atom projection property, then:

∀ a, b ∈ L, b ∈ B n (a) ⇔ b ∈ B L n (a)
Proof We only have to show the following equivalence:

(∀ a ′ ≤ ⊥ a, ∃ a ′′ ≤ ⊥ a ′ : ∃ b ′ ≤ ⊥ b : a ′′ , b ′ ∈ R ∞ ∩ ¬R n ) ⇔ ∀ x ∈ At(a), ∃ y ∈ At(b) : x, y ∈ R At ∞ \ R At n
The ⇐-direction is direct. Conversely, suppose that one has:

∀ a ′ ≤ ⊥ a, ∃ a ′′ ≤ ⊥ a ′ : ∃ b ′ ≤ ⊥ b : a ′′ , b ′ ∈ R ∞ ∩ ¬R n
and let x be an atom of a. Our assumption implies that there exists an element b ′ ≤ ⊥ b such that x, b ′ ∈ R ∞ ∩ ¬R n . Moreover, R ∞ ∩ ¬R n is a lower set so that there exists an atom y of b such that x, y ∈ R ∞ ∩ ¬R n . We conclude by remarking that from proposition 18, one has

R ∞ ∩ At(L #2 ) = R At ∞ and that following proposition 16, x, y ∈ ¬R n ⇔ x, y ∈ R n ⇔ x, y ∈ R At n .
Corollary 19.1 Given an atomic orthomodular lattice L verifying the atom projection property, the topologies T (L) and T L (L) are equivalent.

Proof This a direct consequence of proposition 19.

Theorem 20 Given an Hilbert lattice L H where dim H ≥ 3, the topology T (L H ) is equivalent to the topology induced on L H by the metrics of H.

Proof This follows from corollary 19.1 and proposition 13.

Quantum and Classical Structures

As we have seen in theorem 20, if L is a Hilbert lattice L H where dim H ≥ 3, then the topology T (L H ) is equivalent to the one induced by the metrics of H. In particular, with this topology, there are no isolated points except the extremal ones, i.e. ⊤ and ⊥.

We now explore the topology T (B) of a boolean algebra B. First, following our definitions,

R 0 = ¬¬ a, b a ≤ b ⊥ . Since B is a boolean algebra, one has a ≤ b ⊥ ⇔ a ∧ b = ⊥ and thus: a, b a ≤ b ⊥ = { a, b | a ∧ b = ⊥} ¬ a, b a ≤ b ⊥ = { a, b | ∀ a ′ , b ′ ≤ a, b , (a ′ ∧ b ′ = ⊥ ⇒ a ′ = b ′ = ⊥)} = { a, a | a ∈ At(B)} ∪ { ⊥, ⊥ } ¬¬ a, b a ≤ b ⊥ = ¬ ({ a, a | a ∈ At(B)} ∪ { ⊥, ⊥ }) = { a, b | ∀ a ′ , b ′ ≤ a, b , a ′ , b ′ ∈ { a, a | a ∈ At(B)}} = { a, b | At(a) ∩ At(b) = ∅} We thus have R 0 = { a, b | At(a) ∩ At(b) = ∅}. Now, in order to compute R 1 , we have: ∃ a ′ , b ′ ≥ a, b : a & b ′ , b & a ′ ∈ R 0 ⇔ ∃ a ′ , b ′ ≥ a, b : a ∧ b ′ , b ∧ a ′ ∈ R 0 ⇔ ∃ a ′ , b ′ ≥ a, b : At(a ∧ b ′ ) ∩ At(b ∧ a ′ ) = ∅ ⇔ ∃ a ′ , b ′ ≥ a, b : At(a) ∩ At(b ′ ) ∩ At(b) ∩ At(a ′ ) = ∅ ⇔ At(a) ∩ At(b) = ∅ Thus, we have R 1 = ¬¬R 0 = R 0 and more generally, R ∞ = R n = R 0 .
As a consequence, since for all n, R 0 ∩ ¬R n = { ⊥, ⊥ }, this implies that for a ∈ B, one has B n (a) = ∅. We have shown:

Theorem 21 Given a boolean algebra B, the topology T (B) is the discrete topology on B, that is:

T (B) = ℘(B)
This result shows that the topological construction we have defined on orthomodular lattices is an interesting tool for discriminating some quantum structures from classical boolean structures, since for the topology associated to a boolean algebra is discrete while the topology associated to a Hilbert lattice has only ⊤ and ⊥ as isolated points.

Conclusion and Perspective

In this article, we have shown that it is possible to equip any orthomodular lattice with a topology. Originally, orthomodular lattices were introduced as a generalization of Hilbert lattices which are intrisically topological objects. With the construction that we provide, this suggests that orthomodular lattices can themselves be seen as intrisically topological objects.

The generality of this result is rather striking, since for the case of Hilbert lattices, the topology is induced by the underlying Hilbert space, which is strongly related to mathematical analysis. On the contrary, orthomodular lattices are purely algebraic objects and the mere structure of these objects, in particular their orthomodularity and the properties of the Sasaki projection that derive from it, are sufficient to capture their topological structure. This result can also be related to others, such that Piron's theorem [START_REF] Piron | Axiomatique quantique[END_REF] and Solèr's theorem [START_REF] Solèr | Characterization of hilbert spaces by orthomodular spaces[END_REF][START_REF] Dalla Chiara | Quantum logic[END_REF], which explore the way some properties of Hilbert lattices can be expressed in a lattice-theoretical way.

However, we have just given the definition of this topology and shown a few results in order to motivate a more general study of orthomodular lattices in the light of this topology. Here are some questions for further work:

• Which are the clopen subsets of L with regards to T (L)? And in the case of a Hilbert lattice L H , should the elements of B n (a) have the same dimension as a?

• Under which conditions on L is the topology L(T ) different from the discrete one? More generally, which points are isolated in this topology. Theorems 20 and 21 suggest more particularly that there exists some connection between isolated points and elements of the center of L.

• Which functions are continuous (in particular, we think of orthocomplementation, conjunction, the Sasaki projection, OML morphisms)?

• In the definition of L(T ), R ∞ is a lower set. Is it closed, and if not, what is its closure? The same question also applies to R ∞ ∪ { a, a | a ∈ L}.

• One can define a notion of Cauchy sequence by stating for a sequence {u n } that ∀ m, ∃ a, n ∈ L × N : ∀ k ≥ n, u k ∈ B m (a). This permits to define a sort of topological completion of an OML. Then, how does this type of completion relate to other ways of making an orthomodular lattice complete?

Proposition 8

 8 For any orthomodular lattice L and a, b, c ∈ L, one has: b

Proposition 9

 9 Let L be an atomic OML verifying the atom-projection property, and let a, b and c be three atoms such that b & (a ∨ c) = c and a ≤ b ⊥ . Then one has a & (b ∨ c) = c. Proof As a direct consequence of proposition 8, one has a & (b ∨ c) ≤ c. But having a & (b ∨ c) = ⊥ would mean that a ≤ (b ∨ c) ⊥ so that in particular a ≤ b ⊥ which is not possible. Thus, the only possibility is a & (b ∨ c) = c. Proposition 10 Given a Hilbert space H with dim H ≥ 3, the topology T At (L H ) is equivalent to the topology induced on P H by the metric of H. Proof In proposition 3, we used the notation & # (a, b) = { b & a ′ | a ′ ≥ a}. If a ≤ b ⊥ , for every c in & # (a, b), one has c = b & (a ∨ c) so that using proposition 9, c ∈ & # (b, a) and more generally, & # (a, b) = & # (b, a). With this remark and proposition 3, it follows that for all n ∈ N: R At n = { a, b | d(a, b) ≥ θ n } Thus, for a ∈ P H and n ∈ N, one has B At n (a) = { b ∈ P H | 0 < d(a, b) < θ n }. And since lim θ n = 0, it follows that: O ∈ T At (L H ) ⇔ (∀ x ∈ O, ∃ δ > 0 : ∀ y ∈ P H , d(x, y) < δ ⇒ y ∈ O) Stated another way, O is in T At (L H ) if and only if it is open with regards to the topology induced on P H by the metric of H.

  x∈At(a) d π (x, b), max y∈At(b) d π (y, a) where for x, b ∈ P H × L H , d π (x, b) = min y∈At(b) d(x, y). Proposition 13 Given a Hilbert lattice L H where dim H ≥ 3, the topology T L (L H ) is equivalent to the topology induced on L H by the metrics of H. Proof This is a direct consequence of the definition of the metrics on L H and of proposition 10 where we have shown that B At n (a) = { b ∈ P H | 0 < d(a, b) < θ n }. In order to show a similar result for B L n (a), we have to consider two possibilities: 1. If a is an atom, we have B L n (a) = { b ∈ L | ∀ y ∈ At(b), 0 < d(a, y) < θ n }. In that case, if b is in B L n (a), b has to be an atom too (otherwise, b∧a ⊥ = ⊥ and there exists an atom y of b such that d(a, y) = π