Laëtitia Viau

Arnaud Sallaberry

Nancy Rodriguez

Jean-François Girres

Pascal Poncelet

Polygon vector map distortion for increasing the readability of one-to-many flow maps

Keywords: Geovisualization, one-to-many flow map, map distortion, geospatial network, automated cartography

Cartographers have long been interested in the representation of various movements such as migration, commercial exchanges and transportation. There are several techniques for visualizing this information; this paper focuses on flow mapping. A flow map is a type of map that shows a set of movements through line symbols connecting an origin to a destination. Each link is associated with a value that corresponds to the volume of the movement. However, once a certain volume of data is reached, the maps quickly become cluttered and can be difficult to read and understand. Moreover, the values of the movements must be correctly represented so as not to induce biased interpretations.

The objective of this paper is to create flow maps in which the flows have highly variable thicknesses so that the associated value is correctly represented. The technique used to create the flow paths does not create crossings between flows. In order to remove any visual clutter, such as overlaps between flows and geographic features, some areas of the map are distorted. In other words, our method of map distortion adapts the polygon vector base map to the flows, the central information of the visualization, and not the other way around.

Introduction

Cartographers have long been interested in the representation of various movements such as migration, commercial exchanges and transportation. With the emergence of large volumes of data and globalization, and the corresponding increase in the importance of mobility and exchanges, the visualization of flows remains an issue. A traditional approach from the 1800s involves superimposing links representing the flows onto the maritime areas of a base map representing the geographic features (i.e. land), with the thickness of the links representing the magnitude of the flows. Unfortunately, such an approach presents cluttering issues due to overlaps between flows and geographic features. An alternative is to draw thick lines to represent the CONTACT L. Viau. Email: laetitia.viau@lirmm.fr flows and to use a sequential color scale to represent the associated quantities, but it is difficult for the human eye to perceive the nuances and compare the values [START_REF] Bertin | Sémiologie graphique, les diagrammes, les réseaux, les cartes[END_REF]. Changing the thickness of the flows is much more effective, since the size channel maps quantitative variables much better than the color channel [START_REF] Munzner | GALBE: Adaptive Generalisation. The need for an Adaptive Process for Automated Generalisation, an Example on Roads[END_REF].

The best-known 19th century flow maps are those of Charles Joseph Minard (1781-1870), a French civil engineer [START_REF] Rendgen | The Minard system: The complete statistical graphics of Charles-Joseph Minard[END_REF]. His maps are very easy to interpret and read: each flow has a thickness that corresponds to a value, and no flow overlaps with a geographic feature as a result of base map distortion. Given that the goal of these maps is in no way to measure the surface areas of the geographic features and that the geographic entities remain well recognizable, distortion is a good solution for avoiding cluttering issues while using the thickness of the flows to represent their magnitude. In particular, it makes it possible to have thick flows without any overlap between the geographic information and the flows.

In this paper, we focus on one-to-many flow maps that represent the flow from one origin to several destinations on the maritime areas of a base map representing geographic features, i.e. land (see, for instance, [START_REF] Sun | A spatial one-to-many flow layout algorithm using triangulation, approximate steiner trees, and path smoothing[END_REF] for previous work on the topic). In such maps, representing the magnitude of the flows with their thickness is more effective than with their color. However, it may cause overlaps between flows and geographic features. Geographic information is then lost as some countries are hidden. For example, a large flow passing over a country can mask a smaller flow going towards that country. The objective of this paper is to provide an automated method for generating one-to-many flow maps inspired by those of Minard, i.e. a method that distorts the base map in order to remove overlaps between flows and geographic features, thus preserving all of the advantages of handmade maps while automating their creation.

The contribution of this paper is threefold:

(1) We propose a new approach to one-to-many flow map visualization based on map distortion;

(2) We propose a method for distorting polygon vector base maps in order to remove overlaps between geographic features and flows;

(3) We propose a new method for generating one-to-many flow maps.

In Sec. 2 below, we present related work on flow maps and map distortion. Then, we introduce our flow map distortion method in Sec. 3. In Sec. 4, we propose a new method for creating one-to-many flow maps. Sec. 5 is dedicated to the discussion of the results, and we conclude in Sec. 6.

Related work

The method proposed by this paper relates to the fields of visualizing flows on maps and map distortion. To our knowledge, various methods have been proposed in these two fields, but since Minard, they have never been combined. In this section, we review articles on the creation of flow maps, and then we present the applications of map distortion closest to our method.

Flow maps

Geospatial networks are graphs with nodes that can be associated with geographic locations such as a city, a set of geographic coordinates, or a country. The nodes are connected by links. Drawing these kinds of networks is relevant to mapping flows. Indeed, flows form a graph in which the nodes are the starting and end points of the flows (and sometimes intermediate locations) and the links are the connecting flows [START_REF] Jenny | Force-directed layout of origin-destination flow maps[END_REF].

The geospatial network visualization survey by [START_REF] Schöttler | Visualizing and interacting with geospatial networks: A survey and design space[END_REF] raises the question of how to integrate geographic information and network data, as well as other complementary information. The main challenge in visualizing geospatial networks is to present the information as accurately as possible while maintaining readability and clarity. The user must be able to correctly interpret the information at first glance. The survey proposes a design space composed of four dimensions: Geography Representation, Network Representation, Composition, and Interactivity. For the first two dimensions, the modes of representation are placed on a scale from explicit to abstract. The Network Representation dimension is composed of two sub-dimensions: the representation of nodes and of links. In this paper, we focus on the representation of geographic information, nodes and links for flow maps.

Charles Joseph Minard implemented visualizations of geospatial networks in his flow maps [START_REF] Schöttler | Visualizing and interacting with geospatial networks: A survey and design space[END_REF]. These flow maps manage to integrate geographic information and network data, as well as other complementary information, without clutter via link aggregation. With regard to the geographic information, the mode of representation is an intermediate one, between explicit and abstract, because the geographic space is distorted.

The flow maps of [START_REF] Tobler | Experiments in migration mapping by computer[END_REF] marked the beginning of automatic flow map creation. These maps have the most explicit representation of the links. All the information is present, but the maps can become very cluttered when the number of flows increases because each flow is represented independently of the others. Moreover, according to [START_REF] Bahoken | Représenter la mondialisation par des flux, le rôle de la distance cartographique perçue[END_REF], the representation of exchanges/movements by a straight line is not satisfactory. It is too approximate and does not correspond to the underlying mode of transport (e.g. maritime transport).

In the following years, some automatic methods that overcome the problem of map cluttering by aggregating the links, like Minard did, were proposed. The majority of these methods rely on an explicit representation of geographic information and nodes as well as an aggregation of links. Indeed, the aggregation of links reduces cluttering while representing the desired information.

To reduce the number of edge crossings, [START_REF] Cui | Geometry-based edge clustering for graph visualization[END_REF] proposed one of the first methods suitable for general graphs. The Geometry-Based Edge Bundling method uses a control mesh to enable the edge-gathering process, which involves forcing all the edges to pass through control points defined on the mesh. However, the results obtained present high curvature variation. Another method of edge bundling for general graphs, by [START_REF] Holten | Force-directed edge bundling for graph visualization[END_REF], avoids this problem. Their force-directed method models edges as flexible springs that attract each other. However, this method is slower than the previous one. [START_REF] Debiasi | Force directed flow map layout[END_REF] improve on the method proposed by [START_REF] Holten | Force-directed edge bundling for graph visualization[END_REF]. Their novel method for the automatic generation of flow maps describes the motion and force of attraction or repulsion between points through the laws of physics. [START_REF] Gansner | Multilevel agglomerative edge bundling for visualizing large graphs[END_REF] also address the problem of curvatures created by the control mesh. They propose a "multilevel agglomerative edge bundling method based on a principled approach of minimizing ink needed to represent edges, with additional constraints on the curvature of the resulting splines." [START_REF] Jenny | Design principles for origin-destination flow maps[END_REF] defined design principles for origin-destination flow maps. The findings of a quantitative analysis and the results from user studies in graph drawing led to design principles such as minimizing the overlap between flows, scaling the flow width to represent quantity, and many others. On the basis of these design principles, a force-directed layout method for producing origin-destination flow maps in which the flows are quadratic Bezier curves was defined [START_REF] Jenny | Force-directed layout of origin-destination flow maps[END_REF].

Other methods based on hierarchical clustering or a Steiner tree have been proposed. [START_REF] Phan | Flow map layout[END_REF] use hierarchical clustering to define a flow map layout. The hierarchical clustering creates a tree that connects a root to a set of destinations. The method minimizes edge crossings and distorts node positions. The relative positions of the nodes are well preserved. In this case, the geographic representation is not explicit due to the distortion, and labels are added to nodes instead of a base map. [START_REF] Buchin | Flow map layout via spiral trees[END_REF] use a type of Steiner tree involving logarithmic spirals. It improves flow tracing by avoiding crossings between flows and geographic entities as much as possible. Another method for creating a one-to-many flow map based on a Steiner tree was proposed by [START_REF] Sun | A spatial one-to-many flow layout algorithm using triangulation, approximate steiner trees, and path smoothing[END_REF]. It involves triangulating the space to create a network and using this triangulation to define a Steiner Tree. A series of automatic mapping operations are then performed to improve the lines representing the flows and add color to them according to their value. [START_REF] Bahoken | Représenter la mondialisation par des flux, le rôle de la distance cartographique perçue[END_REF] explains the importance of the "route effect", which should be integrated into the representation of the flows in order to provide a qualitative improvement to the map. Without knowing the route of the movements, it is obviously not possible to transcribe the exact path taken in reality, but these movements must be evoked. On a global scale, representing the flows along possible maritime routes leads to the "route effect".

Map distortion

It is important to keep in mind that maps are always distorted, regardless of how the globe is transformed into a flat surface. This transformation is called a map projection. In this article, the term distortion does not refer directly to the map projection itself, but to the modification of a map already projected onto a flat surface.

Cartograms

One of the main uses of map distortion is for cartograms. Initially, the term cartogram was used to refer to different types of maps such as statistical maps and choropleth maps [START_REF] Tobler | Thirty five years of computer cartograms[END_REF]. As defined in the paper by [START_REF] Nusrat | The state of the art in cartograms[END_REF], the term cartogram now refers to maps that "combine statistical and geographical information in thematic maps, where areas of geographical regions (e.g., countries, states) are scaled in proportion to some statistic (e.g., population, income)". For instance, the diffusion-based method for producing density-equalizing maps of [START_REF] Gastner | Diffusion-based method for producing densityequalizing maps[END_REF] is one of the best-known methods for constructing a cartogram. The technique proposed is based on ideas from elementary physics. [START_REF] Gastner | Fast flow-based algorithm for creating densityequalizing map projections[END_REF] improved this technique by speeding up the process.

Fish-eye

In photography, a fisheye lens is a very wide angle lens that shows nearby objects in great detail while showing objects farther away in less detail. This idea was applied to graphs by [START_REF] Sarkar | Graphical fisheye views of graphs[END_REF]. In cartography, a fisheye means having a variablescale map projection. Fisheye views are very useful for focusing on a particular area containing a lot of information while retaining the most distant elements. The area of interest, called the focus, is enlarged, but the surroundings, called the context, remain visible. This is referred to as a Focus+Context fisheye. The difficulty lies in providing access to all the detailed information in the focus area while preserving the context.

The Focus+Context fisheye view was proposed as a solution for real-time personal navigation by [START_REF] Harrie | A variable-scale map for small-display cartography[END_REF]. A user needs both an overview map and more details around their position. The method uses a scalable map that changes in real time so that the user's position is always in the large-scale part. However, the results deteriorate with increasing magnification intensity. [START_REF] Carpendale | Achieving higher magnification in context[END_REF] present a set of improvements for high magnification by proposing new distortion functions.

The main problems with Focus+Context fisheye views is that the entire area of the map is distorted and the calculation cost is very high. Furthermore, the density of information is too large in the corners of the map. [START_REF] Takahashi | An elastic map system with cognitive map-based operations[END_REF] improved this type of fisheye view by introducing the concept of Focus+Glue+Context. A glue area is included between the Focus and Context areas, and distortions in the Focus and Context are completely removed. The Focus area is large-scale in order to display the details of the map, while the Context area is small-scale. The Glue area links the Focus to the Context. [START_REF] Yamamoto | Focus+glue+context: An improved fisheye approach for web map services[END_REF] proposed a method for generating high-quality maps based on this concept, with Focus+Glue+Context views that can be used in web map services given their generation speed. Similarly, [START_REF] Haunert | Drawing road networks with focus regions[END_REF] try to improve the fisheye views in cartography by arguing that distortion should preferably take place in areas where the information density is lower. Thus, no predefined mapping function is applied. The method is based on an optimization approach to minimize distortions and respects a set of constraints and objectives.

The objective of our Flow Map Distortion method is very similar to that of the fisheye: to make a set of magnifications in particular areas while keeping the whole map visible. (2008) defines map generalization as "the process that simplifies the representation of geographical data to produce a map at a certain scale with a defined and readable legend. To be readable at a smaller scale, some objects are removed; others are enlarged, aggregated and displaced one to another, and all objects are simplified."

Cartographic generalization and schematization

Ruas

The result of a cartographic generalization process can be seen as a distorted but more readable map. Over the last few decades, progress in automated cartographic generalization has allowed geographic data on a map to be automatically adapted according to specific constraints [START_REF] Mackaness | Generalisation of geographic information: Cartographic modelling and applications[END_REF]. For example, specific algorithms were developed to adapt the representation of mountain roads (Mustière 1998) and land use (van Oosterom 1995). Agent-based approaches [START_REF] Ruas | A method vor building displacement in automated map generalisation[END_REF][START_REF] Gaffuri | Field deformation in an agent-based generalisation model: The gael model[END_REF][START_REF] Touya | Collagen: Collaboration between automatic cartographic generalisation processes[END_REF] and least-square adjustment methods [START_REF] Harrie | The constraint method for solving spatial conflicts in cartographic generalization[END_REF][START_REF] Sester | Optimization approaches for generalization and data abstraction[END_REF] were also proposed to resolve conflicts in cartographic generalization.

In this context, cartographic generalization can be seen as a map distortion process since it modifies the initial positions and shapes of geographic features in order to facilitate their visualization on the map.

Finally, map distortion techniques can be illustrated by the application of map schematization approaches (Dykes et al. 2010), used to facilitate the visualization of specific features such as complex networks. The automatic generation of metro maps [START_REF] Stott | Automatic metro map layout using multicriteria optimization[END_REF]) is a classical illustration of map distortion based on cartographic schematization.

Flow map distortion (FMD)

Visualizing flows over maritime areas while using width to represent a quantity generates overlaps between flows and geographic features. We propose a method for removing these overlaps from an existing one-to-many flow map without modifying the width of the flows. Figure 1 illustrates the steps involved: a first distortion, then the drawing of curves, and finally a second distortion that completely removes any overlap.

First distortion

The method proposed in this article involves carrying out a set of magnifications in the areas where the flows pass in order to leave them enough space that they do not overlap the geographic features. To perform the magnifications, we use the formula proposed by [START_REF] Kadmon | A polyfocal projection for statistical surfaces[END_REF]. Let us consider the focus point with coordinates (x f , y f ) and a point P with coordinates (x, y). D is the distance between P and the focus point. (x ′ , y ′ ), the new coordinates of P , are computed as follows:

x ′ = x + A(x -x f ) 1 + C • D 2 y ′ = y + A(y -y f ) 1 + C • D 2 (1)
where A is the level of magnification and C the radial rate of change of this variable. This means that the higher the C, the more the magnification will be localized close to the focus. On the contrary, the lower the value of C, the more the magnification will be spread over the entire map. In our case, the magnification has to be localized, so we set C = 1.

Since our method is dedicated to one-to-many flow maps, the flow can be considered as a weighted tree in which the root is the origin, the leaves are the destinations, and the internal nodes are the control points of the original map (on the left in Figure 2). In this case, the weights of the edges are the widths of the flows. To apply magnifications, we first divide the tree into paths such that each internal node has a degree of 2 in the tree (on the right in Figure 2). We then give these paths a weight corresponding to the weight of one of their edges (the weights of all of a path's edges being equal). We call the ordered sets of nodes of each path sequences. Les us consider the set S of the sequences of the tree such that S = [S 1 , S 2 , ..., S s ]. ∀S i with 1 ≤ i ≤ s, w i is the weight of the sequence. The method involves processing this set of sequences in decreasing order of their weight. For each of these sequences, we have a set of nodes. If there are overlaps between the sequence and a geographic feature, then the node closest to a vertex of the base map is sought. This node then becomes a focus point and we apply the formula (see Algorithm 1) to all the vertices forming the shapes corresponding to the geographic features and to all the sequence nodes. This process is repeated until there is no overlap (see Algorithm 2).

Algorithm 1 Create a new magnification Require: P a set of coordinates 1: function distort(P , f ocus, A, C = 1) 2:

for each p ∈ P do 3:

D ← (x p -x f ocus ) 2 + (y p -y f ocus ) 2 4: x ′ p ← x p + A×(xp-xfocus) 1+C×D 2 5: y ′ k ← y p + A×(yp-yfocus)
1+C×D 2 6: end for

r i ← wi 2 4: while hasOverlap do 5: distM in ← {min distance(u, k)|∀u ∈ S i , ∀k ∈ K} 6: if distM in > r i then 7: hasOverlap ← F alse 8: end if 9: f ocus ← {u ∈ S i | min distance(u, k) ∀u ∈ S i , ∀k ∈ K} 10:
A min ← solution of equation ( 6)

11: A ← A min × ϵ 12: S ← distort(S, f ocus, A) 13: K ← distort(K, f ocus, A) 14:
end while 15: end for Let D ′ be the distance between the focus point and the nearest vertex of a geographic feature after transformation and R be the weight of the sequence divided by 2 (i.e. the radius of the flow width). To eliminate all overlaps, we must define A min , the minimum value of A so that D ′ ≥ R.

D ′ ≥ R ⇔ x + A min (x -x f ) 1 + C • D 2 -x f 2 + y + A min (y -y f ) 1 + C • D 2 -y f 2 ≥ R ⇔ x + A min (x -x f ) 1 + C • D 2 -x f 2 + y + A min (y -y f ) 1 + C • D 2 -y f 2 ≥ R 2 (2)
By isolating the variable A min and expanding the formula we get:

x + A min (x -x f ) 1 + C • D 2 -x f 2 = x -x f 1 + C • D 2 2 A 2 min + 2(x -x f ) 2 1 + C • D A min + (x -x f ) 2
(3)

y + A min (y -y f ) 1 + C • D 2 -y f 2 = y -y f 1 + C • D 2 2 A 2 min + 2(y -y f ) 2 1 + C • D A min + (y -y f ) 2 (4)
By adding ( 3) and ( 4) we can then obtain a single second-degree polynomial of the form αA 2 min + βA min + γ:

α = x -x f 1 + C • D 2 2 + y -y f 1 + C • D 2 2 β = 2 (x -x f ) 2 + (y -y f ) 2 1 + C • D γ = (x -x f ) 2 + (y -y f ) 2 (5)
All that remains is to solve the equation, the result of which corresponds to the minimum value of A that removes any overlap:

αA 2 min + βA min + γ -R 2 = 0 (6)
Initially, very thick sequences can pass through very narrow zones (see Figure 3b). Removing the superpositions with A = A min in a single iteration can create crossings in the shapes of the geographic features (see Figures 3a and3c). Decreasing the value of A by a damping factor ϵ ∈ ]0, 1[ such that A = A min × ϵ moderates the distortion and avoids creating non-simple polygons1 . To avoid any crossings, we propose carrying out the magnifications recursively (Figure 3d).

Nodes are not processed in sequential order, and the same node can be processed several times. This distortion step only ends once the overlaps have been completely removed, but it should not be forgotten that the level of magnification is reduced. We found empirically that this prevented the creation of crossings in the geographic features. In addition, the sequential order would always cause the geographic features to move in the same direction as the flow path, which created more distortion than necessary. Another advantage of processing the node closest to a geographic feature at each iteration is that a node is never processed where a distortion is not necessary.

Let l be the number of vertices forming the shape corresponding to a geographic feature, m the size of the tree, and n the number of sequences. The time complexity of checking if there are overlaps is O(c.l.m), where c is the number of iterations. For setting the focus, the time complexity is also O(c.l.m), and for the distortion it is O(c.m.n + c.l.n). Since the number of sequences is less than the size of the tree, n.l is less than m.n and the time complexity of the first distortion is O(c(l.m+m.n)). If there are many sequences in the tree T , i.e. n tends towards m, the size of the sequences decreases and tends towards 1. The complexity then approaches O(c(l.m + m 2 )). At the end of this first step, the overlaps have been removed while maintaining recognizable and consistent geographic features. We then move on to creating the flow curves.

Curves

In this second step, we transform each sequence of the tree into a curve. This step is performed just after the first distortion because creating the curves involves changes to the tree. Modifying the tree without having created space could cause crossings between the tree and the geographic features.

For this step, we tested and compared different curves. Our method obtained the best results with the Catmull-Rom spline and the basis spline (B-spline).

The Catmull-Rom spline uses the specified control points and a parameter α. Its behavior depends significantly on the choice of the value for α. If α = 0, some selfintersections may occur, and if α = 1, the curve moves too far from the control polygon [START_REF] Yuksel | On the parameterization of catmull-rom curves[END_REF]). In our case, α = 0.5 seems to be the most appropriate. The main advantage of this curve compared to the B-spline is that it passes through the points of the control polygon, i.e. the flow path. This reduces the chances of creating new overlaps since the curve is drawn very close to the original path. However, the B-spline has the advantage of smoothing the path more than Catmull-Rom.

The B-spline is a basis cubic spline straightened with control points. Holten (2006) proposed using it in hierarchical edge bundling to clarify the edges. The spline is straightened according to the β parameter of the curve. β = 1 is the most suitable for our needs (see Figure 4). We therefore opted for this type of curve, which smooths the paths without superimposing them on the geographic features.

Second distortion

During our tests, we noticed that drawing the curves could create slight overlaps in some cases. A second distortion is necessary to make sure that all overlaps are removed. Using the formula (1) and the equation ( 6) presented above, a second distortion is carried out (see Algorithm 3).

The algorithm is almost the same as for the first distortion except that the focus points are not created from the nodes of the sequences but from the nodes of the curves of each sequence. Points along the curve are processed in sequential order. Indeed, at this stage, few changes are necessary and one iteration on a point is enough to completely remove any overlay.

Let l be the number of vertices forming the shapes corresponding to the geographic features and m be the number of nodes in all the curves. The time complexity of checking if there are overlaps is O(l.m). For the distortion, it is O(l.m + m 2 ), and thus the time complexity of the second distortion is also O(l.m + m 2 ).

At the end of these 3 steps, we obtained flow maps without any overlap. The only initial condition was that no edge of the tree should cross a geographic feature. Various Algorithm 3 Second distortion Require: P the set of curves forming the flow, K the set of points forming the shapes of the geographical features Ensure: P the set of curves with updated positions, K the set of points forming the shapes of the geographical features with updated positions 1: for each P i ∈ P do 2:

for each p ∈ P i do

3: dist ← min distance(p, k), ∀k ∈ K 4: r i ← wi 2 5: if dist < r i then 6: f ocus ← p 7:
A min ← solution of equation ( 6)

8:

A ← A min 9:

P ← distort(P, f ocus, A)

10: K ← distort(K, f ocus, A) 11: end if 12:
end for 13: end for methods can be used to create the flows of a flow map. In the next section, we propose one that limits the overlapping surfaces in order to reduce the number of magnifications that must be carried out during the distortion of the geographic features.

Initial one-to-many flow map

In this section, we propose a new method for creating one-to-many flow maps. This method makes it possible to create flows with fluid lines and is the most suitable for applying the FMD method. The flows are drawn along the median axes of the geographic features, which reduces the overlaps between them and the features. Reducing the number of overlaps limits the number of iterations in the FMD process and the distortion of the features.

We will now present the steps for creating a one-to-many flow map: (1) the creation of a graph from a skeleton (see Figures 5a and5b), (2) the definition of a tree representing the flows (see Figure 5c), (3) a simplification of this tree, and (4) the drawing of the flows (see Figure 5d).

Initial graph creation

The first step of our method involves creating a graph on top of the base map with edges that lie as far as possible from the geographic features. This graph must contain paths linking the geographic entities of the dataset. In our case, we consider flows between countries, so there must be a path between each pair of countries lying on a coast. This graph will then be used to rout the flows (see next section). Our method is based on a straight skeleton (see Figure 5a). Introduced by [START_REF] Aichholzer | A novel type of skeleton for polygons[END_REF] and then generalized by [START_REF] Aichholzer | Straight skeletons for general polygonal figures in the plane[END_REF], straight skeletons can be considered as connected graphs whose nodes include the vertices of the polygons (in our case the vertices forming the coasts of the geographic features) and whose edges lie close to the median axes of the features. Unlike the median axes, which can contain parabolic curves, straight skeletons are composed exclusively of straight segments. [START_REF] Felkel | Straight skeleton implementation[END_REF] proposed a method applied to simple polygons and [START_REF] Eppstein | Raising roofs, crashing cycles, and playing pool: Applications of a data structure for finding pairwise interactions[END_REF] a method that builds an improved straight skeleton.

In our implementation, we used a library2 that gives us the skeleton shown in Figure 5a. We built a straight skeleton out of a polygon with "holes" that correspond to the inner faces of the polygons forming the geographic features (i.e. the land).

Since the coastal countries include multiple vertices, there are several paths leading to each country in the skeleton. To simplify the latter, we select one vertex per coastal country (in our implementation, the vertex closest to the biggest port of the country) and iteratively remove non-selected vertices of degree one (and the corresponding edges). This gives us a graph G, shown in Figure 5b.

Tree definition

The second step of our method involves mapping the flows to the graph G in order to create the flow paths. The input flows consist of an origin (the starting country), a set of targets (the destinations), and a quantitative value for each origin/destination pair. From the graph G, we calculate the shortest path between the origin and each destination using Dijktra's algorithm. For these calculations, there are 3 possibilities for defining the weight w of the edges:

• w = 1; there is no weighting.

• w = d, with d the distance between the two nodes. Long edges are penalized.

• w = d + (p × (180 -a)); the smaller the value a of the angle (in degrees) with the previous edge, the higher the weight w. Small angles are thus penalized.

The results presented are based on the last definition, with p = 5. We will discuss this parameter in Section 5.

We then create a graph T = (V, E) via the union of the shortest paths (see Figure 5c). T is connected, acyclic and edge directed; it is thus a directed tree in which the root represents the origin of the flows and the leaves represent the targets of the flows.

Simplification of the tree

When we visualize the tree computed in the previous step, we can observe many unnecessary bends along the paths (see Figure 6a). This step consists in improving the appearance of the flows by smoothing the paths defined by the tree. There is a very wide variety of solutions for simplifying the tree. What is essential is to not create any crossings between the nodes and the geographic features. We propose making a series of successive improvements (see Figure 6). We will describe each of the steps for achieving a better appearance of the flows in detail (see the final result in Figure 6e).

First, we remove the nodes that form detours (from Figure 6a to Figure 6b). Let us consider four successive nodes v 0 , v 1 , v 2 , v 3 and three edges (v 0 , v 1 ), (v 1 , v 2 ) and (v 2 , v 3 ) (see Figure 7a). We compute the polar coordinates of the nodes v 1 , v 2 and v 3 from the node that precedes each of them on the interval [0, 2π[. For example, for v 2 we will use v 1 . Let us note the polar coordinates as (r 1 , θ 1 ), (r 2 , θ 2 ) and (r 3 , θ 3 ) and consider 4 intervals: (v 0 , v 2 )). This is the case in Figure 7b : θ 1 and θ 3 are in the interval [0, 3π 2 [ while θ 2 is in the interval [ 3π 2 , 2π[. Thus, v 1 is removed (see Figure 7c). The second improvement consists in smoothing the paths when the flows split, i.e. when the nodes have a degree greater than 2. The idea is to produce a smooth path by avoiding bends forming an angle of less than π 2 (from Figure 6b to Figure 6c). Let us consider a split composed of four nodes v 1 , v 2 , v 3 , v 4 and three edges (v 1 , v 2 ), (v 2 , v 3 ) and (v 2 , v 4 ) (see Figure 8a). If the polyline made by the path v 1 , v 2 , v 4 forms an angle of less than π 2 (this is the case in the Figure 8a), we remove the edge (v 2 , v 4 ) and add the edge (v 1 , v 4 ) (see Figure 8b). If the angle produced by the parent v 0 of v 1 , v 1 , and v 4 is less than π 2 , we delete (v 1 , v 4 ) and add an edge between v 0 and v 4 (see Figure 8c). We repeat this process as long as there is a parent node and until the angle is greater than π 2 . To eliminate the effects of "half-turns" on the tree, we reposition the 2-degree nodes for which the angle formed is below a threshold (from Figure 6c to Figure 6d). Let us consider three successive nodes v 1 , v 2 , and v 3 and two edges (v 1 , v 2 ) and (v 2 , v 3 ). If the polyline made by the path v 1 , v 2 , v 3 forms an angle of less than 0.6π (see Figure 9a), we reposition the node v 2 . For this, we first compute the polar coordinates of v 2 and v 3 from v 1 on the interval ] -π, π]. The polar coordinates of v 2 and v 3 are respectively noted as (r 2 , θ 2 ) and (r 3 , θ 3 ). Then, we redefine the coordinates of v 2 as (r 2 , θ2+θ3 2 ) (see Figure 9b). All that remains is to convert these polar coordinates into cartesian coordinates. The node v 2 is then repositioned (see Figure 9c). Finally, we delete the 2-degree nodes that are not significant (from Figure 6d to Figure 6e)3 . Let us consider three successive nodes v 1 , v 2 , and v 3 and two edges (v 1 , v 2 ) and (v 2 , v 3 ) (see Figure 10a). D 1 is the distance between v 1 and v 2 , D 2 the distance between v 2 and v 3 , and D min the distance between v 2 and the nearest vertex of a geographic polygon. Let us note the larger of D 1 and D 2 as D (see Figure 10b). If D < D min , we remove the node v 2 (see Figure 10c). v 1 , v 2 ) and(v 2 , v 3 ). D min , in blue, is the distance between v 2 and the nearest vertex of a geographic polygon. Since D < D min , v 2 is removed.

[0, π 2 [, [ π 2 , π[, [π,
(c) Final path without the node v 2 , which was not significant.

Figure 10. Procedure for removing nodes that are not significant.

Drawing the flows

After the simplification step, the flows can be drawn using the tree. We draw each path, the nodes of which have a degree of at most 2, with a B-spline and add a thickness corresponding to the sum of the quantities of the flows that pass through it. The result is shown in Figure 5d.

In this section, we have provided a new solution for creating a one-to-many flow map. Using this method is not mandatory but highly recommended. The flow paths follow the median axes between the geographic features, which minimizes the number and size of overlapping surfaces and therefore the magnifications to be carried out. This method can be used alone, without FMD, just to create a one-to-many flow map.

Discussion

In this section, we discuss the results obtained on three different datasets. First, we present the characteristics of the datasets and the computation times. Then, we analyze the maps created by the one-to-many flow map method and the FMD method and the impact of the parameter p. Finally, we report expert feedback on the strengths and limitations of our method.

Datasets

The first dataset concerns tea exports from China (CHN), the second soybean exports from the United States (USA), and the third vanilla exports from Madagascar (MDG). The data is available in FAOSTAT4 . For each input dataset, the number of destinations (i.e. countries in these examples) is reported in

Computation time

The initial flow maps for each dataset were created as presented in Section 4. For this, we initially had 6996 vertices forming the polygons of the geographic features. The skeleton created from these polygons had 14002 vertices, 42006 edges and 6890 faces.

The derived graph had 3697 nodes and 3808 edges. The creation of the skeleton and of the graph took about 5 minutes (deleting unnecessary nodes and edges). Regardless of the dataset, the time to create the initial flow map from the graph and the data source was about 2 minutes. All processing was done on a laptop with a 2.3 GHz quad-core CPU and 32GB memory. The process was implemented in Python, except for the creation of the skeleton, which was done using the C++ library mentioned in Section 4. Then, we executed the FMD method; the times5 are indicated in Table 2. We used the same computer but the implementation was carried out in JavaScript. 11, 12, and 13 show the results obtained for each of the 3 datasets with and without distortion. The results without distortion were obtained by the method proposed in Section 4. The results with distortion were obtained using the FMD method of Section 3 on the results without distortion.

Dataset

Figures

In some areas of the initial map (i.e. the map without distortion), a thick flow can hide small flows and geographic features such as islands. For example, Figure 14 compares the Mediterranean area with and without distortion. Without the distortion, the island of Crete, south of Greece, is not visible at all, while Sicily and Sardinia are partly hidden. With the distortion, the three islands are all fully visible and well recognizable. Likewise, several smaller flows to various countries in North Africa, such as Tunisia and Algeria, are barely or not at all visible. After the distortion, we can distinguish these flows much better.

The risk with these distortions is that the geographic entities may no longer be identifiable. In our examples, despite the more or less significant distortions, it is still possible to recognize the geographic entities quickly. Figure 15 illustrates an example for Indonesia that compares the geographic features with and without distortion: they keep their positions relative to each other, the deformations are localized, and the features remain identifiable. We performed a small experiment with 5 participants. It consisted in matching several countries labeled on a map before distortion to the map after distortion (without the flow drawing). Two areas were studied: the Mediterranean Sea (as in Figure 14) and the area of Figure 15. Participants had no specific knowledge of mapping and all obtained 100% correct answers.

To obtain all the maps presented above, we used the parameter p = 5 (except in Figure 14). As a reminder, this parameter corresponds to the penalty that we apply to the weight of the edges of the graph when computing the tree (see Section 4.2). With p = 0.5 and p = 5, the flows do not necessarily follow the shortest path in terms of Euclidean distances, but favor the longest edges and then the widest spaces (see Figure 16). This limits the distortions to be performed and makes the computation time of FMD faster. As we can see in Table 3, when p increases, the number of relocated vertices decreases, and the total distance, i.e. the sum of the distances between the initial and the final positions of the relocated vertices, is much smaller. Whatever the value of p, the average distance travelled by the relocated vertices remains in the same range. Therefore, the total of the distances between the initial and final vertices depends mainly on the number of vertices relocated and thus decreases when we have a penalty (i.e. p > 0). As a result, we recommend defining the tree via a straight skeleton as presented in Section 4 and penalizing the weight of the edges when computing the shortest paths in order to avoid areas that are too narrow (Section 4.2). Table 3. Modification of the positions of the vertices of the polygons with the FMD method.

Expert feedback

In this section, we report the feedback of an expert in cartography who was not involved in the development of our method. We solicited his expertise and provided him with the previous sections of this article and an online application containing all the datasets presented above and the possibility to run our method on them step by step. We then asked him to write a report on the strengths and limitations of our approach:

The FMD method makes it possible to automatically create Minard-style flow maps At the global level, the flow maps presented in Figures 11, 12 and 13 make it possible to clearly visualize the main destinations from an origin, as well as the quantities associated with each flow. The overall objective of the flow map is therefore achieved: the reader is able to (1) clearly prioritize the destinations corresponding to the flows from an origin, and (2) associate a quantity with each flow. Graphically, the use of splines offers comfort in terms of the legibility of major maritime routes. When dividing a flow towards several destinations, the orientation of the curvatures in the direction of the flow guarantees consistency in the dynamic reading of the map. On the other hand, the general realism of the proposed routes can be discussed. Indeed, the skeletonization method builds roads far away from the coasts and major shipping lanes. The reader's perception of the actual organization of flows may therefore be impacted. Nevertheless, keeping flows away from the coasts offers an advantage, since it limits the overlapping conflicts between flows and geographic features, and avoids the consequent distortions of geometries.

At the local level, the proposed method makes it possible to avoid overlaps between large flows (materialized by a significant width) and geographic features located near the coasts (which constitute important landmarks for the map reader). The example in the Mediterranean Sea in Figure 14 clearly illustrates the interest of the method in order to visualize islands masked by the symbolized flows. On the other hand, the realism of specific distortions (associated with the proximity of large flows to small geographic features) can be discussed. For example, unrealistic distortions are generated on the islands of Java and Borneo in Figure 12d. Here, the map reader can lose important reference points on the base map, due to the excessive modification of the original shape of geographic features.

In this context, the integration of constraints to preserve the initial shape of geographic features could constitute an interesting perspective in order to maintain a compromise between visibility and realism. Indeed, such constraints would generate distortions that minimize the visual impact on small geographic features, which can be seen as important landmarks for the map reader.

Conclusion and future work

In this paper, we have presented a method, FMD, for removing the overlaps between the flows and the geographic features in a one-to-many flow map. We have also presented a method for generating one-to-many flow maps with few overlaps that can be employed before running FMD. Through several examples, the discussion shows that our method reveals hidden objects while maintaining the recognizability of the geographic features. The results meet the main objective, which was to automatically create distorted maps inspired by the hand-drawn maps of Charles Joseph Minard in the 19th century.

In future work, we plan to apply FMD to other kinds of base maps, such as cities. In this case, the geographic features could be the buildings and the empty areas through which the flows must pass could be the streets. Our method could be useful for better visualizing traffic.

The future work will also concern the processing of many-to-many flow maps, which will certainly be the most complex enhancement. Indeed, this will surely lead to an increase in computation time since the number of sequences will be higher. Moreover, several questions arise, in particular for the definition of the tree: should the sequences be grouped together or treated independently depending on the starting point? How should the overlapping of flows be managed? cations, data mining, and visual analytics. He received his PhD in Computer Science from the Université de Nice-Sophia Antipolis, France.

Figure 1 .

 1 Figure 1. An Overview of the main steps of the FMD method.

Figure 2 .

 2 Figure 2. Tree (left) and the sequences of the tree (right). Each color corresponds to a sequence (8 total) and the numbers represent their weight.

  First distortionRequire: S the set of sequences forming the flow, K the set of points forming the shapes of the geographical features, ϵ ∈ ]0, 1[ a damping factor Ensure: S the set of sequences with updated positions, K the set of points forming the shapes of the geographical features with updated positions 1: for each S i ∈ S do 2: hasOverlap ← T rue 3:

  (a) Example of repositioning p 1 and p 2 by distortion with A = 14 and A = 3, where F is the focus point. In the first case, the value of A is too high: the two points do not keep their positions relative to one another. (b) Initial geographical features: there is no distortion. (c) Example of distortion in one iteration with A = A min : crossings (circled in blue) are created in the geographic features. The polygons are not therefore simple. (d) Example of distortion carried out recursively: for each magnification, we set A = A min ×0.2.The polygons are simple and there are no crossings. In this example, not all iterations are illustrated.

Figure 3 .

 3 Figure 3. Comparison of the results of distortion according to the value of A and the number of iterations.

Figure 4 .

 4 Figure 4. Comparison of the rendering of the B-spline curve with different values of β.

Figure 5 .

 5 Figure 5. Creation of a one-to-many flow map.

  Figure 6. Simplification steps. o is the root of the tree (the origin of the flows) and t 1 and t 2 are the leaves (the targets of the flows).

  (a) Initial nodes and edges. There are four successives nodes : v 0 , v 1 , v 2 , v 3 , and three edges: (v 0 , v 1 ), (v 1 , v 2 ) and (v 2 , v 3 ). (b) Definition of the angular coordinate of the nodes v 1 , v 2 and v 3 based on the preceding node in the interval [0, 2π[.θ 1 ∈ [0, π 2 [, θ 2 ∈ [ 3π 2 , 2π[ and θ 3 ∈ [0, π 2 [. (c)The node v 1 is removed because θ 1 and θ 3 are in the same interval and θ 2 in a different one.

Figure 7 .

 7 Figure 7. Procedure for removing detours.

  (a) Initial flow paths : the path v 1 , v 2 , v 4 forms an angle of less than π 2 . (b) The edge (v 2 , v 4 ) is removed and replaced by (v 1 , v 4 ). The new path v 0 , v 1 , v 4 forms an angle of less than π 2 .(c) The edge (v 1 , v 4 ) is removed and replaced by (v 0 , v 4 ).

Figure 8 .

 8 Figure 8. Path smoothing procedure when flows split.

  (a) Initial position of nodes v 1 , v 2 , v 3 . The path v 1 , v 2 , v 3 forms an angle of less than 0.6π: the node v 2 will be repositioned. (b) Definition of the polar coordinates of v 2 and v 3 on the basis of v 1 on the interval ] -π, π]. The coordinates are noted as (r 2 , θ 2 ) and (r 3 , θ 3 ). The new position of the node v 2 is shown in red. (c) New path formed by the nodes v 1 , v 2 and v 3 .

Figure 9 .

 9 Figure 9. Procedure for eliminating "half-turns" effects.

  (a) Initial path: three successive nodes v 1 , v 2 and v 3 are processed. (b) D, shown in orange, is the larger edge length out of (

  Figure 11. Results of CHN dataset.

  Figure 12. Results of USA dataset.

Figure 16 .

 16 Figure 16. Results with different p values for USA

Table 1 .

 1 This table also shows the minimum, maximum, mean, median, and standard deviation of the values associated with the flows.

	Dataset Nb of dest. Min	Max	Median	Mean	Std
	CHN	115	1	74260	280	3189.57	8134.21
	USA	74	1	22575972 23183.5 707951.15 2717691.15
	MDG	23	1	625	9	64.35	145.24
			Table 1. Input dataset characteristics

We set ϵ = 0.2. Empirically, this value was the most effective in moderating the distortion and avoiding the creation of non-simple polygons.

The Computational Geometry Algorithms Library (CGAL): https://www.cgal.org (accessed: 2022-09)

When using the FMD method, we recommend performing this last step after the first distortion, just before drawing the curves, because these nodes, which are not significant in terms of appearance, can be significant for the first distortion step. On the other hand, if the objective is just to create a one-to-many flow map without distortion, then this last step can take place in the order described before drawing the curves and thicknesses of the flows like in Figure5d.

https://www.fao.org/faostat/en/#data/TM(accessed: 2022-11)

The times indicated are the average of 10 different runs.

Funding

This study was partially funded by EU grant 874850 MOOD and is catalogued as MOOD 065. The contents of this publication are the sole responsibility of the authors and not necessarily reflect the views of the European Commission.

This work has also been supported by the STIC-AmSud program, code: 21-STIC-06.

Notes on contributors

Laëtitia Viau is a PhD student at the Université de Montpellier, France. She is a member of the data mining and visualization research team (ADVANSE) at the Laboratory of Computer Science, Robotics and Microelectronics of Montpellier (LIRMM).

Her research interests include geospatial data visualization and visual analytics.

Arnaud Sallaberry is an Assistant Professor at the Université Paul Valéry Montpellier 3, France. He is the head of the applied mathematics and computer science research team (AMIS). He is also a member of the data mining and visualization research team (ADVANSE) at the Laboratory of Computer Science, Robotics and Microelectronics of Montpellier (LIRMM). His research interests include information visualization and visual analytics. He received his PhD in computer science from the Université de Bordeaux, France.

Data and codes availability statement

The data and codes that allowed us to obtain the results presented in this paper are available in figshare at the permanent link https://doi.org/10.6084/m9.figshare. 21501999.

Nancy Rodriguez is an Assistant Professor at the Université de Montpellier, France. She is a member of the data mining and visualization research team (ADVANSE) at the Laboratory of Computer Science, Robotics and Microelectronics of Montpellier (LIRMM). Her research interests include virtual and augmented/mixed reality, interaction, accessibility and visualization. She received her PhD in Computer Science from the Université de Toulouse, France.

Jean-François Girres is an Assistant Professor at the Université Paul Valéry Montpellier 3, France. He is a member of the applied mathematics and computer science research team (AMIS) and is also a member of the ESPACE-DEV research unit. His research interests include spatial data quality, spatial analysis and geovisualization. He received his PhD in Geographic Information Science from the Université Paris-Est, France.

Pascal Poncelet is a Full Professor at the Université de Montpellier, France, and head of the data mining and visualization research team (ADVANSE) at the Laboratory of Computer Science, Robotics and Microelectronics of Montpellier (LIRMM). His research interests include advanced data analysis techniques for emerging appli-


Dr Elodie Bertrand
email: elodie.bertrand1@gmail.com

Daniel C Mograbi
email: danielmograbi@puc-rio.br

Richard G Brown
email: richard.g.brown@kcl.ac.uk

Jesus Landeira- Fernandez

Robin G Morris
email: robin.morris@kcl.ac.uk

Prof Richard

Heterogeneity of anosognosia in Alzheimer's disease according to the object of awareness

Keywords: anosognosia, awareness, Alzheimer's disease, cognition, ADL, behavioral disturbance

Anosognosia or unawareness of neuropsychological deficits, other symptoms or the illness itself is frequent in Alzheimer's disease (AD). The aim of this study was to explore the multidimensionality of awareness in AD, comparing directly the awareness of patients in relation to several different objects: executive functions, depression, apathy, disinhibition and overall condition. Awareness was determined by discrepancy between patient-and informant-report. The results showed that the level of awareness differs according to the object studied, with awareness for overall condition and executive functions being more impaired, and relative more preservation of awareness for disinhibition and apathy. Correlational analysis suggested that awareness for executive function impairment, apathy and condition correlate strongly, while awareness of depression appears to be an independent construct. These findings have several clinical implications, highlighting the importance of a more thorough evaluation of the patients' awareness.

-Introduction

The term anosognosia refers to the lack of awareness about the illness itself or deficits, including motor, sensory, behavioural and cognitive alterations. It has been used in the context of neurological conditions such as stroke, hemiplegia and dementia [START_REF] Mograbi | Anosognosia[END_REF]. Indeed, recent studies showed that anosognosia is a common characteristic of Alzheimer's disease (AD) [START_REF] Morris | Anosognosia, autobiographical memory and self knowledge in Alzheimer's disease[END_REF]. Here, it has important clinical implications because of the impact on treatment compliance [START_REF] Patel | Ageing and mental health in a developing country: who cares? Qualitative studies from Goa, India[END_REF], the capacity to deal appropriately with potential risk situations [START_REF] Starkstein | Insight and danger in Alzheimer's disease[END_REF] and its contribution to caregiver burden [START_REF] Seltzer | Awareness of deficit in Alzheimer's disease: relation to caregiver burden[END_REF].

There is variability in the presentation and severity of anosognosia in patients with AD, with unawareness ranging from slight minimization to complete denial of problems [START_REF] Clare | Awareness in dementia: A review of assessment methods and measures[END_REF]. In addition, the reported prevalence of anosognosia in AD varies widely, depending on the study and recruitment setting [START_REF] Lopez | Awareness of cognitive deficits and anosognosia in probable Alzheimer's disease[END_REF][START_REF] Mograbi | Unawareness of memory impairment in dementia: a population-based study[END_REF][START_REF] Sousa | Awareness of disease is different for cognitive and functional aspects in mild Alzheimer's Disease: A one-year observation study[END_REF]. One possible explanation for this variability, both in terms of severity and prevalence, may be the fact that awareness is a multidimensional phenomenon (Clare et al., 2011). Indeed, as highlighted by studies with different neurologic populations (e.g., cerebrovascular accident, Huntington's disease, AD), patients may have anosognosia for some deficits but preserved awareness of other difficulties [START_REF] Hoth | Patients with Huntington's disease have impaired awareness of cognitive, emotional, and functional abilities[END_REF][START_REF] Kotler-Cope | Anosognosia in Alzheimer disease[END_REF][START_REF] Young | Unawareness of impaired face recognition[END_REF]. This has led to the notion that awareness cannot be considered a unitary concept, but rather a phenomenon with varied presentations depending on the specific object (e.g. deficits, activities of daily living, the illness itself; [START_REF] Marin | Reliability and validity of the Apathy Evaluation Scale[END_REF]).

In this regard, studies exploring anosognosia in patients with AD showed that awareness differs according to domain assessed. Using the Cognitive Behavior Rating Scales (CBRS) self-and informant-report [START_REF] Williams | Cognitive behavior rating scales -research edition manual[END_REF][START_REF] Williams | Cognitive Behavior Rating Scalesself-report form[END_REF], [START_REF] Kotler-Cope | Anosognosia in Alzheimer disease[END_REF] suggested a difference between the ability to recognize neuropsychological deficits and behavioral and psychological symptoms in people with AD. They demonstrated a significant discrepancy between ratings by patients and informants only for the five cognitive scales: language disorder, higher cognitive deficits, memory disorder, dementia, apraxia. For the four scales evaluating the behavioral deficits (agitation, need for routine, depression, disorientation), the difference was not statistically significant. [START_REF] Vasterling | Unawareness of deficit in Alzheimer's Disease: Domain-specific differences and disease correlates[END_REF] published similar results also pointing that patients with AD present better awareness of behavioral difficulties relative to cognition.

Nevertheless, [START_REF] Verhülsdonk | Anosognosia and depression in patients with Alzheimer's dementia[END_REF] highlighted in a recent study that patients with AD demonstrate difficulty recognizing their own depressive symptoms, leading them to propose the term "affective anosognosia."

Further support for the multidimensional nature of anosognosia in AD is given by evidence that the correlation of anosognosia with neuropsychiatric symptoms also differs according to the domain of awareness assessed (for review, [START_REF] Starkstein | Anosognosia in Alzheimer's disease: Diagnosis, frequency, mechanism and clinical correlates[END_REF]. [START_REF] Starkstein | Two domains of anosognosia in Alzheimer's disease[END_REF] indicated that awareness of cognitive difficulties is associated with depression, while awareness of behavioral problems is correlated with disinhibition. More recently, [START_REF] Spalletta | Anosognosia and neuropsychiatric symptoms and disorders in mild Alzheimer disease and mild cognitive impairment[END_REF] found a correlation between awareness of behavioral deficits and apathy.

In support of these findings, [START_REF] Starkstein | Two domains of anosognosia in Alzheimer's disease[END_REF] proposed, based on a factor analysis of the Anosognosia Questionnaire-Dementia (AQ-D) [START_REF] Migliorelli | Anosognosia in Alzheimer's disease: a study of associated factors[END_REF], a division of the concept of anosognosia into two domains: lack of cognitive awareness and lack of behavioral awareness. More recently, they extended these findings by analysing four domains of anosognosia in a larger series of patients with AD (n=750), including awareness for basic activities of daily living, for instrumental activities of daily living, for depression and for disinhibited behaviors [START_REF] Starkstein | A diagnostic formulation for anosognosia in Alzheimer's disease[END_REF]. Analysing another instrument evaluating awareness in dementia, the Assessment Scale of Psychosocial Impact of the Diagnosis of Dementia (ASPIDD) [START_REF] Dourado | Awareness of disease in dementia Development of a multidimensional rating scale[END_REF], [START_REF] Dourado | Awareness of disease in dementia: factor structure of the assessment scale of psychosocial impact of the diagnosis of dementia[END_REF] presented also a fourfactor division: awareness for activities of daily living, awareness of cognitive functioning and health condition, awareness of emotional state, and awareness of social functioning and relationships.

In summary, there are three types of argument supporting the idea of anosognosia as a multifaceted phenomenon: firstly, studies highlighting that patients with AD show varying degrees of awareness in relation to different 'objects' of awareness [START_REF] Kotler-Cope | Anosognosia in Alzheimer disease[END_REF][START_REF] Vasterling | Unawareness of deficit in Alzheimer's Disease: Domain-specific differences and disease correlates[END_REF][START_REF] Verhülsdonk | Anosognosia and depression in patients with Alzheimer's dementia[END_REF]; secondly, correlational analysis indicating differences in the relation between anosognosia and neuropsychiatric symptoms according to the nature of deficits evaluated [START_REF] Spalletta | Anosognosia and neuropsychiatric symptoms and disorders in mild Alzheimer disease and mild cognitive impairment[END_REF][START_REF] Starkstein | Two domains of anosognosia in Alzheimer's disease[END_REF]; and finally, factorial analysis showing that instruments measuring unawareness can be divided into various factors [START_REF] Dourado | Awareness of disease in dementia: factor structure of the assessment scale of psychosocial impact of the diagnosis of dementia[END_REF][START_REF] Starkstein | A diagnostic formulation for anosognosia in Alzheimer's disease[END_REF][START_REF] Starkstein | Two domains of anosognosia in Alzheimer's disease[END_REF].

Despite the growing understanding of the complexity of awareness, to the best of our knowledge no study has been published comparing directly awareness for cognitive deficits, behavioral problems and mood disorder. The purpose of the present study is to focus on the heterogeneity of anosognosia in AD patients according to the object, comparing directly awareness for five factors: executive functioning, depression, apathy, disinhibition and the condition itself. Considering the findings from prior studies, we hypothesized that the patients' awareness would differ according to the different domains evaluated. [START_REF] Folstein | Mini-mental state". A practical method for grading the cognitive state of patients for the clinician[END_REF])) scores of 18 or above [START_REF] Mungas | In-office mental status testing: a practical guide[END_REF][START_REF] Nice | NICE clinical guideline 42 -Dementia: supporting people with dementia and their carers in health and social care[END_REF], in order to focus the present study on mild to moderate dementia. Consecutive patients who fulfilled the study eligibility criteria were approached.

-Methods

Exclusion criteria were history of neurological disorder (other neurological disorder than AD, also excluding cases with mixed AD and vascular dementia); history of head injury resulting in loss of consciousness for more than an hour; history of alcohol or substance abuse (based on ICD-10 criteria); and history of diagnosed major psychiatric illness or current psychological comorbidity (for example, mood disorder).

Measures

Frontal Systems Behavior Scale (FrsBe)

The FrSBe [START_REF] Grace | Frontal Systems Behavior Scale: Professional manual[END_REF], formerly Frontal Lobe Personality Scale (FLoPS), was designed to measure behavioral impairment in various neurologic populations including people with dementia (e.g. AD, dementia in Parkinson's disease, and vascular dementia), stroke and head injury [START_REF] Malloy | A review of rating scales for measuring behavior change due to frontal systems damage[END_REF]. The 46-item scale evaluates three components: executive dysfunction, disinhibition and apathy. Every item is rated on a 5-point Likert scale and higher ratings correspond to greater abnormal behavior. In order to assess awareness of cognitive/behavioral deficits, self and informant versions were used.

Geriatric Depression Scale (GDS)

Depression was assessed using the GDS 15-item scale, which was developed to evaluate depressive symptoms specifically in older adults [START_REF] Sheikh | Geriatric Depression Scale (GDS) Recent evidence and development of a shorter version[END_REF]. This short version of the questionnaire consists of 15 items (range 0-15 points), with higher scores indicating more-severe depressive symptoms. In order to measure awareness regarding depressive symptoms, self and informant versions were used.

Anosognosia questionnaire

As an assessment for broader awareness into their condition, AD patients were examined with the Anosognosia Questionnaire-Dementia (AQ-D; [START_REF] Migliorelli | Anosognosia in Alzheimer's disease: a study of associated factors[END_REF]). The AQ-D was specifically designed to measure anosognosia in dementia, covering functional changes commonly found during the course of this illness. The AQ-D is a 30-item questionnaire with patient and informant version, divided into two sections; the first section assesses performance of basic and instrumental activities of daily living, whereas the second section examines changes in mood and behavior in relation to daily activities.

Scoring for each item ranges from 0 (never experiences impairment in the activity) to 3

(always experience impairments), with anosognosia determined by calculating discrepancies between patient and informant report. In addition to the ACE-R subscale, memory was tested using the word list memory task from the Consortium to Establish a Registry for Alzheimer's disease (CERAD) battery [START_REF] Morris | The Consortium to Establish a Registry for Alzheimer's Disease (CERAD). Part I. Clinical and neuropsychological assessment of Alzheimer's disease[END_REF]. A ten-item word list is presented to the participant over three trials, each trial immediately followed by a recall phase (immediate recall score; min-max: 0-30). After a short delay (approximately five minutes), the participant is asked to recall as many of the 10 words as they can (delayed recall score; min-max: 0-10). Finally, immediately after the Delayed Recall task, a list of twenty words, with ten new words and ten from the previous list, is shown to participants. The participants respond by identifying (forced choice yes-no) which words had been presented before (recognition score; min-max: 0-20).

To screen for levels of depression and apathy clinically, clinician ratings were used. For depression the selected assessment was the Cornell Scale for Depression in Dementia (clinician; [START_REF] Alexopoulos | Cornell Scale for Depression in Dementia[END_REF]). The Cornell is a 19-item clinician administered interview that relies on information from both the patient and an informant. Each item represents a common symptom of depression in older adults and is scored as absent (0), mild (1) or severe (2). Total scale scores vary from 0 to 38, with higher scores indicating greater severity of depression. Apathy was measured using the clinician version of the Apathy Evaluation Scale (AES; [START_REF] Marin | Reliability and validity of the Apathy Evaluation Scale[END_REF]). The AES-clinician is a semi-structured interview about the participant's interests and daily activities. In the current study, information from the interview was confirmed with an informant in the case of the patients. It is scored in an 18-item scale with a Likert-like system (1 to 4 for each item), with some items being reverse scored; scores range from 18 to 72 and higher scores mean more apathy.

Ethical issues

All participants provided informed consent, with caregivers also giving their agreement for the patient to take part. The project was approved by the South London and Maudsley/ Institute of Psychiatry Ethics Committee (Research Ethics Committee number 08/H0807/6).

Statistical analysis

For each measure (FrsBe subscales [apathy, disinhibition and executive functions], GDS and AQ-D), corrected discrepancy ratio scores [START_REF] Clare | Appraisal of memory functioning and memory performance in healthy ageing and early-stage Alzheimer's disease[END_REF] were calculated using the following formula: e.g. for the FrsBe, (FrsBe-Self -FrsBe-Informant/((FrsBe-Self + FrsBe-Informant)/2)). This method was applied in order to control scaling effects relating to between-subject differences in level of rating or performance. It also provides direct comparison of awareness of domains measured with questionnaires with varying number of items. Corrected discrepancy scores close to zero indicate good agreement participant-informant. Positive scores suggest that participants underestimate their abilities and negative scores suggest that self-rating of difficulties is lower than informant rating (overestimation of ability).

A repeated-measures ANOVA was used to compare the discrepancy ratios across five domains: condition (AQ-D), cognitive ability (FrsBe executive functions), depression (GDS), apathy (FrsBe apathy subscale) disinhibition (FrsBe disinhibition subscale).

Pairwise comparisons with post-hoc t-tests followed the ANOVA. To avoid inflation of type I error, comparisons were adjusted with Bonferroni-Hochberg corrections.

Pearsons correlations investigated the relationship between different forms of awareness.

To explore associations of awareness according to object, Pearson correlations were calculated between each awareness score and demographic (age, gender and educational level) and clinical variables (clinician ratings of apathy and depression, cognitive level [total score in the ACE-R] and memory [total immediate recall]). In the case of gender, point-biserial correlations were used. Considering the exploratory nature of this analysis, corrections for familywise error rate were deemed unnecessary.

-Results

Sample characteristics

Socio-demographic and clinical characteristics of the sample can be seen in Table 1.

Regarding the level of anosognosia of the sample, an unpaired t-test showed that the mean AQ-D score was within the range of, and not significantly different from, the values reported in other studies using the same measure [START_REF] Migliorelli | Anosognosia in Alzheimer's disease: a study of associated factors[END_REF][START_REF] Starkstein | Two domains of anosognosia in Alzheimer's disease[END_REF] (respectively, t(121) = 1.35, p = .18; t(188)= 1.02, p = .31).

PLEASE INSERT TABLE 1 HERE

Awareness of symptom or condition

The repeated-measures ANOVA indicated significant differences between awareness types (F (4, 76) = 15.47, p < .001, ηp 2 = .45). Pairwise comparisons suggested that awareness of condition is significantly more impaired than all other forms of awareness (p < .001), with the exception of awareness of mood disorder (p = .096). Awareness of executive functions was also significantly worse than awareness of disinhibition (p =

.017) and of apathy (p = .024). There were no other significant differences. Results can be seen in Figure 1.

PLEASE INSERT FIGURE 1 HERE

Correlational analysis

Correlations between awareness scores can be seen in Table 2. The majority of awareness

PLEASE INSERT TABLE 2 HERE

There were no significant associations between types of awareness and educational level, gender, age, memory, cognitive level or apathy. There was a significant positive correlation between awareness of condition and depression (r = .50, p = .025) and a trend for a positive correlation between depression and awareness of executive functions (r = .43, p = .058).

-Discussion

The results showed, in a typical AD sample, that the level of awareness differs according to the object studied. Using a ratio method, which allowed direct comparison of unawareness as measured by different scales and controlled between-subject variation due to level of rating or performance, unawareness was shown to be higher for overall condition and executive functions, with relatively more preserved awareness for disinhibition and apathy. Correlational analysis suggested the existence of an association between awareness of condition and awareness for other objects, with the exception of depression. Awareness of depressive symptoms was not related to awareness of condition, nor of disinhibition, and the correlation with awareness of executive functions and apathy was weak.

Awareness was lower for condition, activities of daily living and consequences of the illness, which characterises unawareness in AD as a form of anosognosia, considering the etymology of the term (anosognosia; unawareness of illness). This highlights the important clinical impact of unawareness in AD, suggesting why it is linked to exposure to risks and treatment compliance [START_REF] Patel | Ageing and mental health in a developing country: who cares? Qualitative studies from Goa, India[END_REF][START_REF] Starkstein | Insight and danger in Alzheimer's disease[END_REF].

Unawareness of executive functions was also higher than for other objects, and this may be linked to difficulties in goal-directed activities, leading to unawareness of instrumental activities of daily living, and loss of autonomy and quality of life.

Of note, the results showed better awareness for apathy and disinhibition than for executive functions. The role of the frontal lobes in awareness has been discussed [START_REF] Harwood | Frontal lobe hypometabolism and impaired insight in Alzheimer disease[END_REF][START_REF] Salmon | Neural correlates of anosognosia for cognitive impairment in Alzheimer's disease[END_REF], but the current results suggest that the nature of the frontal function may also be relevant when determining awareness. Alternatively, even though executive functions rely on the frontal lobes, the role of other brain areas has been emphasised [START_REF] Collette | Exploring the unity and diversity of the neural substrates of executive functioning[END_REF][START_REF] Robinson | Neuroanatomical correlates of executive functions: a neuropsychological approach using the EXAMINER battery[END_REF], while apathy and disinhibition may be more strictly linked to frontal lobe dysfunction. The better awareness for a cognitive frontal symptom than for behavioral frontal symptoms can be linked to results highlighting that, generally, unawareness of cognitive deficits is greater that of behavioral disturbance [START_REF] Kotler-Cope | Anosognosia in Alzheimer disease[END_REF][START_REF] Vasterling | Unawareness of deficit in Alzheimer's Disease: Domain-specific differences and disease correlates[END_REF].

The results suggest that awareness for executive functions, apathy and condition correlate strongly, while it seems that awareness of depression is a different construct. With this respect, previous studies have emphasised how awareness of emotional state differs from awareness for activities of daily living and for cognitive functioning [START_REF] Dourado | Awareness of disease in dementia: factor structure of the assessment scale of psychosocial impact of the diagnosis of dementia[END_REF][START_REF] Nelis | Awareness of social and emotional functioning in people with early-stage dementia, and implications for carers[END_REF]. The results of the present study are in line with findings from [START_REF] Verhülsdonk | Anosognosia and depression in patients with Alzheimer's dementia[END_REF], who highlighted a lack of relationship between anosognosia (awareness of condition) and awareness of depressive symptoms.

Most studies exploring the relationship between depression and anosognosia in AD suggest a link between these variables, with greater awareness associated with more depressive symptoms (e.g [START_REF] Clare | Multidimensional assessment of awareness in early-stage dementia: A cluster analytic approach[END_REF][START_REF] Harwood | Impaired insight in Alzheimer disease: association with cognitive deficits, psychiatric symptoms, and behavioral disturbances[END_REF][START_REF] Mograbi | On the relation among mood, apathy, and anosognosia in Alzheimer's disease[END_REF]. The current study provides further evidence for this, with awareness for condition and executive functions showing moderate positive correlations with depression. This relationship was not observed in the case of behavioral and mood disturbance variables, such as apathy, depression and disinhibition. This suggests that depression impacts differentially on awareness according to the object and also contributes to the discussion regarding the direction of causality between depression and awareness [START_REF] Mograbi | On the relation among mood, apathy, and anosognosia in Alzheimer's disease[END_REF]. It has been proposed that depression leads to more awareness because of general over-reporting of problems or depressive realism, but the current findings do not support this view, since some forms of awareness did not correlate with depression. A more plausible explanation, in the context of current findings, is that awareness leads to more depressive symptoms, with awareness of condition and executive functions contributing more to depression arguably because of a greater impact in everyday life.

A control group was not used in the comparison of awareness concerning different objects, because, by definition, a healthy control group would not show the cognitive impairments or behavioral disturbance associated with dementia. Also regarding the awareness measure used in the present study, it is important to note that a discrepancy between caregivers and patients reports does not mean necessarily that there is a deficit in the specific domains. However, in the literature, patient-carer discrepancy is a widely used method to assess awareness [START_REF] Clare | Awareness in dementia: A review of assessment methods and measures[END_REF]. Another limitation of the study refers to the small sample size, which may affect generalizability and statistical power. Regarding generalizability, the sample is similar to other studies exploring awareness in AD and typical of patients in the mild to moderate range of the illness. In addition, the study found a number of significant results, and where significant differences were not found, effect sizes were typically small (e.g. some of the correlations).

In summary, the results of the present study underline the complexity of anosognosia

  2.3.4. Background variablesParticipants were assessed on the revised Addenbrooke's Cognitive Examination (ACE-R;[START_REF] Mioshi | The Addenbrooke's Cognitive Examination revised (ACE-R): A brief cognitive test battery for dementia screening[END_REF]). The ACE-R covers a wider range of cognitive functions, being divided into five domains: attention and orientation, memory, verbal fluency, language and visuospatial skills. Higher scores indicate better cognitive ability and the maximum total score is 100 (min=0).

  scores showed positive correlations, with the coefficients varying from strong significant correlations (e.g. awareness of executive functions and awareness of apathy [r = .88, p < .001] or awareness of condition [r = .89, p < .001]) to weaker correlations (e.g. awareness of disinhibition and awareness of condition [r = .46, p < .05] or awareness of apathy [r = .53, p < .05]), with two non-significant correlation coefficients (awareness of depression and awareness of disinhibition [r = .25, p = .283] and awareness of depression and awareness of condition [r = .03, p = .902]).

Figure 1 --

 1 Figure 1 -Discrepancy ratios for different domains of awareness

  In terms of clinical implications, the study suggests that awareness affects differentially behavioral and cognitive symptoms, indicating the importance of a more extensive evaluation of the patients' capacity to appreciate the presence of deficits and symptoms. Despite having low correlations with other awareness scores, unawareness for depressive symptoms was found to be high. This has important clinical implications in terms of depression diagnosis in the context of dementia. It suggests, for instance, that if diagnosis is based on patients' report, depressive symptoms can be underestimated and the condition remain untreated. Future studies should explore further the correlations between awareness for different objects and implications for quality of life of patients and caregivers.

	Variable	AD group (n=20)
		Mean (SD)
	Age	80.2 (6.7)
	Gender*	12 / 8
	Years of education	10.6 (2.4)
	CERAD Immediate recall	12 (3.9)
	Delayed recall	1.3 (1.2)
	Recognition	15.2 (3.6)
	# of Intrusions	2.0 (1.7)
	ACE-R Total	64.1 (10)
	Attention and	14.2 (2.3)
	orientation Memory	9.4 (4.1)
	Fluency	7.2 (3.0)
	Language	20.2 (3.4)
	Visuospatial
	MMSE	23.3 (2.9)
	AES	40.9 (9.2)
	CORNEL	4.0 (3.7)
	AQ-D	20.3 (12.1)

and the relative independence of awareness for depressive symptoms in relation to other objects. * # female/ male.

Table 2 -

 2 Correlations between awareness variables

	Variable	Disinhibition	Executive functions	Apathy	Depression
	Executive functions	.56
	Apathy	.53	.88
	Depression	.25	.52	.52
	Condition	.46	.89	.83	.03

Significant results are presented in bold.

Acknowledgements

This paper represents independent research part funded by the National Institute for Health Research (NIHR) Biomedical Research Centre at South London and Maudsley NHS Foundation Trust and King's College London. The views expressed are those of the authors and not necessarily those of the NHS, the NIHR or the Department of Health. The authors also acknowledge support from the CAPES foundation.