Alireza Rezaei

Sylvie Le Hégarat-Mascle

Emanuel Aldea

Piercarlo Dondi
email: piercarlo.dondi@unipv.it

Marco Malagodi
email: marco.malagodi@unipv.it

One step clustering based on a-contrario framework for detection of alterations in historical violins

Preventive conservation is an important practice in Cultural Heritage. The constant monitoring of the state of conservation of an artwork helps us reduce the risk of damage and number of necessary interventions. In this work, we propose a probabilistic approach for the detection of alterations on the surface of historical violins based on an a-contrario framework.

Our method is a one step NFA clustering solution which considers grey-level and spatial density information in one background model. The proposed method is robust to noise and avoids parameter tuning and any assumption about the quantity of the worn-out areas. We have used as input UV induced fluorescence (UVIFL) images for considering details not perceivable with visible light. Tests were conducted on image sequences included in the "Violins UVIFL imagery" dataset. Results illustrate the ability of the algorithm to distinguish the worn area from the surrounding regions. Comparisons with state-of-the-art clustering methods show improved overall precision and recall.

I. INTRODUCTION

Preventive conservation, a crucial procedure in Cultural Heritage, consists in the constant monitoring of artworks and monuments to reduce the risk of damages and to minimise restorations [START_REF] Bradley | Preventive conservation research and practice at the british museum[END_REF], [START_REF] Lucchi | Review of preventive conservation in museum buildings[END_REF]. This practice is particularly complex and requires an interdisciplinary approach to correctly interpret and to manage the effects of chemical, physical and biological alterations [START_REF] Ghedini | Atmospheric aerosol monitoring as a strategy for the preventive conservation of urban monumental heritage: The florence baptistery[END_REF], [START_REF] Ortiz | Vulnerability index: A new approach for preventive conservation of monuments[END_REF], [START_REF] Perles | An energy-efficient internet of things (IoT) architecture for preventive conservation of cultural heritage[END_REF]. Historical wood musical instruments (such as violins or violas) are a peculiar kind of artwork because they are both held in museums and played (even today), leading to a major risk of mechanical wear in the areas in direct contact with the musicians. In particular, the monitoring of historical violins presents various complexities: (i) the instruments have undergone multiple restorations during centuries, leading to a stratified surface difficult to analyse; (ii) varnish wear can evolve in different ways depending on the initial conditions of the surface and on the different substances being present; (iii) the rounded morphology and the high reflectance of the varnishes generate noisy reflections during photo acquisition (almost impossible to avoid completely) that can be confused with potential alterations; (iv) in order to avoid damages to the varnishes, violins cannot be rigidly fixed to a support, thus, slight misalignment between photographic sessions can occur. If we consider the entire surface of an instrument these last two issues are not critical; however, for detecting small alterations (especially in the initial steps) we need to focus on the areas always in contact with the musicians (mainly top left and bottom right of the back plate), taking close-up pictures of these details that are naturally more subject to this kind of acquisition noise. The combined use of multiple analytical techniques proved to be a valid approach for handling these issues [START_REF] Fichera | Innovative monitoring plan for the preventive conservation of historical musical instruments[END_REF], [START_REF] Rovetta | The case of antonio stradivari 1718 ex-san lorenzo violin: History, restorations and conservation perspectives[END_REF], however, mixing more techniques is a very slow and time-consuming process. A more efficient procedure will consist of regular analysis of images in order to quickly identify possible altered areas, and then apply only on them spectroscopic techniques as confirmation.

In this study, we have used multi-temporal UV induced fluorescence (UVF or UVIFL) images collected in the "Violins UVIFL imagery" dataset1 [START_REF] Dondi | Segmentation of multi-temporal uv-induced fluorescence images of historical violins[END_REF]. UVIFL photography is a widely adopted non-invasive diagnostic technique that allows to see details not perceivable using visible light [START_REF] Janssens | Non-destructive micro analysis of cultural heritage materials[END_REF]. In particular, for the study of historical violins, UVIFL images are generally used to highlight possible restorations or interesting areas [START_REF] Dondi | Automatic analysis of uv-induced fluorescence imagery of historical violins[END_REF] and to decide where to apply more precise but slower diagnostic techniques, like X-Ray Fluorescence (XRF) [START_REF] Rovetta | The elemental composition of stradivari's musical instruments: new results through non-invasive edxrf analysis[END_REF] or Fourier Transform Infrared (FTIR) spectroscopy [START_REF] Invernizzi | A noninvasive stratigraphic study by reflection ft-ir spectroscopy and uvinduced fluorescence technique: The case of historical violins[END_REF]. The possibility to see "hidden" details is particularly valuable in our scenario since it potentially allows for an early detection of new alterations. In fact, visible light can be deceptive. When a new alteration is clearly identifiable under a standard illumination, it is likely that an irreversible damage to the surface varnish has already occurred. Moreover, substances used for cleaning the instruments can temporarily hide the presence of alterations, and, thus, slow down the detection. On the contrary, the effectiveness of UVIFL photography to characterise variations in varnishes is well known in the Cultural Heritage field [START_REF] Thoury | Nondestructive varnish identification by ultraviolet fluorescence spectroscopy[END_REF].

Then, wear detection can be viewed as a semantic segmentation problem with two semantic classes, namely the intact areas and the wear region(s). However, wear features are highly variable both in terms of radiometry and in terms of geometric shape so that we would rather cast the wear detection problem as a change detection problem assuming we have at our disposal a reference image taken from the initial state of the instrument. Then, a famous framework to handle 2-class problems with one of the class unknown (the change one) is the a-contrario framework [START_REF] Desolneux | Gestalt theory and computer vision[END_REF]. It takes inspiration from the statistical rejection tests with the null hypothesis (H 0 ) being "the pixel does not correspond to a change". However, acontrario approaches differ on the two following points: (i) the null hypothesis (i.e., the one to reject) is represented through a "naive" model that corresponds to unstructured data; (ii) the decision is taken based on the Number of False Alarms (or significance). In this work, we describe a new method for early detection of superficial alterations on historical violins based on the a-contrario framework [START_REF]Meaningful alignments[END_REF], [START_REF] Desolneux | A grouping principle and four applications[END_REF]. This method can be applied with minor modifications to any wooden instruments. Also, by replacing the acquisition and pre-processing steps, we can cover a wider variety of change detection problems. The main contribution of the proposed approach is to encompass in a single model both criteria that wear areas present: a high density of pixels having different radiometry with respect to the reference image. It also allows us to be free from any threshold decision and cluster feature specification.

II. RELATED WORKS

The a-contrario framework introduced by Desolneux et al. [START_REF]Meaningful alignments[END_REF] has been used in image analysis in a variety of applications such as texture analysis [START_REF] Grosjean | A-contrario detectability of spots in textured backgrounds[END_REF], motion detection [START_REF] Veit | An a contrario decision framework for region-based motion detection[END_REF], [START_REF]Space-time a contrario clustering for detecting coherent motions[END_REF], edge and line detection [START_REF] Widynski | A contrario edge detection with edgelets[END_REF], [START_REF] Akinlar | Edlines: A real-time line segment detector with a false detection control[END_REF], [START_REF] Aldea | Robust crack detection for unmanned aerial vehicles inspection in an a-contrario decision framework[END_REF], [START_REF] Liu | A contrario comparison of local descriptors for change detection in very high spatial resolution satellite images of urban areas[END_REF], or reconstruction from motion [START_REF] Moulon | Adaptive structure from motion with a contrario model estimation[END_REF]. In all these studies, the detection is performed by rejecting a naive model which describes the statistic of the unstructured data. At the same time, grouping principles have been used for tasks related to the higher level perceptual organization of scenes [START_REF] Michaelsen | Self-organizing maps and gestalt organization as components of an advanced system for remotely sensed data: An example with thermal hyper-spectra[END_REF], [START_REF] Yan | Unsupervised image saliency detection with gestalt-laws guided optimization and visual attention based refinement[END_REF], thanks to their general nature.

In our case, we are interested in the applications dealing with the detection of changed areas across multi-temporal images. In early studies, the a-contrario framework and spectral invariant features have been used to detect meaningful changes between two satellite images of the same area taken at different times [START_REF] Lisani | Detection of major changes in satellite images[END_REF]. An a-contrario approach has also been proposed for change detection in three dimensional multi-modal medical images such as Magnetic Resonance sequences [START_REF] Rousseau | An a contrario approach for change detection in 3d multimodal images: application to multiple sclerosis in mri[END_REF]. In 2010, Robin et al. use the a-contrario framework for the definition of a criterion assessing the level of coherence in a sequence of images for detecting sub-pixel changes in a time-series of satellite images [START_REF] Robin | An a-contrario approach for subpixel change detection in satellite imagery[END_REF]. Flenner et al. further investigated this approach by using exchangeable random variables instead of relying on the independent and identically distributed (IID) assumption [START_REF] Flenner | A helmholtz principle approach to parameter free change detection and coherent motion using exchangeable random variables[END_REF]. All these works focus on the grey level values (and their changes) so that the considered naive model deal with grey level discrepancy. Now, numerous works have been proposed to deal with objects characterized by their spatial feature consistency. Searching for a pattern using a-contrario framework can be done either by computing the significance of every pattern corresponding to a parametric shape (e.g., circle [START_REF] Akinlar | Edcircles: A real-time circle detector with a false detection control[END_REF], line, ellipse [START_REF] Pȃtrȃucean | Joint a contrario ellipse and line detection[END_REF]) or by checking possible clusters of points regardless of their shape [START_REF] Palma | Detection of masses and architectural distortions in digital breast tomosynthesis images using fuzzy and a contrario approaches[END_REF], [START_REF] Zair | A-contrario modeling for robust localization using raw GNSS data[END_REF]. In these works, even if one may consider more complex models which allow for taking into account dependence in the image [START_REF] Myaskouvskey | Beyond independence: An extension of the a contrario decision procedure[END_REF], the usual naive model that represents unstructured data is the uniform distribution, for instance of the gradient orientations in [START_REF] Von Gioi | Lsd: A fast line segment detector with a false detection control[END_REF] or simply of the location of the 1-valued pixels in [START_REF]Meaningful alignments[END_REF] and [START_REF] Hégarat-Mascle | Efficient evaluation of the number of false alarm criterion[END_REF] that process binary images.

Based on all previous cited works, a straightforward solution considered in [START_REF] Aldea | Robust crack detection for unmanned aerial vehicles inspection in an a-contrario decision framework[END_REF], [START_REF] Rezaei | Detecting alterations in historical violins with optical monitoring[END_REF] is to first use an a-contrario approach to detect points that, according to their grey level values, are likely to belong to a change area, to store these points as a binary image, and then group these points (1-valued in the binary image) together [START_REF] Desolneux | A grouping principle and four applications[END_REF]. However, with such an approach the two criteria on grey level values and spatial features of the researched objects or areas are considered sequentially and there is an implicit threshold (at the end of the first step) that makes us lose the information about the intensity of the change. Our proposal is then to combine both steps and cluster a greylevel image by using a single naive model.

III. PROPOSED APPROACH

A. Notations and key idea

Let K N + denote a set of indices, and let |K| = K denote its cardinality. A series of images I i , i ∈ K, are captured from a wood sample with pre-defined time intervals. They are in conventional RGB format. For notation convenience, in the following we assume K = {0, . . . , K -1} so that I 0 is the original image which serves as a reference. Comparison of each I i , i ∈ {1, . . . , K -1} with I 0 using a colour difference formula [START_REF] Luo | The development of the cie 2000 colourdifference formula: Ciede2000[END_REF] gives a difference map ∆I i . ∆I i is a grey-level image defined on the pixel domain P ⊂ N 2 Our problem boils down to segmenting each image ∆I i , i ∈ {1, . . . , K -1}, with respect to semantic classes, one of which representing the wear area(s). Note that, even if, according to expert knowledge the temporal evolution of the wear areas includes some information, in the perspective of the evaluation of a new change detection algorithm, it is left aside for the wear estimation and only considered for qualitative validation of our results. Now, as already said, we will encompass both criteria characterising a wear area in ∆I i images (radiometric and spatial) in a single naive model.

The basic idea is to extend the meaningfulness concept specifying that a cluster is all the more significant that it is very dense (i.e., its points are 'surprisingly' close) not only spatially but also in terms of grey-level differences. Now, considering grey-level differences, low values correspond (mainly) to no change and extended high values (mainly) to changes, so that a grey-level transform is required to meet the assumption that a change can be detected as surprisingly structured or dense values. Then, using the cluster NFA based on distance, the proposed method also needs the specification of the considered distance. These two points are presented in the next subsections before the NFA computation and cluster detection algorithm.

B. Grey-level transformation

Let us first enumerate the desirable properties for the researched grey-level transformation for ∆ pixel values: after transformation, (i) grey level values of pixels belonging to unchanged areas shall be stretched, (ii) grey level values of pixels belonging to change areas shall be similar and (iii) close to zero. This last property aims at controlling not only the relative values of grey-level differences but also their absolute value. Then, the grey-level function (f ) that we can consider has to:

• be decreasing;

• spread no significant grey-level differences so that uniform distribution will be acceptable. In this study, two reasonable f functions were evaluated: the inverse function and the tanh function, with satisfactory results. Both provide expected discrepancy of the grey-value cloud as shown in Fig. 1

C. Distance between two points

The cluster detection using a-contrario is based on point distance calculation [START_REF] Desolneux | From gestalt theory to image analysis: a probabilistic approach[END_REF]. In order to take into account both spatial distance and (transformed) grey-level differences, the handled distance is a weighted sum of two terms: the 2D spatial proximity and a term representing the modified greylevel of each point. In this way, we can enforce that points with higher (f -transformed) grey-level values are considered closer together compared to points with lower grey-level. Since the grey-level values and spatial distance are inherently in different scales, we use the constant scale factor c to make them of the same order of magnitude. Note that, even if appearing as a parameter, this factor c is very easy to set based on image resolution features (spatial and radiometric). Denoting by y i the value at pixel i ∈ P, and by D sp (i, j) the 2D spatial distance between location of pixels i and j, ∀(i, j) ∈ P 2 ,

D(i, j) = (D sp (i, j)) 2 + c × (f (y i )) 2 + (f (y j )) 2 (1)
Finally, for any distance value d, a cluster C ⊆ P is a set

of points i such that ∀i ∈ C, ∃j ∈ C s.t. D(i, j) ≤ d, ∀j ∈ P \ C, D(i, j ) > d.
Note that for a given d there may be several distinct clusters C i verifying previous definition. Inversely, for a given cluster C, there is a range of distances leading to C that allows us to associate an inner border and an outer border to cluster C. In the following, we denote d min (C) and d max (C) the bounds of this interval.

D. Number of False Alarms

The Number of False Alarms (NFA) is based on the considered naive model. In our case, this latter is the uniform distribution:

Definition 1 (Naive model M). The set of points S is a random set of |S| independent uniformly distributed variables over the 3D (2D+Greylevel) space of the image.

Note that a keypoint of a-contrario approaches is that the naive model has not to be exact, but it has only to be contradicted in the case of the researched structured data (wear in our application).

The Number of False Alarms is computed extending the NFA proposed in [START_REF] Desolneux | A grouping principle and four applications[END_REF], defined for 2D cluster detection. Considering here a 3D space, the 2D surface areas are replaced by 3D volumes and the 2D distance by the distance defined in Eq. ( 1), so that, for any cluster C of 3D (2D+Greylevel) points,

N F A M (C, M ) = N test M i=k M i V i C (1 -V C ) M -i ( 2
)
where k is the number of points in the cluster, M is the total number of points and V C and V C are the lower and upper bounds of the relative volume of the cluster compared to the whole image cube. These regions are obtained by performing morphological operations on the union of every point in the cluster [START_REF] Desolneux | A grouping principle and four applications[END_REF]. In our application, for simplicity of computation, we estimate both V C and V C by utilising the area of the 2D surfaces associated with these 3D regions:

   V C = A C × max i∈C (f (y i )) /V P V C = A C × max i∈C (f (y i )) /V P (3)
where y i is the grey-level of point i, f is the transformation function, V P is the volume of the image cube, and A C (A C ) is the upper (lower) bound of the area of the cluster in the 2D image.

IV. IMPLEMENTATION

The implementation process (Figure 2) begins with an application dependent step. This step can be switched out to adapt our proposed algorithm to new applications. Once the acquisition is complete, it is imperative to make sure the two images we compare are spatially registered and have the same general lightness level. Indeed, the images can be taken in different conditions and by different operators. The algorithm should remain robust to these changes.

Next, we compute a difference map between a given frame and the reference frame (t 0 ). The difference map between t 0 and t i , i ∈ {1, . . . , K -1}, is created by computing the color difference between the corresponding pixels. We have utilized the CIEDE2000 formula, after transforming the RGB values into the CIELAB color space.

It is worth noting that other approaches may also be used to create the difference map for the rest of the detection process, from a simple Euclidean distance between RGB values to more complicated color difference models. However, in our case the employed difference provides consistent results which agree with the expected wear areas.

Application dependent steps

Image acquisition

Illumination correction Spatial Registration

Difference map

Color difference

Transformation

Change detection and clustering

NFA computation

Gray level image

Two RGB images

Ranked Clusters

Time 0 and i

Fig. 2: General steps of the proposed algorithm

The difference image is then used for finding the clusters and computing their significance. First, we create a minimum spanning tree of the points in the difference image based on the distances computed by Eq. ( 1). Each sub-tree then is considered a potential cluster and by computing the meaningfulness value we decide which ones to keep and which ones to discard. Algorithm 1 describes the process step by step.

Calculating the Eq. ( 1) can be difficult without an approximation. In images with small pixel numbers (around 1000 or lower) we can calculate the exact values using BigInt data types. In other cases, an approximation can be viable if we restrict the area possible for a cluster compared to the whole image. Since the wear area is not covering the image, this restriction will not hinder the process.

V. RESULTS

A. Dataset

The "Violins UVIFL imagery" dataset is a public dataset containing multi-temporal UVIFL images of historical and sample violins. The former regularly acquired during a short period of use, the latter artificially altered in laboratory to simulate various possible alterations over a long-term use. Currently, the dataset includes image sequences of two historical violins held in Museo del Violino in Cremona (Italy), "Carlo IX" (c.1566) made by Andrea Amati and "Vesuvio" (1727) made by Antonio Stradivari, and two artificially created sample sequences. The alterations were created scrubbing the surface with a cloth damped with alcohol to reproduce, as faithfully as possible, the effect of mechanical wear during playing.

For creating the first artificial sequence, called WSO1 (Figure 3(a) and (b)), a wood sample was slowly altered several times to simulate an alteration in an area with intact varnish. This set contains one reference image of the initial state of the sample and 20 altered frames.

The second sequence, called SV01 (Figure 3(c) and (d)), contains images of the lower part of the back plate of a sample violin. This set simulates the growing of wear starting from an Algorithm 1 Change detection between frame i and the reference frame 1: Perform the pre-processing 2: Compute the color difference map ∆I between I i and I 0 3: for each pixel j in ∆I do 4:

∆I(j) = f (∆I(j)) 5: end for 6: Normalize ∆I between 0 and 255 7: Compute the distance D sp between each pair of points in ∆I 8: for each pair of pixels j and j in ∆I do Compute NFA value (up to scale owing to N test ) according to Eq. ( 2) using values k, M , V C and V C 17: end for 18: Find disjoint sub-trees with maximum NFA 19: Sort sub-trees according to their NFA area already ruined and consists of one reference image and 20 altered frames.

All the images were acquired following our previously defined acquisition protocol [START_REF] Dondi | An interactive tool for speed up the analysis of UV images of Stradivari violins[END_REF], using a Nikon D4 full-frame digital camera with a 50 mm f/1.4 Nikkor objective, 30s exposure time, aperture f/8, ISO 400 and two wood lamp tubes (Philips TL-D 36 W BBL IPP low-pressure Hg tubes, 40 Watt, emission peak ∼ 365nm) as UV-A lighting source.

For the purpose of this paper we focused only on the two artificial sequences, since the period of examination for the two historical violins was relatively short and only "Vesuvio" showed a very slight alteration on its back plate.

B. Outputs

Figure 4 shows the result of clustering for four sample frames of WS01. As we can see, small noises change from frame to frame, big artefacts have a constant size and location, and the wear area grows over time. In all cases, small noises have been ignored and significant high change areas have been identified.

C. Evaluation

For the purpose of evaluation, we need a viable existing process as an alternative for our proposal. As mentioned before, our proposal deals with a certain set of limitations and assumptions; so, it is imperative that the comparison is made with a process which can also work in those conditions. Therefore, we are looking for these features: • The available number of images in the problem domain is very limited. • Number of clusters is unknown beforehand.

• The general shape of a wear cluster is unknown.

• There is no clear threshold between background and foreground.

• The data contains noise and artifacts.

• The process should be automatic and not interactive.

• There should be a way to rank the resulted clusters. For separating the background from the foreground, we use FRFCM proposed by Lei et al. [START_REF] Lei | Significantly fast and robust fuzzy c-means clustering algorithm based on morphological reconstruction and membership filtering[END_REF], which is a modified fuzzy cmeans algorithm. They incorporate local spatial information by using morphological reconstruction, which improves the classic fuzzy cmeans to help dealing with the different types of noise. Applying this method gives us consistently good separation between background and foreground which makes it a good candidate for the evaluation of our algorithm.

To spatially cluster the points produced from FRFCM we have chosen a density based algorithm proposed by Campello et al. [START_REF] Campello | Hierarchical density estimates for data clustering, visualization, and outlier detection[END_REF]. HDBSCAN (or Hierarchical Density-Based Spatial Clustering, with Application with Noise) is a framework for density-based cluster analysis. The algorithm produces a complete clustering hierarchy of all possible density based clusters and crucially for us -it also provides a global, optimal non-hierarchical solution which maximizes the overall stability of the proposed clusters. This means we can evaluate our automatic process (for the number of clusters) with a direct comparison of resulted clusters from both methods.

For the comparison to be fair, we also need to rank the produced clusters based on a specific feature. The meaningfulness value which we assign to each cluster does not directly translate to the grey-level value or shape of the cluster; but it can be mimicked generally by the area of the cluster. Therefore, we sort the results of HDBSCAN based on the number of pixels in each cluster in descending order.

The ground truth for each frame has been created manually with our best estimation as to where the wear region is. In the absence of any exact (due intrinsically to the wear construction process and to the inability to apply physico-chemical analysis on a large scale) knowledge on the boundaries of wear regions, they have been defined intentionally very loose.

We directly compared our k(k ≤ 4) most significant clusters with the first k clusters we get from the mentioned process in 4 separate comparisons. The ideal scenario is that the wear region appears as the rank 1 cluster; but depending on the position of the image in the series and on the amount of artifacts being present, we may get lower rank wear regions.

Tables I and II show three sample frames from each dataset: their ground truth and output of each method. A qualitative consideration of the results indicates:

• The proposed method has segmented the difference image with regards to both grey-level values and spatial density in one step. The background separation is done seamlessly and correctly. • We have successfully dealt with background noise and artifacts from UV reflections in the majority of cases. For example, in dataset two, reflections on the border of the violin are very close to the actual wear region. The NFA clustering has managed to avoid them completely or to a significant degree while FRFCM+HDBSCAN have group them together with the wear in a few cases. Inherently the NFA clustering allows for controlling the number of false alarms. This results in globally better wear detection (lower false positive values by accepting more false negative values).

• The wear region has been identified without any prior assumption about its position or shape. In addition, the most significant clusters have been detected without the need for number of clusters as an input.

For quantitative evaluation, precision and recall are computed regarding three cases: our proposal, the FRFCM [START_REF] Lei | Significantly fast and robust fuzzy c-means clustering algorithm based on morphological reconstruction and membership filtering[END_REF], and the combination of FRFCM [START_REF] Lei | Significantly fast and robust fuzzy c-means clustering algorithm based on morphological reconstruction and membership filtering[END_REF] and HDBSCAN [START_REF] Campello | Hierarchical density estimates for data clustering, visualization, and outlier detection[END_REF] as mentioned before.

Figure 5 shows the precision/recall charts for sequences WS01 and SVO1. For these plots the first cluster in the ranked order has been considered. In both sets, the proposed NFA clustering has better precision while maintaining an acceptable recall in most cases:

• For set WS01, in average, we achieve 87% precision and 89% recall which is higher than the average scores for FRFCM+HDBSCAN (50% precision and 75% recall). Number of failures to identify the wear region in rank one is one for NFA and three for FRFCM+HDBSCAN. • For set SV01, which contains more noise and artifacts, on average, we achieve 77% precision and 69% recall as opposed to 46% precision and 90% recall for FR-FCM+HDBSCAN. The number of failures to identify the wear region in rank one is zero for NFA and one for FRFCM+HDBSCAN.

VI. CONCLUSION

In this paper, we introduced a probabilistic method for detecting significant changes located in clusters within a series of images. The proposed method performs an automatic clustering process directly on the grey-level difference image, while dealing with the background noise and artifacts. Comparisons with recent clustering methods show meaningful improvements while having the benefit of an inherent ranking criterion for the produced clusters.

This approach can be used in preventive conservation as a fast, preliminary examination of the surface of a violin able to identify the most likely altered areas. Thus, a verification using more precise but slower techniques (like spectroscopic analyses) will be done only on the detected areas, optimising the monitoring procedure.

For future studies, we intend to perform a long-term monitoring process considering real historical violins which are played weekly. Beside creating a valuable dataset for our community, this will allow us to validate our algorithm as well on real wear patterns.

Lastly, while in the current work we have only used one pair of images at a time to track the changes, it is interesting and useful to incorporate time information in our model. Inherently, the evolution of the wear region through time is different from that of artifacts being present in the image; this can help us differentiate wear regions better. Also, in an ongoing monitoring process it can help us detect a newly created wear at an earlier stage.

  .

Fig. 1 :

 1 Fig. 1: 3D point cloud (a) before applying the function f , (b) after applying f (x) = 1x and (c) after applying f (x) = tanh(th -x). The vertical axis represents the (transformed) grey-level values while the other two axes are for the spatial data

9 :

 9 Compute D(j, j ) according to Eq. (1) 10: end for 11: Compute the minimum spanning tree T for the points in ∆I 12: for each subtree in T defining a cluster C do 13:Compute V C and V C according to Eq. (3) 14:M ← the total number of points in the spanning tree15: k ← the number of points in C 16:

Fig. 3 :Fig. 4 :Fig. 5 :

 345 Fig. 3: Samples UVIFL images contained in the dataset: (a) and (b) from set WS01, (c) and (d) from set SV01.

TABLE I :

 I Comparison of the result of proposed NFA clustering, FRFCM+HDBSCAN clustering and the ground truth for examples drawn from set WS01.

	No.	Ground truth	NFA clustering	FRFCM+HDBSCAN
	S 1 : 9
	S 1 : 15
	S 1 : 20

TABLE II :

 II Comparison of the result of proposed NFA clustering, FRFCM+HDBSCAN clustering and the ground truth for examples drawn from set SV01.

	No.	Ground truth	NFA clustering	FRFCM+HDBSCAN
	S 2 : 9
	S 2 : 15
	S 2 : 20

https://vision.unipv.it/research/UVIFL-Dataset/

• Segmentation should be done using both the grey-level and the density information.


Francesco Sarti

Niccolo Caselli

Federico La China

Francesco Biccari

Ughetta Torrini

Francesca Intonti

Anna Vinattieri

Elena Duran Valdeiglesias

Weiwei Zhang

Adrien Noury

Niccolò Caselli

Elena Duran-Valdeiglesias

Niccolò Caselli

Coupling of semiconductor carbon nanotubes emission with silicon photonic micro ring resonators

teaching and research institutions in France or abroad, or from public or private research centers.

HAL

  is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come


Thi Thuy

Duong Dinh

Xavier Le Roux

Natnicha Koompai

Daniele Melati

Miguel Montesinos-Ballester

David González-Andrade

Pavel Cheben

Aitor V Velasco

Eric Cassan

Delphine Marris-Morini

Thi Thuy Duong Dinh
email: thi-thuy-duong.dinh@c2n.upsaclay.fr

Laurent Vivien

Carlos Alonso-Ramos

Mid-infrared Fourier-transform spectrometer based on metamaterial lateral cladding suspended silicon waveguides

teaching and research institutions in France or abroad, or from public or private research centers.

Integrated mid-infrared micro-spectrometers have a great potential for applications in environmental monitoring and space exploration. Silicon-on-insulator (SOI) is a promising platform to tackle this integration challenge, due to its unique capability for large volume and low-cost production of ultra-compact photonic circuits. However, the use of SOI in the mid-infrared is restricted by the strong absorption of the buried oxide layer for wavelengths beyond 4 μm. Here, we overcome this limitation by utilizing metamaterial-cladded suspended silicon waveguides to implement a spatial heterodyne Fourier-transform (SHFT) spectrometer operating near 5.5μm wavelength. The metamaterial-cladded geometry allows removal of the buried oxide layer, yielding measured propagation loss below 2 dB/cm between 5.3μm and 5.7μm wavelengths. The SHFT spectrometer comprises 19 Mach-Zehnder interferometers with a maximum arm length imbalance of 200 μm, achieving a measured spectral resolution of 13cm -1 and a free-spectral range of 100 cm -1 near 5.5μm wavelength.

Optical spectrometers are widely used in diverse applications such as food safety, medical diagnosis, indoor air quality monitoring, astrophysics science and resource exploration [START_REF] Ferrari | A brief review on the history of human functional nearinfrared spectroscopy (fNIRS) development and fields of application[END_REF][START_REF] Zhang | Research progress on on-chip Fourier transform spectrometer[END_REF]. The mid-infrared (mid-IR, 2-20 μm wavelengths) is a particularly interesting wavelength range for spectroscopic applications as it contains the absorption fingerprints of many chemical and biological substances of interest [START_REF] Soref | Mid-infrared photonics in silicon and germanium[END_REF][START_REF] Hu | Silicon photonic platforms for mid-infrared applications[END_REF]. Emerging applications in planetary exploration and environmental monitoring also require the development a new generation of integrated spectrometers, providing high-sensitivity, real-time monitoring with a compact and low-cost technology [START_REF] Zhang | Research progress on on-chip Fourier transform spectrometer[END_REF]. In this context, multiaperture spatial heterodyne Fourier-transform (SHFT) spectrometers [START_REF] Cheben | A new wavelength dispersive device based on a Fourier transform Michelson type arrayed waveguide grating[END_REF][START_REF] Florjanczyk | Multiaperture planar waveguide spectrometer formed by arrayed Mach-Zehnder interferometers[END_REF][START_REF] Velasco | High-resolution Fourier-transform spectrometer chip with microphotonic silicon spiral waveguides[END_REF][START_REF] Bock | Subwavelength grating Fourier-transform interferometer array in silicon-on-insulator[END_REF][START_REF] Nedeljkovic | Mid-infrared silicon-on-insulator Fourier transform spectrometer chip[END_REF][START_REF] Podemore | Demonstration of a compressive-sensing Fouriertransform on-chip spectrometer[END_REF][START_REF] Liu | Integrated broadband dual-polarization Ge-rich SiGe mid-infrared Fourier-Transform spectrometer[END_REF][START_REF] Heidari | On-chip Fourier transform spectrometer on silicon-on-sapphire[END_REF][START_REF] Montesinos-Ballester | On-chip Fouriertransform spectrometer based on spatial heterodyning tuned by thermo-optic effect[END_REF][START_REF] Wang | On-chip polarization-insensitive Fourier transform spectrometer[END_REF][START_REF] Kita | Highperformance and scalable on-chip digital Fourier transform spectroscopy[END_REF][START_REF] Dinh | Silicon photonic on-chip spatial heterodyne Fourier transform spectrometer exploiting the Jacquinot's advantage[END_REF] provide key advantages in terms of optical throughput, resolution and robustness against fabrication imperfections, compared to dispersive devices like arrayed waveguide grating (AWGs) [START_REF] Cheben | Wavelength dispersive planar waveguide devices: echelle gratings and arrayed waveguide gratings[END_REF][START_REF] Akca | Interleaved silicon nitride AWG Spectrometers[END_REF][START_REF] Cheben | A high-resolution silicon-on-insulator arrayed waveguide grating micros-pectrometer with sub-micrometer aperture waveguides[END_REF] and echelle gratings [START_REF] Janz | Planar waveguide echelle gratings in silica-on-silicon[END_REF][START_REF] Ma | High-resolution compact on-chip spectrometer based on an Echelle grating with densely packed waveguide array[END_REF]. Furthermore, the combination of the SHFT architecture and machine-learning algorithms enable advanced data analysis, yielding improved performance in terms of resolution and robustness against environmental variations such as temperature fluctuations [START_REF] Kita | Highperformance and scalable on-chip digital Fourier transform spectroscopy[END_REF][START_REF] Herrero-Bermello | On-chip Fouriertransform spectrometers and machine learning: a new route to smart photonic sensors[END_REF]. Silicon photonics have a great potential for addressing the integration challenges of compact and low-cost integrated spectrometers in the mid-IR, and particularly of SHFT spectrometers, due to its unique high-index contrast and mass-fabrication advantages [START_REF] Soref | Mid-infrared photonics in silicon and germanium[END_REF][START_REF] Hu | Silicon photonic platforms for mid-infrared applications[END_REF]. Mid-IR SHFT spectrometers implemented with the SiGe technology have been recently demonstrated reaching 8.5 μm wavelength with a resolution of 15 cm -1 [START_REF] Liu | Integrated broadband dual-polarization Ge-rich SiGe mid-infrared Fourier-Transform spectrometer[END_REF]. Yet, SHFT spectrometer demonstrations based on silicon-on-insulator (SOI) technology are limited to 3.75 μm wavelength (2 cm -1 resolution) [START_REF] Nedeljkovic | Mid-infrared silicon-on-insulator Fourier transform spectrometer chip[END_REF]. The main reason for this is the strong absorption of the silica cladding for wavelengths above 4 μm [START_REF] Hu | Silicon photonic platforms for mid-infrared applications[END_REF]. This limitation could be partially overcome by using the silicon-on-sapphire technology to reach 5 μm wavelength (transparency limit of sapphire). However, the only demonstration of an SHFT spectrometer in the silicon-onsapphire technology operated near 3.3 μm wavelength, with a resolution of 10 cm -1 [START_REF] Heidari | On-chip Fourier transform spectrometer on silicon-on-sapphire[END_REF]. Suspended silicon membrane waveguides have been identified as a promising solution to exploit the full transparency of silicon (1.1-8 μm wavelength range), as they enable the selective removal of the buried oxide layer under the waveguide core, while taking advantage of the mature SOI technology [START_REF] Wang | High-responsivity graphene/siliconheterostructure waveguide photodetectors[END_REF][START_REF] Soler Penadés | Suspended SOI waveguide with sub-wavelength grating cladding for mid-infrared[END_REF][START_REF] Soler Penades | Suspended silicon mid-infrared waveguide devices with subwavelength grating metamaterial cladding[END_REF][START_REF] Soler Penadés | Suspended silicon waveguides for longwave infrared wavelengths[END_REF][START_REF] Zhou | Fully suspended slot waveguide platform[END_REF][START_REF] Liu | Suspended silicon waveguide platform with subwavelength grating metamaterial cladding for long-wave infrared sensing applications[END_REF][START_REF] Kou | Mid-IR broadband supercontinuum generation from a suspended silicon waveguide[END_REF][START_REF] Nader | Infrared frequency comb generation and spectroscopy with suspended silicon nanophotonic waveguides[END_REF]. However, this approach has not been yet exploited for the implementation of mid-IR SHFT spectrometers in silicon. In this letter, we use suspended silicon waveguides with metamaterial-grating cladding to implement an SHFT spectrometer operating near 5.5 μm wavelength, effectively circumventing operational wavelength limitations caused by BOX absorption. The metamaterial-grating cladding provides mechanical stability and effective lateral index contrast required to confine the optical mode, while allowing fabrication with a single Si etch step [START_REF] Soler Penadés | Suspended SOI waveguide with sub-wavelength grating cladding for mid-infrared[END_REF]. The suspended waveguides, with a core thickness of 700 nm, exhibit measured propagation loss below 2 dB/cm in the 5.3-5.7 μm wavelength range. We experimentally demonstrate an SHFT spectrometer comprising 19 Mach-Zehnder interferometers (MZI) with a measured resolution of 13 cm -1 and a free-spectral range of 120 cm -1 near 5.5 μm wavelength.

SHFT spectrometers are generally formed by an array of MZIs, each with a different optical path length difference, thereby creating a spatial interferogram from which the input spectrum is retrieved. The wavelength resolution (dl) and the free spectral range (FSR) of the SHFT spectrometer are determined by [START_REF] Velasco | High-resolution Fourier-transform spectrometer chip with microphotonic silicon spiral waveguides[END_REF]:

𝛿𝜆 = 𝜆 𝑜 2 𝑛 𝑔 𝛥𝐿 𝑀𝑎𝑥 , (1)
𝐹𝑆𝑅 = 𝛿𝜆×𝑁 2 , (2)
where 𝜆 𝑜 is the central wavelength, 𝛥𝐿 𝑀𝑎𝑥 the maximum arm length imbalance, 𝑛 𝑔 is the group index of the waveguides and 𝑁 is the number of MZIs.

Figure 1 shows a schematic view of the proposed suspended Si waveguide with metamaterialgrating lateral cladding. The silicon thickness is 700 nm, and the buried-oxide layer thickness is 3 μm. The waveguide width of 2.5 μm and the cladding width of 5 μm minimize losses due to optical leakage to the lateral silicon slabs in the 5-6 μm wavelength range. The grating serving as lateral cladding has a period of 350 nm, operating in the subwavelength regime, i.e. with a period shorter than half of the wavelength [START_REF] Cheben | Subwavelength integrated photonics[END_REF]. We set a gap length of 200 nm to allow penetration of hydrofluoric (HF) acid vapor for substrate removal. To characterize the performance of the fabricated device, light from a tunable quantum cascade laser was injected and extracted from the chip using aspheric ZnSe lenses [START_REF] Liu | Integrated broadband dual-polarization Ge-rich SiGe mid-infrared Fourier-Transform spectrometer[END_REF]. A polarization controller was used to select transverse-electric (TE) polarized light into the chip. The output signal was recorded with a mercury-cadmiun-tellurite (MCT) photodetector. We use 20-μm-wide input/output waveguides at the chip facets, with measured insertion loss of ~10 dB per facet. Propagation loss of the suspended waveguide has been determined using the cut-back method by measuring the optical transmission for 4 different waveguides with length varying between 1.4 cm and 2.58 cm. As an example, Figure 3(a) shows the transmittance as a function of the waveguide length for a wavelength of 5.5 μm. The linear fitting shows a propagation loss of 0.7 dB/cm. Figure 3(b) shows the measured propagation loss as a function of the wavelength. Our waveguides exhibit propagation losses of 1-2 dB/cm between 5.2 μm and 5.7 μm wavelength which is a substantial improvement compared to state-of-the-art metamaterial-cladded Si membrane waveguides. This improvement can be mainly attributed to optimized lithography and etching process. For comparison, previous demonstrations of metamaterial-cladded Si suspended waveguides yielded 0.8 dB/cm near 3.8 μm wavelength for 500nm Si thickness [START_REF] Soler Penades | Suspended silicon mid-infrared waveguide devices with subwavelength grating metamaterial cladding[END_REF], 4.3 dB/cm near 6.6 μm wavelength for 1.5 μm Si thickness [START_REF] Liu | Suspended silicon waveguide platform with subwavelength grating metamaterial cladding for long-wave infrared sensing applications[END_REF], and 3.1 dB/cm near 7.6 μm wavelength for 1.4μm Si thickness [START_REF] Soler Penadés | Suspended silicon waveguides for longwave infrared wavelengths[END_REF].

We use the pseudo-inverse transfer matrix method to calibrate the SHFT spectrometer and retrieve the input spectrum from the output interferogram, I, formed by the outputs of the MZIs [START_REF] Velasco | High-resolution Fourier-transform spectrometer chip with microphotonic silicon spiral waveguides[END_REF]. The pseudo-inverse retrieval method allows numerical correction of amplitude and phase errors produced by fabrication imperfections. The output interferogram I, can be described as I = B × T, where B is the input spectrum and T is the calibration matrix formed by the measured transmittance of each MZI. Then, the input spectrum is retrieved by multiplying the output interferogram, I, by the pseudo-inverse of the calibration matrix T. Figure 4(a) shows the measured calibration matrix, T, comprising the transmittance spectra of the 19 MZIs near 5.5 μm wavelength. For signal retrieval we use a bandwidth of 100 cm -1 , corresponding to the theoretical FSR. In Figure 4(b-d) we plot the retrieved spectrum for monochromatic input and doublets with peak-to-peak separation of 13 cm -1 for wavelengths near 5.44 μm (Fig. 4(b)), 5.5 μm (Fig. 4(c)) and 5.6 μm (Fig. 4(d)). The spectral resolution is ~13 cm -1 in the three cases.

In summary, we demonstrated an integrated mid-IR SHFT spectrometer realized using suspended silicon waveguides with metamaterial-grating cladding. This strategy circumvents the absorption constrain of silica cladding for wavelengths above 4 μm, overcoming the major limitation of the SOI technology in the mid-IR. The metamaterial-cladded suspended waveguides exhibit measured propagation loss below 2 dB/cm near 5.5 μm wavelength. The SHFT spectrometer comprises an array of 19 MZIs with a maximum imbalance length of 200 μm, achieving a measured resolution of 13 cm -1 and a bandwidth of 100 cm -1 , near 5.5 μm wavelength. This is to the best of our knowledge the longest wavelength reported for an integrated silicon SHFT spectrometer. These results open a promising route for the implementation of integrated spectrometers harnessing of the large throughput and robustness against fabrication imperfections of the SHFT architecture, exploiting the full silicon transparency range (1.1-8 μm wavelength) while taking advantage of the mature SOI technology.

FUNDING: French Industry Ministry (Nano2022 project under IPCEI program); Agence Nationale de la Recherche (ANR-MIRSPEC-17-CE09-0041); Spanish Ministry of Science and Innovation (RTI2018-097957-B-C33, PID2020-115353RA-I00), Community of Madrid -FEDER funds (S2018/NMT-4326).

Fig. 1 .

 1 Fig. 1. (a) 3D Schematic view of the suspended silicon waveguide with metamaterial lateral cladding; (b) detail view showing the main geometrical parameters.

Fig. 2 .

 2 Fig. 2. (a) Optical images of the fabricated silicon SHFT spectrometer comprising 20 MZIs, (b) Mach-Zehnder interferometer. Scanning electron microscope images of: (c) Y-junction splitter, and (d) metamaterial-cladded suspended waveguide.

Fig. 3 .

 3 Fig. 3. (a) Measured transmittance for different waveguide lengths between 1.4 cm and 2.58 cm for a wavelength of 5.5 μm. Dashed line is the linear fitting, yielding a propagation loss of 0.7 dB/cm. (b) Measured propagation loss of the Si suspended waveguide as a function of the wavelength.

Fig. 4 .

 4 Fig. 4. (a)Measured calibration matrix of the SHFT spectrometer comprising the transmittance spectra of 19 MZIs. Retrieved spectrum of a monochromatic input and a doublet with peak-to-peak separation of 13 cm -1 for wavelengths near (b) 5.45 μm, (c) 5.5μm and (d) 5.6 μm.


Alireza Rezaei

Sylvie Le Hégarat-Mascle

Emanuel Aldea

Piercarlo Dondi
email: piercarlo.dondi@unipv.it

Marco Malagodi
email: marco.malagodi@unipv.it

Analysis of multi-temporal image series for the preventive conservation of varnished wooden surfaces

Keywords: A-contrario methods, Cultural heritage, Preventive conservation

Preventive conservation is a vital practice in Cultural Heritage that consists in the constant monitoring of the state of conservation of an artwork, with the final goal of minimizing the risk of damages and thus, to reduce the need of restorations. In this work, we propose a probabilistic approach for analyzing alterations appearing on varnished wooden surfaces (such as those of historical violins) based on an a-contrario framework. Our method works on time series of images, it is robust to noise and avoids parameter tuning as well as any assumption about the quantity and the shape of the worn-out areas. Furthermore, the proposed algorithm is adapted to the context of preventive conservation, with few data samples and imprecise annotations. As test set, we used image sequences included in the "Violins UVIFL imagery" dataset. Results illustrate the capability of the proposed method to distinguish altered areas from the surrounding noise and artifacts.

Introduction

In the Cultural Heritage field, preventive conservation, namely the constant monitoring of the state of conservation of artworks and monuments, is becoming a common practice, fundamental to reducing the risk of damages and the need of restorations [START_REF] Bradley | Preventive conservation research and practice at the British museum[END_REF][START_REF] Lucchi | Review of preventive conservation in museum buildings[END_REF]. Handling an effective preventive conservation is particularly complex and, generally, requires an interdisciplinary approach to properly interpret and manage the effects of chemical, physical, and biological alterations [START_REF] Ghedini | Atmospheric aerosol monitoring as a strategy for the preventive conservation of urban monumental heritage: The florence baptistery[END_REF][START_REF] Ortiz | Vulnerability index: A new approach for preventive conservation of monuments[END_REF][START_REF] Perles | An energy-efficient internet of things (IoT) architecture for preventive conservation of cultural heritage[END_REF]. In this context, image processing can be helpful to simplify the overall procedure. Even if images alone are not enough to perfectly characterize the state of an artwork, they are ideal to provide a quick preliminary examination and to identify regions of interest on a surface. This paper addresses this specific problem, focusing in particular on varnished wooden surfaces.

The varnishes applied on wooden surfaces of historical relics are generally delicate and can be easily damaged if the artifact is not properly handled. Historical musical instruments, such as violins or violas, are a common example of artworks with a varnished wooden surface that can be subject to wear. In fact, differently from other artworks that are simply held in museums, historical musical instruments are occasionally played even today, leading to a clear risk of mechanical wear, especially in those areas in direct contact with the musician.

Varnished wooden surfaces present various complexities: firstly, they are generally highly reflective, thus, noisy reflections, that can be confused for alterations, are common during photo acquisition; secondly, varnish wear can evolve in different ways depending on both the initial conditions of the surface and the different substances present; finally, the surface can be very complex and stratified due to multiple restorations occurred during centuries (which is very common for historical violins).

The combined use of multiple analytical techniques has proved beneficial for handling these issues [START_REF] Fichera | Innovative monitoring plan for the preventive conservation of historical musical instruments[END_REF][START_REF] Rovetta | The case of Antonio Stradivari 1718 ex-San Lorenzo violin: History, restorations and conservation perspectives[END_REF], however, as previously said, this "traditional" preventive conservation approach is very time consuming. A more efficient procedure should consist of regular but rapid optical analysis of the images to quickly identify possible altered areas. The result of this analysis will indicate where a more thorough multi-modal analysis is needed. Image acquisition is very fast compared to other chemical-physical examinations, thus, ideally, using image processing, it would be possible to frequently examine the state of conservation of one or more artworks in a limited amount of time.

In this work, we describe a probabilistic method for early detection and analysis of alterations through a multi-temporal image series. We specifically work with varnished wooden surfaces subjected to mechanical wear over time. Our proposition is based on the a-contrario framework [START_REF] Desolneux | Meaningful alignments[END_REF][START_REF] Desolneux | A grouping principle and four applications[END_REF] and it improves the method presented in [START_REF] Rezaei | One step clustering based on a-contrario framework for detection of alterations in historical violins[END_REF] which deals with detection of changes between two single frames: a reference image and an up-to-date acquisition. In this study, we extend the detection framework in order to process a temporal sequence of changes and to account thus for their significance with respect to the spatial plus time dimensions. The main novelty of our work is that the analysis is focused on the evolution of the alterations in time, which in turn allows us to detect the wear earlier and more reliably. This way, we can successfully handle noise and artifacts that would have been detrimental in change detection between only two frames.

The remainder of the paper is structured as follows: Section 2 provides a brief overview of the state of art; Section 3 discusses the proposed approach; Section 4 discusses the experiments; and finally, Section 5 draws the conclusions.

Related Works

Change and damage detection

In recent years, change detection has been tackled with deep neural networks for a variety of applications, among which remote sensing, traffic and medical monitoring, or quality control in manufacturing [START_REF] Varghese | Changenet: A deep learning architecture for visual change detection[END_REF][START_REF] Khelifi | Deep learning for change detection in remote sensing images: Comprehensive review and meta-analysis[END_REF][START_REF] Mandal | An empirical review of deep learning frameworks for change detection: Model design, experimental frameworks, challenges and research needs[END_REF][START_REF] Li | Siamese neural networks for continuous disease severity evaluation and change detection in medical imaging[END_REF][START_REF] Sturari | Robotic platform for deep change detection for rail safety and security[END_REF][START_REF] Verma | Qfabric: Multi-task change detection dataset[END_REF]. Typically, the feature extraction modules of these architectures are fine-tuned on an appropriate quantity of samples of the target domain in order to reach high levels of task accuracy. In our considered application however, the relative scarcity of the data coupled with the high level of imprecision of the expert annotations favor statistical approaches in which prior information may be more readily integrated into the models. This is a common problem in the Cultural Heritage field. Even if machine and deep learning approaches are used, their adoption is not widespread as in other fields, mainly due to the limited availability of data [START_REF] Fiorucci | Machine learning for cultural heritage: A survey[END_REF].

In Cultural Heritage, methods for automatic wear and damage detection mainly rely on traditional computer vision techniques, but nevertheless they proved to be very useful and applicable in different scenarios. A widely studied topic is the detection of crack in paintings. Notable approaches in this field include the use of content-based analysis [START_REF] Abas | Classification of painting cracks for content-based analysis[END_REF], the extension of morphological operations to hyper-spectral images [START_REF] Deborah | Hyperspectral crack detection in paintings[END_REF] and the combination of elongated filters, multiscale morphology and K-SVD [START_REF] Cornelis | Crack detection and inpainting for virtual restoration of paintings: The case of the Ghent altarpiece[END_REF], subsequently improved using multimodal data [START_REF] Pizurica | Digital image processing of the ghent altarpiece: Supporting the painting's study and conservation treatment[END_REF].

Another related topic is the damage detection in stone artifacts. For example, Gelli et al. [START_REF] Gelli | Surface analysis of stone materials integrating spatial data and computer vision techniques[END_REF] used Shape from Shading for identifying degradation in historical buildings; Cerimele and Cossu [START_REF] Cerimele | Decay regions segmentation from color images of ancient monuments using fast marching method[END_REF] identified decay in ancient monuments using fast marching numerical method; while Manfredini et al. [START_REF] Manferdini | An integrated and automated segmentation approach to deteriorated regions recognition on 3D reality-based models of cultural heritage artifacts[END_REF] used a combination of 2D and 3D segmentation to discriminate among different types of deterioration.

Image processing techniques are also used for monitoring the decay of frescoes [START_REF] Guarneri | 3D remote colorimetry and watershed segmentation techniques for fresco and artwork decay monitoring and preservation[END_REF] and even for detecting damaged archaeological sites from space [START_REF] Cerra | Cultural heritage sites in danger-towards automatic damage detection from space[END_REF].

Despite their wide applicability, computer vision methods are still rarely used for wear detection in varnished wooden surfaces, as those of historical musical instruments. Attempts have been made to detect areas with established wear, that had peculiar chromatic characteristics [START_REF] Dondi | Automatic identification of varnish wear on historical instruments: The case of Antonio Stradivari violins[END_REF]. More recently, Rezaei et al. [START_REF] Rezaei | Detecting alterations in historical violins with optical monitoring[END_REF][START_REF] Rezaei | One step clustering based on a-contrario framework for detection of alterations in historical violins[END_REF] proposed a method for detecting changed pixels between two multi-modal images and then grouping them together, while Dondi et al. [START_REF] Dondi | Segmentation of multi-temporal UVinduced fluorescence images of historical violins[END_REF] presented a segmentation approach based on histogram quantization and genetic algorithm. However, all these works study the evolution of alterations using no more than two frames at a time.

A-contrario framework

Since wear areas are highly variable, we look for some approaches which can handle the difficulties in wear detection (unknown shape, unknown distribution, high amount of noise) and at the same time can give us a criterion to judge how surprising a certain arrangement of points is. Both spatially and temporally extraordinary closeness of points could indicate a wear region. The a-contrario framework introduced by Desolneux et al. [START_REF] Desolneux | Meaningful alignments[END_REF] has the potential to satisfy the mentioned requirements. Across the computer vision field, it has been used in variety of areas such as texture analysis [START_REF] Grosjean | A-contrario detectability of spots in textured backgrounds[END_REF], motion detection [START_REF] Veit | An a contrario decision framework for region-based motion detection[END_REF][START_REF] Veit | Space-time a contrario clustering for detecting coherent motions[END_REF], edge and line detection [START_REF] Widynski | A contrario edge detection with edgelets[END_REF][START_REF] Akinlar | Edlines: A real-time line segment detector with a false detection control[END_REF][START_REF] Aldea | Robust crack detection for unmanned aerial vehicles inspection in an a-contrario decision framework[END_REF], or reconstruction from motion [START_REF] Moulon | Adaptive structure from motion with a contrario model estimation[END_REF]. At the same time, grouping principles have been used for tasks related to the higher-level perceptual organization of scenes as well [START_REF] Michaelsen | Self-organizing maps and gestalt organization as components of an advanced system for remotely sensed data: An example with thermal hyper-spectra[END_REF][START_REF] Yan | Unsupervised image saliency detection with gestalt-laws guided optimization and visual attention based refinement[END_REF], owing to their general nature.

In our case, we are interested in the applications dealing with the detection of changed areas across multi-temporal images. In early studies, Lisani and Morel [START_REF] Lisani | Detection of major changes in satellite images[END_REF] used a-contrario framework and spectral invariant features to detect meaningful changes between two satellite images of the same area taken in different times. In 2007, Rousseau et al. [START_REF] Rousseau | An a contrario approach for change detection in 3D multimodal images: application to multiple sclerosis in mri[END_REF] proposed an a-contrario approach for change detection in three dimensional multi-modal medical images such as Magnetic Resonance sequences. In 2010, a similar approach was proposed by Robin et al. [START_REF] Robin | An a-contrario approach for subpixel change detection in satellite imagery[END_REF] for sub-pixel change detection in a timeseries of satellite images. Flenner and Hewer [START_REF] Flenner | A helmholtz principle approach to parameter free change detection and coherent motion using exchangeable random variables[END_REF] further investigated this approach by using exchangeable random variables instead of independent and identically distributed (IID) assumption. As a later example, Liu et al. [START_REF] Liu | A contrario comparison of local descriptors for change detection in very high spatial resolution satellite images of urban areas[END_REF] applied a-contrario framework for comparing very high-resolution images of urban areas.

In the field of preventive conservation, a-contrario framework has been used for comparison of two single images to detect candidate points (seed detection) and then group them together (clustering) [START_REF] Rezaei | Detecting alterations in historical violins with optical monitoring[END_REF]. Later, Rezaei et al. [START_REF] Rezaei | One step clustering based on a-contrario framework for detection of alterations in historical violins[END_REF] introduced a noise model to detect in one step very unlikely (under noise model assumption) arrangements of pixels based both on their location and their gray-level value. They assume the graylevel values to be a third axis and introduce a custom 3D distance to relate all three axes together. While this approach handles some of the aforementioned problems; it still only works with two frames of the series at a time, missing some potentially very useful information present in the evolution of the clusters through time to make the distinction between wear and noise more evident. The current paper aims to introduce to notion of time into the a-contrario framework through the definition of the noise model and the computation of the number of false alarms.

Proposed approach

Let {I t } n t=0 be a series of n multi-temporal RGB images taken from a varnished wooden sample; I 0 being the reference image. The size of each I t is w × h. We perform the change detection in each I t (t > 0) image with respect to I 0 using the method described in [START_REF] Rezaei | One step clustering based on a-contrario framework for detection of alterations in historical violins[END_REF]. This results in a set of binary images denoted {B t } n t=1 . It is important to note that this step is not part of the algorithm; therefore, it can be performed using any proper 2D change detection and clustering algorithm. If the 2D change detection method outputs a change map, it needs to be converted to a binary image prior to being used in the following steps. The proposed algorithm gets this series of binary images as input.

The following sections summarize each step of the algorithm that produces as output, three dimensional clusters ranked based on their significance.

Distance matrix

Considering the whole series {B t } n t=1 , we create the 3D point cloud P with the two spatial lines and columns axes of each image alongside time as the third axis (Fig. 1): P = {(x, y, z), x ∈ {1, 2, . . . , w}, y ∈ {1, 2, . . . , h}, z|z/c t ∈ {1, 2, . . . , n}}, (1) where c t is the time coefficient; a real valued constant that controls the importance of a distance in time compared to a distance in image space. If c t = 1, then we give the same importance to both. After constructing P, we aim to segment it into 3D clusters and rank them based on their visual significance in the 3D (2D + time) space. The cluster detection using the a-contrario approach is based on point distance [START_REF] Desolneux | From gestalt theory to image analysis: a probabilistic approach[END_REF]. A Euclidean distance between the points of P automatically considers their spatial proximity as well as their closeness in time. The distance matrix D is constructed by computing the 3D distance between each pair (x a , y a , z a ), (x b , y b , z b ) ∈ P:

D a,b = (x a -x b ) 2 + (y a -y b ) 2 + (z a -z b ) 2 .
(2)

Number of False Alarms

In order to define and evaluate the significance of the clusters formed by any subset of points in P, we use an a-contrario criterion. Our naive model M represents spatial and temporal inconsistency for the points in P. Specifically, Definition 1 (Naive model M) The set of points P is a random set of |P| independent uniformly distributed variables over the 3D cube of the image series (

[1, w] × [1, h] × [1, n × c t ]).
Let us consider a sub-volume O in the image cube; the probability that a point belongs to O, under the naive model M, is given as:

p O = V O w × h × n × c t , (3)
where V O is the volume of O. The probability P M (p O , |P| , κ) of observing κ points within O by chance, is given by the tail of the binomial distribution. The Number of False Alarms (NFA) can then be derived by extending the formulation proposed in [START_REF] Desolneux | A grouping principle and four applications[END_REF] to 3D clusters:

N F A (p O , |P| , κ) = N test × |P| i=κ |P| i p i O (1 -p O ) |P|-i . (4)
In practice, the volume V O needed for the computation of the term p i O , is estimated as the lower bound of the volume of the points in O which can form any 3D shape. The term (1 -p O ) on the other hand represents the relative volume of the region that is definitely outside the cluster i.e., outside the upper bound of the cluster. These two volumes are calculated by morphological operations extending the approach detailed in [START_REF] Desolneux | A grouping principle and four applications[END_REF] to work with 3D data. It is worth noting that the distance between each two points indicated in D is defined as a 3D Euclidean distance; therefore, the morphological operations used here are usual dilation and erosion with a 3D kernel.

In Eq. ( 4), N test , called the numbers of tests, controls the average number of false alarms [START_REF] Desolneux | From gestalt theory to image analysis: a probabilistic approach[END_REF]. In this work, we make this number constant for a given image cube (image series), i.e., independent of the considered cluster O, so that it is not involved in NFA minimization.

Maximal clusters

Using Eq. ( 2) a minimum spanning tree is created for the points in P. Using this tree, we group the points from smallest cluster(s) (nearest couple of points) to the biggest cluster (all points). For each cluster, we compute the NFA using Eq. ( 4). The used ranking criterion for clusters (O containing κ points) is their meaningfulness S defined as:

S(O) = -log(N F A(p O , |P|, κ)). (5)
In our case, we want to partition P; therefore, from all the possible clusters of points in P, only the maximal clusters are considered, with maximal clusters defined according to [START_REF] Desolneux | A grouping principle and four applications[END_REF]:

Definition 2 A cluster C ⊂ P is said maximal if ∀L C,

S(L) < S(C) and ∀L C, S(L) ≤ S(C).

The list of maximal clusters sorted by their S value is the output of the algorithm. The cluster with the highest S is most significant cluster in the image series.

Experiments

Dataset

As test set, we used the "Violins UVIFL imagery" dataset5 [START_REF] Dondi | Segmentation of multi-temporal UVinduced fluorescence images of historical violins[END_REF], a collection of multitemporal UV induced fluorescence (UVF or UVIFL) images, referring to both real instruments and artificially altered samples created in laboratory.

UVIFL photography is a well-known non-invasive diagnostic technique that allows us to observe details not perceivable using visible light [START_REF] Janssens | Non-destructive micro analysis of cultural heritage materials[END_REF]. For example, in the case of historical violins, UVIFL images are commonly used to highlight possible restorations or regions of interest where it would be advisable to apply more precise but slower diagnostic techniques [START_REF] Dondi | Automatic analysis of UV-induced fluorescence imagery of historical violins[END_REF], such as X-Ray Fluorescence (XRF) [START_REF] Rovetta | The elemental composition of Stradivari's musical instruments: new results through non-invasive EDXRF analysis[END_REF] or Fourier Transform Infrared (FTIR) spectroscopy [START_REF] Invernizzi | A non-invasive stratigraphic study by reflection FT-IR spectroscopy and UV-induced fluorescence technique: The case of historical violins[END_REF]. The capability to see "hidden" details of a surface is particularly valuable for our goal since it potentially allows for an early detection of new alterations. Please note that UV induced fluoresce is a phenomenon due to the specific properties of substances commonly used in the varnishes, that react to the light in the UV-A spectrum (315 -400 nm) re-emitting radiations in the visible spectrum (400-700 nm), and thus, the image acquisition is performed with a standard RGB camera [START_REF] Dondi | Automatic analysis of UV-induced fluorescence imagery of historical violins[END_REF].

In our experiments we focused only on the artificially created sequences present in the dataset, since those referring to real historical violins contain only three sessions each, too few for performing a proper multi-temporal analysis. Each sequence consists in one to three images of the initial state of the sample and then a series of subsequent frames where the surface was altered, and the wear starts and grows. We use the following sequences from the dataset: WS01, a 20-frame sequence that displays the appearing and the growing of an alteration in an area with intact varnish; and SVO1, a 20-frame sequence that shows the growing of an already worn-out region on a sample violin back plate.

Results

Using the sequences WS01 and SV01, for each frame, we generate binary change images respect to the reference frame, applying the method described in [START_REF] Rezaei | One step clustering based on a-contrario framework for detection of alterations in historical violins[END_REF]. Figures 2 and3 show, respectively for WS01 and SV01, four sample frames and their corresponding binary image.

No. UVIFL image Input binary image Detection results

Ground truth 10 14 16 20 Fig. 2: Four sample frames from the sequence WS01 and their corresponding cluster detection results. At each frame, the first, second and third cluster according to decreasing order of meaningfulness are shown in red, green, and blue, respectively.

No. UVIFL image Input binary image Detection results

Ground truth 8 12 16 20 Fig. 3: Four sample frames from the sequence SV01 and their corresponding cluster detection results. At each frame, the first, second and third cluster according to decreasing order of meaningfulness are shown in red, green, and blue, respectively.

Figure 1 shows the point cloud created by accumulating the change detection results for each frame of both sequences. The coefficient c t is set to 2 (this gives slightly more importance to spatial proximity over time proximity, penalizes the clusters which disappear in some frames and keeps the size of the representation domain small enough for a better performance). The time domain is the upward axis in all visualizations. As we can see, there is a considerable amount of noise present in both point clouds. In addition, there are two sets of artifacts in the left and right sides of both sequences, which are related to the UV reflections from the the wooden sample.

In practice, the optical monitoring of a varnished wooden surface is an ongoing and never-ending process. This means that the number of available frames and the accuracy of the detection increases as the time goes on. Of course, if the sequence of available images becomes too large, we only keep the last n frames with n chosen as a compromise between computation resources and detection performance. In our case, we keep every frame available from the first until the last one.

To simulate a practical optical monitoring process and to analyze different detection results in any given point in time, we started the experiment with 10 frames for the sequence WS01 and 8 for the sequence SV01; then, we ran the algorithm repeatedly increasing the considered number of frames in each run until processing all frames.

(a) t ∈ [1, 10] (b) t ∈ [1, 14] (c) t ∈ [1, 16] (d) t ∈ [1, 20]
Each run gives us a list of clusters sorted by their meaningfulness value. The rank of each cluster among the whole set of clusters can change from one run to another.

In general, assuming that the wear region expands over time, our expectation is that the wear starts in lower ranks and steadily rises to the first rank. We can also follow the evolution of the meaningfulness value from one run to the next. Again, we expect that the significance of the wear region increases as we get more frames. On the other hand, the meaningfulness value for the noise and artifact clusters should remain nearly unchanged or go up and down randomly. Another difference between the wear region and the rest is that it shall be present in every frame after its first appearance. Therefore, it is possible that a detected area is divided into two clusters along the time axis. In that case, we only take into account the one with the highest meaningfulness. Figures 4 and5 illustrate the detection results through time, i.e., based on the number of frames used as input. In each case, the top three clusters have been shown. As one may notice, firstly, only clusters which are consistent have been detected. Minor artifacts and small noises have been disregarded, a fact which is a very desirable behavior when multi-temporal information is available. Secondly, the region of interest which is the actual changed area has been chosen by the end of the sequence as the most significant cluster. The evolution of meaningfulness of each cluster for the two sequences is depicted in Fig. 6.

For a more in-depth analysis, we can consider the four main clusters present in the sequence WS01 (see Fig. 7 for their location on a sample frame) and as a result in P:

-Cluster 1: the wear area.

-Cluster 2: the artifact related to the right border.

-Cluster 3: the artifact related to the left border.

-Cluster 4: the noise present in the upper part of some frames.

Considering the left chart in Fig. 6, which illustrates the meaningfulness values for these four clusters in each run of the algorithm, we can deduce the following:

-The values for both clusters 3 and 4 remain fairly constant in each run which indicates artifacts with the same size appearing consistently in all frames. On the other hand, the values for cluster 2 behave more erratically and go up and down. This is due to appearing and disappearing of the artifact in some frames. Therefore, the detection is divided in multiple clusters along the time axis. -Only cluster 1, the wear region, has values that consistently increase over time. This is in line with our expectation about an "ideal" wear that keeps growing. In practice, the wear may not expand over multiple acquisitions which will show itself as a plateau. However, it will never shrink or disappear so the values should remain increasing overall. Moreover, even if the wear region becomes the most meaningful cluster only at frame 16, we can notice that it is the only growing cluster already at frames 11-12, meaning that we can trigger a "possible alteration" alert at this early stage.

The same deductions for WS01 can be observed for SV01, as well. This time we have three main clusters (see Fig. 6, right): cluster 1, the wear, and clusters 2 and 3, which are reflections present on most of the samples. In this case the alteration is correctly detected early than in sequence WS01. In fact, at frames 7-8, cluster 1 is both the most meaningful cluster and the most rapidly growing. This is coherent with the type of input data. Sequence WS01 displays the growing of a completely new damaged area, which is initially small and slowly growing, while SVO1 shows the worsening of an already worn-out region, thus it is reasonable that the new alteration appears earlier and grows faster.

Overall, results clearly show that, with respect to previous works leveraging single change maps [START_REF] Rezaei | One step clustering based on a-contrario framework for detection of alterations in historical violins[END_REF], this analysis based on the temporal dynamics of the changes is able to pinpoint at an earlier stage and more reliably the emergence of a wear, in full accord with the preventive conservation principle.

Conclusions

In this paper, we introduced a probabilistic method for detecting significant changes located in clusters within a noisy multi-temporal series of images. The proposed method can be used in preventive conservation, as a fast preliminary examination of a varnished wooden surface able to identify the most likely new altered areas. Thus, a verification using more precise but slower techniques (like spectroscopic analyses) will be done only on these detected areas, optimizing the monitoring procedure. This approach, rooted in the Gestalt theory, is particularly interesting for the task of historical violin analysis with respect to other statistical learning-based and deep learning methods, given the limited amount of available data and also the difficulty of performing reliable annotations.

Future works will involve more tests, to further validate our approach, as well as refinements in the clustering method to achieve an even earlier detection of wear (ideally, as close as possible to the beginning of the alteration, when only a few pixels change).

Fig. 1 :

 1 Fig. 1: Point cloud P derived from the sequence WS01 (left) and SV01 (right). X and Y axes are in pixels, while the time dimension (Z axis) depends on the factor c t (in these experiments c t = 2).

Fig. 4 :

 4 Fig. 4: Evolution of the detection by increasing the number of used time frames t in the sequence WS01. Color code gives the rank according to significance: red first, green second, blue third. The time domain is the upward axis in all visualizations.

Fig. 5 : 3 Fig. 6 :

 536 Fig. 5: Evolution of the detection by increasing the number of used time frames t in the sequence SV01. Same conventions of Fig. 4.

Fig. 7 :

 7 Fig. 7: Location of clusters 1 to 4 on a sample frame (frame 8) from the sequence WS01.

https://vision.unipv.it/research/UVIFL-Dataset/

Acknowledgments

The authors gratefully acknowledge support from the French-Italian Galileo PHC partnership (project 44391VL/G20-160, MUSical Instrument Conservation with Optical Monitoring (MUSICOM)).