Julie Digne

Raphaëlle Chaine

Sébastien Valette

Self-similarity for accurate compression of point sampled surfaces

Keywords: (according to ACM CCS): I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling-Curve, surface, solid, and object representations I.4.2 [ImageProcessing and Computer Vision]: Compression (Coding)-Approximate methods

Most surfaces, be it from a fine-art artifact or a mechanical object, are characterized by a strong self-similarity. This property finds its source in the natural structures of objects but also in the fabrication processes: regularity of the sculpting technique, or machine tool. In this paper, we propose to exploit the self-similarity of the underlying shapes for compressing point cloud surfaces which can contain millions of points at a very high precision. Our approach locally resamples the point cloud in order to highlight the self-similarity of the shape, while remaining consistent with the original shape and the scanner precision. It then uses this self-similarity to create an ad hoc dictionary on which the local neighborhoods will be sparsely represented, thus allowing for a light-weight representation of the total surface. We demonstrate the validity of our approach on several point clouds from finearts and mechanical objects, as well as a urban scene. In addition, we show that our approach also achieves a filtering of noise whose magnitude is smaller than the scanner precision.

Introduction

Recent years have witnessed a drastic improvement in scanner acquisition devices which now yield point sets of tens of millions of points at a high precision. The counterpart of this development are datasets requiring ever higher storage capacity. In the absence of efficient yet accurate compression techniques, lower-resolution representations of the shape are commonly used, but at the cost of losing precision. Furthermore, scanner acquisition devices are often supplied with software packages that exploit proximity and local redundancy of points to reconstruct a polygonal mesh from the point set, with the idea that a triangle is worth a set of points. Therefore research has mainly focused on mesh compression.

Yet, some applications require only a point cloud and do not need a mesh. For example, dense point clouds are better suited to some operations like registration. An acquisition campaign might result in incomplete data, and prematurely building a mesh from these acquisitions is not an optimal strategy. Finally, the global meshing operations are sometimes not accurate enough, and oversmooth the data. Thus, there is a dire need for compression methods that preserve the acquisition device precision.

In this paper, we propose a solution to the compression of raw surfaces represented as unorganized point clouds. Unlike meshes, input point sets correspond to samplings of shapes, without any interpretation. From this observation, we build a new compression approach that turns a dense point set surface into another one while keeping the distortion below the input accuracy. We only locally change the sampling on the surface, so as to favor compression strategies, while remaining within the input precision.

The very reason for locally changing the sampling on the surface is to highlight the self-similarity inherent to the shapes, and to use it for compression purpose. The discrete analysis that we perform requires that the different areas of the same object be represented with comparable sampling schemes. Moreover, our goal is not to use a local constraining model as is sometimes done to visualize point set surfaces (Q-Splat [RL00], MLS [ABCO * 01]) but to remain closer to the data (shape and sampling), while assuming that each point has a 2-manifold neighborhood at the scale where the self-similarity is analyzed. The outcome is a very efficient and faithful compression approach that performs much better than previous algorithms to compress unstructured surface point sets. Strictly speaking, this method is not lossless since the exact input point cloud cannot be recovered. Yet, the distortion introduced by the compression can be set below the scanner accuracy, as we will illustrate experimentally.

To sum up, our contributions are the followings:

• We introduce a way of encoding point clouds based on shape self-similarity, while being respectful of the scanner accuracy. • This compression is done at the resolution of the scanner enabling improved control of the point cloud resolution. • The input point clouds can be oriented or non-oriented.

Figure 1 shows the difference between an original point set and its decompression. Our method achieves a 0.59 bits per point (bpp) bitrate on this model while keeping the reconstruction error below the scanner precision for 99.14% of the points. As a side-effect, noise and registration artifacts are filtered out.

The remainder of this paper is organized as follows: section 2 reviews the related work on 3D compression and selfsimilarity analysis. Section 3 presents the working assumptions for our framework. Sections 4 and 5 describe respectively the compression and decompression methods, and, finally, section 6 shows quantitative and qualitative results and comparisons.

Related Work

3D Compression

3D compression has been deeply explored during the two last decades. For our purpose, we can split 3D compression in 3 different categories : point cloud compression, mesh compression and shape compression. Point cloud compression approaches have mostly dealt with coordinates quantization via recursive space partitioning [SK06, GD02, HPKG06, SPS12]. In a nutshell, these approaches consist in inserting the points in a space partitioning data structure (e.g. octree, kd-tree) of given depth, and to replace them by the center of the cell they belong to.

Other approaches encode the shape represented by the point cloud and use this encoding to generate a different point cloud in the decompression phase. Kalaiah et al. [KV05] propose to define a level-of-detail hierarchy of the point cloud by computing the PCA of the points and splitting along the axis corresponding to the largest variation. The final points are then generated by Gaussian sampling. Schnabel et al. [START_REF] Schnabel R | Fast vector quantization for efficient rendering of compressed point-clouds[END_REF] segment the point cloud into canonical geometric shapes. Each part is then efficiently encoded individually using vector quantization. Our approach also uses local height maps, but no segmentation or model regression is needed. Instead, the shape will define its own analysis space by itself. Compression based on surface similarity was proposed in [HMHB08], and we will come back extensively on it in the next sections. Our approach will also encode the shapes and generate a different point cloud from this shape encoding. Deering [Dee95], and has been greatly improved over the years. The rationale for mesh compression is that some applications need the original mesh connectivity unchanged, e.g. for simulation applications, or for cases when attribute data are present. Valencebased coding [TG98] has proven to be nearly optimal for single-resolution compression. Progressive approaches such as [AD01, MCAH12] aim at refining the mesh geometry and connectivity in a progressive way, until full resolution is reached. Karni & Gotsman [KG00] decompose the input geometry on a spectral basis and the complexity of this approach is alleviated via mesh partitioning. Spacepartitioning approaches have also been proposed, such as [GD02, PK05] and have the ability to encode meshes with complex topologies. However, these approaches give mixed performance at low bitrates.

Mesh compression was initiated by

When only the shape needs to be transmitted, without the need to keep acquisition or sampling data, resampling approaches for shape compression have proven to be the most efficient approaches. Prior to compression, the input model is remeshed into a new structured surface (e.g. mesh with subdivision connectivity). This remeshing step has the advantage to remove tangential information which does not contribute to the general aspect of the shape. Wavelets [KSS00, GVSS00, GGH02] and bandelets [PM05] submitted to COMPUTER GRAPHICS Forum (4/2014). exploit the fact that meshes exhibit smooth parts, and offer good performance at very low bitrates. However, few robust approaches were proposed for structured remeshing like [LSS * 98] and dealing with data generated from complex meshes might be problematic.

Self-similarity for images

Self similarity of measured signals has gained interest over the past decade: research on signal processing as well as image processing has accomplished outstanding progress by taking advantage of the self-similarity of the measure. In the image processing field, the idea originated in the non-local means algorithm [BCM05]: instead of denoising a pixel using its neighboring pixels, it is denoised by exploiting pixels of the whole image looking similar. The similarity between pixels is computed by comparing patches around them. Behind this powerful tool lies the idea that pixel far away from the considered area might entail information that will help processing it, because of the natural self-similarity of the image. This idea gave birth to a thread of work proposing patch-based methods for a wide variety of problems including denoising, super-resolution or inpainting. Recently, the Patch Match algorithm made the similar patch search much faster [BSFG09]. Following the development of the compressive sensing theory [START_REF] Candès | Stable signal recovery from incomplete and inaccurate measurements[END_REF], the idea arose that there exists spaces, in which the signals would be sparsely represented, that are especially well suited for processing the signals. In particular, [AEB06] introduces the idea of designing dictionaries over which each patch of the image is represented as a sparse linear combination of the dictionary atoms. We will describe more precisely this method below.

As a consequence of its success in image processing, this idea was also introduced for surfaces, defined as meshes or point clouds.

Self-similarity for surfaces

Self-similarity of surfaces has mainly been exploited for surface denoising applications: the non local means filter has been adapted for surfaces be it meshes [YBS06] or point clouds [START_REF] Gelfand | Gaussian kd-trees for fast high-dimensional filtering[END_REF]Dig12]. It was also used to define a Point Set Surface variant [GAB12] exhibiting better robustness to noise. Self-similarity of surfaces is obviously not limited to denoising purposes. For example, analyzing the similarity of a surface can lead to detect symmetries or repetition structures in surfaces [MGP06], [PMW * 08]. An excellent survey of methods exploiting symmetry in shapes can be found in [MPWC12]. The similarity can also be analyzed between surfaces [BLSC * 11]. More related to our approach is the method of [ZSW * 10] where self-similarity is used to consolidate scans of urban scenes.

To the best of our knowledge, self-similarity for compression has only been proposed once before and in a different setting [HMHB08]. After decomposition of the surface into patches, these are clustered by similarity, and are replaced by the representative of each cluster during decompression. Our approach is more pliant: each patch is represented as a sparse linear combination over a dictionary, allowing for a better depiction of the variety of the patches, and permitting a very high resolution. We will compare our approach to this method and show a clear gain both numerically and visually.

K-SVD

The K-SVD algorithm is a method for building representations of finite discrete signals as sparse linear combinations over an ad hoc dictionary. It was introduced in [AEB06], and has since then be used for various purposes including denoising [EA06] and image analysis [MBP * 09].

Let Y be a k × n matrix, whose columns are n training signals (y i ) i=1,•••n , each of them represented by k samples. The idea is to find a dictionary D, composed of d signal atoms, over which each signal y i can be represented as a linear combination of the dictionary atoms d j . In other words, one can find a vector of coefficients x i such that y i = D • x i . Let us call X the d × n matrix whose columns are the x i , then both X and D (k × d) are solved for by computing:

min D,X Y -DX s.t. ∀i, x i 0 ≤ T 0
The 0 represents the number of nonzero coefficients in x i . It is sometimes abusively called the L 0 norm, though it is not a norm (it is not positive-homogeneous). It measures the sparsity of a decomposition (a decomposition is said to be sparse if a lot of the coefficients are zero and thus its L 0 "norm" is small). T 0 constrains the number of nonzero coefficients.

The K-SVD alternates between finding the best sparse representation for Y on D using a pursuit algorithm (e.g. orthogonal matching pursuit) and updating the dictionary. The dictionary update consists in considering iteratively each atom d k from the previous step: consider the group of samples that use d k in their representation and compute the representation error E R k of the samples that use atom

d k on dic- tionary D \ d k . Then E R k is decomposed using Singular Value Decomposition E R k = U∆V T .
The updated atom d k is the first column of U while the updated coefficients is the first column of V multiplied by ∆(1, 1) (first singular value). This iterative process needs an initial dictionary which is a random subset of the input training signals

(y i ) i=1•••N .
In this paper we will use a fast version of this algorithm, with complexity O(2n 2 d), that was introduced in [RZE08].

Working assumptions

Our algorithm takes as input a point cloud P which is supposed to be dense enough to unambiguously represent an submitted to COMPUTER GRAPHICS Forum (4/2014).

interpolating surface M at a fixed resolution coarser than the point set density. It can handle possibly non-oriented point clouds, by estimating possibly inexact normal directions from local neighborhood. If the input data is provided with oriented normals, instead of estimating the normal directions, the provided ones are used.

The algorithm also requires a radius R that corresponds to the scale at which the self-similarity of the surface is to be analyzed. Our framework is based on some precise assumptions on R with respect to the input point set.

• Topological condition: R must be set such that P can be covered by the set of R-neighborhoods corresponding to a subset of seed points in P. Additionally, each R-neighborhood should delimit a topological disk on the underlying surface M (to enable parameterization over the tangent plane). • Sampling condition: The R-neighborhood of a seed point must contain enough points so that a meaningful patch of surface can be computed. This implies R to be sufficiently large with respect to the scanner resolution. • Noise level: The method assumes that noise magnitude is strictly below radius R.

In the examples provided in this paper, the Rneighborhood of a seed point on the surface is approximated by the R-neighborhood in the ambient space. Therefore, the topological condition amounts to taking the radius R below the reach of the underlying shape, i.e. below the distance between a surface point and the shape medial axis. In the case of sharp edges, it is sometimes possible to find a valid covering with nonzero radius. Otherwise, the approximation of the R-neighborhood on the surface should be replaced by a more sophisticated approach. However, it should be noted that the current approximation approach performs well in practice, even when there are sharp edges and invalid neighborhoods (Figures 7,12), or when dealing with surfaces with boundaries.

Compression

The compression algorithm consists in three successive steps: the first one selects a subset of points (the seeds) that will serve as center points to cover the surface with local patches and obtain an atlas description. The second one computes a discrete description of the local neighborhood around a seed. This patch description is obtained by sampling the local surface height over a local frame: each neighborhood is resampled and described as a height map over a radial grid (Fig. 2). Finally, the third step compresses the description of the patches by exploiting their self-similarity and building a custom dictionary, over which all descriptors will be decomposed sparsely.

Seeds selection

The first step of the compression algorithm selects a subset of seed points. This subset S has to satisfy the following property: its dilatation of radius R must contain the whole point cloud P. More formally:

∀p ∈ P, ∃s ∈ S, p -s ≤ R.
Each point p can be covered several times. The subset is constructed in a dart-throwing fashion: we pick a first seed, tag all the other points in its R-neighborhood as covered and continue traversing the points until a non-covered point is found, then a new seed is added and the process continues until all points are covered. In order to discard outliers in the point cloud, we require that a point with few neighbors cannot be considered as a seed, at the cost of breaking the covering condition above, and omitting some points (in most cases these outliers represent around 0.1% of the points). In fact, the outliers will only be problematic if there is a cluster of outliers resulting in a spurious patch, however it is very unlikely in real-life cases. On the Lovers pointset (Fig. 1) containing 15.8 million points, using a radius of 0.5mm, 89256 seeds are selected and each point is covered in average 2.6 times.

Neighborhood description

Once the seeds are selected, their local R-neighborhoods are sampled on a local pattern, yielding a sampling of the surface local graph on a radial grid (see Fig. 2). To perform this resampling properly and consistently over the surface, the normal n and first principal direction t 1 are first estimated from local covariance analysis on each local neighborhood. If the normal is available (in case of an oriented point set), then only t 1 is computed. It should be noted that during decompression, n and t 1 can be predicted using covariance analysis on the seeds only, and only three difference angles need to be stored for correction.

Once both n and t 1 are available, a radial grid (r i , θ j ) i, j=0•••N bins -1 is computed such that:

r i = ( 1 2 + i) • R N bins ; θ j = j • 2π N bins
A different local sampling pattern could have been used (e.g. rectangular grid, Poisson sampling), provided the same pattern is used for the whole surface and the resolution of the pattern is close to the point cloud resolution. This radial pattern is well adapted since more points are sampled near the seed, where it is likely that there is less multiple coverage. The seeds coordinates are encoded using a kd-tree based approach similar to [GD02]. The difference angles are quantized and compressed using arithmetic coding. For the Lovers of Bordeaux, bitstream sizes are 312KB for the seeds coordinates and 297KB for the angle differences, both quantized on 8 bits. submitted to COMPUTER GRAPHICS Forum (4/2014).

Self-similarity compression

To exploit self-similarity between neighborhoods, our approach relies on building a dictionary, a set of atoms over which the patches will be decomposed. Both the dictionary atoms and the coefficients of the patch decomposition are solved for by the K-SVD algorithm [AEB06], which minimizes the error while maintaining the sparsity of the coefficients, as explained in section 2.4. This algorithm yields sparse linear decompositions of the patches (i.e. with few nonzero coefficients). In addition, the nonzero coefficients distribution presents itself with a sharp peak centered around 0, making them suitable for compression. Therefore we use scalar quantization followed by entropy coding to compress the coefficients. In addition to its purpose of reducing the file size, this quantization improves the sparsity of the coefficients. On the Lovers pointset, the percentage of nonzero coefficients is about 30% before quantization and 20% after quantization (on 8 bits).

Our content-based self-similarity approach compares favorably with the Discrete Cosinus Transform (DCT) which is known to concentrate the energy on the low frequencies but is fixed independently of the signal. This DCT is performed on the same neighborhood description and the same amount k of nonzero coefficient is kept (corresponding to the k DCT basis functions accumulating the highest energy). Then the coefficients are quantized using the same scalar quantization on 8 bits. The Root Mean Square Error (RMSE) is higher for the DCT: 0.14 against 0.01 for our approach.

Figure 3 shows examples of built dictionaries for two shapes. For the Lovers of Bordeaux, the dictionary and coefficients are respectively encoded on 18KB and 507KB.

Tuning parameters and controlling the accuracy

The proposed approach is driven by 5 parameters : the patch radius R, the patch discretization size N bins , the number of atoms in the dictionary Natoms, and the quantization for the seeds coordinates and patch coefficients.

In all our experiments, we set N bins = 16 (i.e. the average number of points per neighborhood is around N 2 bins ) and set the radius accordingly, so that the resolution of the sampling is close to the resolution of the scanner. This gives a rough way of computing the radius parameter.

The number of atoms is also easy to set: it will impact directly on the preservation of the details. If the surface has low detail variation (for example a mechanical artifact or a shape where all details have roughly the same size), 16 atoms is enough to represent the shape. In case of more detailed surfaces (e.g. the Tanagra with details at various size) we took 32 atoms. For a more precise control on the accuracy of the compression, the number of atoms in the dictionary can also be gradually increased until the k-SVD approximation error drops below the accuracy of the scanner. As long as the target accuracy is not reached, atoms are added to the current dictionary which is further updated. This approach is valid since it should be noted that the approximation error decreases towards zero for a dictionary in which every seed is an atom.

Quantization was set to 16 bits for the seeds coordinates and 8 bits for the patch coefficients.

Decompression

The decompression algorithm consists in two steps: the first one decompresses the patches given the dictionary and the coefficients. The second step translates the set of seeds and the decompressed patches into the final point cloud.

Given the dictionary D and the coefficients matrix X, the reconstructed patches are computed as Yrec = D • X, therefore for each seed, the corresponding decompressed patches are recovered. The next step is to go from the set of decompressed patches to the final point cloud. In order to avoid high variations of sampling density in areas of overlap between neighboring patches, we perform a consolidation of the samplings in those areas.

Each bin (r, θ) of the sampling pattern Fs(r, θ) corresponding to seed s yields potentially a point, with coordinates in the local frame (s,t 1 (s),t 2 (s), n(s)):

(r cos(θ), r sin(θ), Fs(r, θ)) which are translated in the global coordinate system as: s + r cos(θ)t 1 (s) + r sin(θ)t 2 (s) + Fs(r, θ)n(s). Since the patches overlap, one needs a merging strategy in overlapping areas (Fig. 4). More precisely, when adding a decompressed point p, we count the number of seeds whose neighborhood contains p. If there is a single seed the point is kept as is. Otherwise, p is projected onto the parameterized tangent planes of each of the seeds. Each of those seeds s can then predict the position of p from its own patch, yielding a position p s , and the final position of p is the mean of the predicted positions p s . On a side note, using radial grids makes the overlap computation easier than with square grids. Figure 5 shows the detail recovery as well as the denoising obtained through a compression/decompression cycle. One can see that the details are very well-preserved, while the noise is removed. One can check that only noise is removed by comparing the error (obtained with the fixed parameters proposed) with the scan resolution: in most cases this error is below the scanner resolution. Figure 6 shows the compression/decompression result on the dense St Matthew pointset [LPC * 00], which has been reduced to 93,4M points due to memory limitations (4GB). One can see that the details are very well preserved. For the following comparisons, we measured decompression error as the root mean square normal distance between two point clouds (except for the curves of Fig 10, where it is measured as the root mean square distance to the nearest point, to be consistent with the results presented in the corresponding papers). We have compared our method with a kd-tree based approach similar to [GD02] which compresses unoriented point clouds. At 4.13bpp, [GD02] yields a RMSE of 0.066mm. In comparison our approach gives a RMSE of 0.010mm at 0.6bpp. The results are compared visually on figure 8. We also have compared our approach with an approach close to [HMHB08], by computing selfsimilarity clusters and replacing each patch by the representative of its cluster. We allowed for 16 clusters (the same size as our dictionary). We can see that numerous artifacts appear (Figure 9) which translates into a higher error (0.12mm vs 0.01mm with our method for the Lovers point set). Figure 10 compares our approach with previous Comparison with an approach similar to [HMHB08] (right). Our method (left) avoids creating artifacts by being more pliant. Quantitatively, the RMSE for the compared method is 0.12mm (0.01mm in our case). R = 0.5mm, N bins = 16 works [KV05,SMK08,HMHB08] in terms of rate/distortion. For our approach, we obtained several bitrates simply by varying the patch radius R. Note that our approach clearly outperforms previous works. Finally, Table 1 shows quality measures for the compression/decompression of several point clouds. In particular, it shows that the amount of points that have higher decompression error than the initial resolution is low, illustrating therefore that the compression does not hinder precision.

Figure 11 shows the behavior of the algorithm with respect to added outliers. The only outliers that hinder the process are the ones that fall within a distance R to the surface. Their neighborhoods might indeed contain enough points to create a patch and therefore decompression artefacts. Those outliers can be considered as large noise. This confirms our working assumptions (section 3). Outliers lying far from the surface are well discarded by the algorithm.

On figure 12 we show the compression results on a very challenging data set of 70 million points that clearly breaks our working assumption: some linear structures exist and submitted to COMPUTER GRAPHICS Forum (4/2014). there are some low density areas. The result is still valid in densely sampled areas. However, sparsely sampled areas may be filtered out by the preprocessing step (section 4.1), showing the limits of the algorithm.

Conclusion

We introduced a very efficient framework for compressing dense point sets describing a surface. First, the method selects a subset of the points, it then computes local descriptions of the selected points and use the similarity between the descriptions to encode them.

Our approach encodes tens of millions of points scanned over a surface with less than 1 bit per point. Furthermore, the difference with the input point set is related to noise. So, the decompressed point set behaves as well, if not better, than the original point set, for the purposes of a reconstruction algorithm or visualization.

There are several ways in which our compression scheme could be improved:

• Exploiting patch-based representation, artifacts may appear in case of boundaries, which could be dilated throughout decompression. One could mitigate this issue by adjusting the patch size (clipping some outer grid cells) along boundaries. This would require to store one small integer for each patch, at a small cost. • Other seed picking strategies could be implemented, for example by placing the seeds so that they minimize the local error, in the spirit of [OBS06]. • Encoding per-point attribute such as normals and colors is possible with the same similarity-based coder.

Perspectives : Although our algorithm is based on the exploitation of self-similarity on the whole surface, most of the involved treatments remain local. This is a good prospect for handling data of ever increasing size, using streaming processes. This is particularly important at a time when the geometric digitization campaigns sometimes cover entire cities.

Figure 1 :

 1 Figure 1: The Lovers of Bordeaux (15.8 million points). Exploiting self-similarity in the model, we compress this representation down to 1.15 MB. The resulting model (right) is very close to the original one (left), as the reconstruction error is less than the laser scanner precision (0.02mm) for 99.14% of the input points.

n t 1 Figure 2 :

 12 Figure 2: Local neighborhood description: a height map over a radial grid

Figure 3 :Figure 4 :

 34 Figure 3: Dictionaries built for two different shapes: a geometrical one (the mire, left) and a fine art one (the Lovers, right). The atoms are shown by order of importance (total absolute weight in the linear decompositions).

Figure 5 :

 5 Figure 5: Compression and decompression of the Anubis point set (Left:original, right: decompression). R = 0.7mm, N bins = 16

Figure 6 :

 6 Figure 6: Rendering of the original (left) and decompressed Stanford St Matthew (right), both have 93.4 million points. R = 3mm, N bins = 16

  Figure5shows the detail recovery as well as the denoising obtained through a compression/decompression cycle. One can see that the details are very well-preserved, while the noise is removed. One can check that only noise is removed by comparing the error (obtained with the fixed parameters proposed) with the scan resolution: in most cases this error is below the scanner resolution. Figure6shows the compression/decompression result on the dense St Matthew pointset [LPC * 00], which has been reduced to 93,4M points due to memory limitations (4GB). One can see that the details are very well preserved. Figure7shows the behavior of the decompressed surface with respect to a standard reconstruction algorithm [KH13]. Quasi-identical results are submitted to COMPUTER GRAPHICS Forum (4/2014).

Figure 7 :

 7 Figure 7: Original mire pointset (top) and decompression (bottom). Both pointsets were reconstructed using Screened Poisson Reconstruction [KH13]. R = 0.6mm, N bins = 16

Figure 8 :

 8 Figure 8: Comparison with kd-tree based coding. Left : original point cloud. Right : comparative decompression. Even with more bits per point (4.83 against 0.6 in our method), the right part encoded with [GD02] is less accurate than our approach (left part). R = 0.7mm, N bins = 16

Figure 10 :

 10 Figure 10: Comparison with previous works in terms of rate/distortion on the David model. The different bitrates were obtained by increasing the radii of the patches and the size of the descriptors.

Figure 11 :

 11 Figure 11: Compressing the Lovers with 10% of outliers (Left:decompressed; Right:initial)

Figure 12 :

 12 Figure 12: Left : the Bremen point cloud (bottom) and the decompressed result (top). Right : closeup views : original (bottom) and decompressed (top) R = 18cm, N bins = 16. In sparsely sampled areas, two phenomena occur: if the sampling is too low, the points are removed during the pre-processing step and if the sampling is above the limit, there is a dilatation effect clearly visible in the closeups (right).

Table 1 :

 1 Number of bytes compressed versus decompressed. The last column uses the acquisition device precision, information we lacked for both the Bremen and St Matthew point clouds. The St Matthew pointset (Fig 6) was first reduced to 93,4M points due to memory limitations (4GB).

		number of		compressed size	RMSE	Percentage of points
	Pointset	points	R	(bytes)	(% of diagonal)	with error above	bpp
						sampling precision
	Anubis	9, 9M	0.7mm	1, 201, 636	0.01mm (0.003%)	1.23%	0.96
	Lovers of Bordeaux	15, 8M	0.5mm	1, 152, 245	0.01mm(0.006%)	0.86%	0.59
	Mire	16, 1M	0.6mm	1, 480, 118	0.03mm (0.011%)	1.30%	0.73
	Tanagra	16, 4M	0.7mm	1, 238, 271	0.01mm (0.004%)	1.56%	0.60
	David	28, 2M	10mm	2, 150, 711	0.24mm (0.004%)	0.75%	0.61
	Bremen	69, 9M	18cm	6, 699, 915	1.48cm (0.005%)	not available	0.76
	St Matthew	93, 5M	3mm	9, 780, 886	0.05cm (0.002%)	not available	0.83

submitted to COMPUTER GRAPHICS Forum (4/2014).

Acknowledgements

The authors wish to thank Pierre-Marie Gandoin for providing his compression code and valuable discussions. The Bremen dataset (Fig. 12) is courtesy of Dorit Borrmann and Jan Elseberg from Jacobs University Bremen, Germany (the dataset is available from the Robotic 3D Scan Repository of Universität Osnabrück). The Saint Matthew (Fig. 6) and David (Fig. 10) point sets are courtesy of the Digital Michelangelo Project, Stanford University. This work was partially supported by the Région Rhône-Alpes (ARC6).


Luis Gonzalez-Mestres
email: luis.gonzalez-mestres@cosmology.megatrend.edu.rs

Pre-Big Bang, space-time structure, asymptotic Universe Spinorial space-time and a new approach to Friedmann-like equations

Planck and other recent data in Cosmology and Particle Physics can open the way to controversial analyses concerning the early Universe and its possible ultimate origin. Alternatives to standard cosmology include pre-Big Bang approaches, new spacetime geometries and new ultimate constituents of matter. Basic issues related to a possible new cosmology along these lines clearly deserve further exploration. The Planck collaboration reports an age of the Universe t close to 13.8 Gyr and a present ratio H between relative speeds and distances at cosmic scale around 67.3 km/s/Mpc. The product of these two measured quantities is then slightly below 1 (about 0.95), while it can be exactly 1 in the absence of matter and cosmological constant in patterns based on the spinorial spacetime we have considered in previous papers. In this description of space-time we first suggested in 1996-97, the cosmic time t is given by the modulus of a SU(2) spinor and the Lundmark-Lemaître-Hubble (LLH) expansion law turns out to be of purely geometric origin previous to any introduction of standard matter and relativity. Such a fundamental geometry, inspired by the role of half-integer spin in Particle Physics, may reflect an equilibrium between the dynamics of the ultimate constituents of matter and the deep structure of space and time. Taking into account the observed cosmic acceleration, the present situation suggests that the value of 1 can be a natural asymptotic limit for the product H t in the long-term evolution of our Universe up to possible small corrections. In the presence of a spinorial space-time geometry, no ad hoc combination of dark matter and dark energy would in any case be needed to get an acceptable value of H and an evolution of the Universe compatible with observation. The use of a spinorial space-time naturally leads to unconventional properties for the space curvature term in Friedmannlike equations. It therefore suggests a major modification of the standard cosmology based on general relativity. In the new cosmology thus introduced, the contribution of the space curvature to the value of H 2 is positive definite independently of the apparent sign of this curvature, and has a much larger value than the standard curvature term. Then, a cosmological constant is no longer required. The spinorial space-time also generates automatically for each comoving observer a privileged space direction that, together with parity violation, may explain the anisotropies observed in WMAP and Planck data. Contrary to frequent statements, such a signature would not be a strange phenomenon. The effect emerges directly from the use of two complex space-time coordinates instead of the conventional four real ones, and the privileged direction would correspond to the phase of the cosmic spinor. We remind here our previous work on the subject and further discuss some cosmological implications of the spinorial space-time.

A puzzling situation

The present situation in Cosmology and Particle Physics is intriguing. It can raise many exciting and unconventional debates leading to new ideas and approaches that should not be ignored.

WMAP [START_REF]Wilkinson Microwave Anisotropy Probe (WMAP[END_REF] and Planck [START_REF] Gonzalez-Mestres | [END_REF] analyses claim basically to have confirmed the validity of standard cosmological models. At the same time, the confirmation by CERN of the existence of the Higgs boson has not been followed by any clear indication suggesting the possible detection of supersymmetric particles. LUX results [START_REF]First results from the LUX dark matter experiment at the Sanford Underground Research Facility[END_REF] do not seem to confirm dark-mater patterns based on low-mass WIMPs (Weakly Interacting Massive Particles), and the recent ACME collaboration result on the electron dipole model [START_REF] Acme | Advanced Cold Molecule Electron EDM) Collaboration, Order of Magnitude Smaller Limit on the Electric Dipole Moment of the Electron[END_REF] does not favour supersymmetric dark matter. The possible interpretation of the AMS (Alpha Magnetic Spectrometer) results [START_REF] Aguilar | First Result from the Alpha Magnetic Spectrometer on the International Space Station: Precision Measurement of the Positron Fraction in Primary Cosmic Rays of 0.5-350 GeV[END_REF] is still being debated [START_REF] See | The Price of a Dark Matter Annihilation Interpretation of AMS-02 Data[END_REF].

Clearly, Planck and WMAP data require further studies after the 2013 Planck results have been made available (see [START_REF]Planck 2013 results. I. Overview of products and scientific results[END_REF][START_REF]Planck 2013 results. XVI. Cosmological parameters[END_REF] and other recent papers by the Planck Collaboration). Although WMAP and Planck have systematically used the ΛCDM model as the basic tool for data analysis, other possible cosmological approaches [START_REF] Gonzalez-Mestres | Pre-Big Bang, fundamental Physics and noncyclic cosmologies[END_REF][START_REF] Gonzalez-Mestres | Planck data, spinorial space-time and asymptotic Universe, mparc 13-33[END_REF] must be explored including pre-Big Bang patterns. Suitable new cosmologies can be related to new ultimate constituents of matter such as superbradyons (superluminal preons [START_REF] Gonzalez-Mestres | Cosmological Implications of a Possible Class of Particles Able to Travel Faster than Light[END_REF]) beyond the original preon idea [START_REF] Salam | Gauge Unification of Fundamental Forces, Nobel lecture[END_REF] and to new physics at ultra-high energy [START_REF] Gonzalez-Mestres | High-energy cosmic rays and tests of basic principles of Physics[END_REF][START_REF] Gonzalez-Mestres | Ultra-high energy physics and standard basic principles[END_REF] possibly generated beyond Planck scale. Actually, it is even not obvious that the notion of Planck scale still makes sense or that quantum mechanics, relativity and grand unification of standard interactions are indeed relevant at these energy and distance scales [START_REF] Gonzalez-Mestres | Cosmic rays and tests of fundamental principles[END_REF][START_REF] Gonzalez-Mestres | Vacuum Structure, Lorentz Symmetry and Superluminal Particles[END_REF]. The actual preon-like dynamics and the effective space-time geometry may obey substantially different physical principles [START_REF] Gonzalez-Mestres | Pre-Big Bang, fundamental Physics and noncyclic cosmologies[END_REF][START_REF] Gonzalez-Mestres | Cosmological Implications of a Possible Class of Particles Able to Travel Faster than Light[END_REF].

Similar questions can be raised about the meaning of standard concepts such as dark matter and dark energy that, according to Planck analysis, would account for 95% of the energy in our Universe. Even assuming that the recent AMS results correspond to a dark matter signature, alternatives to fashionable theories already exist [START_REF] Gonzalez-Mestres | Superbradyons and some possible dark matter signatures[END_REF] and dark energy is an ambiguous concept used ad hoc to help standard cosmology to explain data. As already emphasized in [START_REF] Gonzalez-Mestres | Pre-Big Bang, fundamental Physics and noncyclic cosmologies[END_REF][START_REF] Gonzalez-Mestres | High-energy cosmic rays and tests of basic principles of Physics[END_REF], fundamental physics at ultrahigh energy and very small distances can be different enough from standard relativity and quantum field theory (QFT) to radically change the situation concerning the cosmological constant and the cosmic forces contributing to the expansion of the Universe. Vacuum structure may be substantially different from that considered in standard QFT.

In particular, the dark energy term in the standard Friedmann equations may be hiding a leading contribution of purely geometric origin unrelated to any cosmological constant or conventional quantum field [START_REF] Gonzalez-Mestres | Pre-Big Bang, fundamental Physics and noncyclic cosmologies[END_REF][START_REF] Gonzalez-Mestres | Cosmic rays and tests of fundamental principles[END_REF]. The question of the role of cosmic space curvature is in this respect a major one, but measuring the actual curvature at large cosmological scales can be a very difficult task if the Universe is much larger than the observable one. The global space curvature can even be masked by more local phenomena. Even so, such a small cosmic space curvature generated beyond standard matter and without the conventional space-time geometry may turn out to provide the leading contribution to the expansion of the Universe [START_REF] Gonzalez-Mestres | Pre-Big Bang, fundamental Physics and noncyclic cosmologies[END_REF][START_REF] Gonzalez-Mestres | Spinorial space-time and Friedmann-like equations (I), mparc[END_REF] contrary to standard expectations.

Inflation is often presented as having been confirmed by Planck and WMAP data, but the actual situation seems extremely unclear in spite of the remarkable effort in model building for more than thirty years. Pre-Big Bang cosmologies with a new fundamental space-time geometry and new ultimate constituents of matter can naturally provide suitable alternatives to inflation [START_REF] Gonzalez-Mestres | Pre-Big Bang, fundamental Physics and noncyclic cosmologies[END_REF][START_REF] Gonzalez-Mestres | Cosmological Implications of a Possible Class of Particles Able to Travel Faster than Light[END_REF] avoiding all the standard problems that led to such a hypothesis within the conventional cosmological framework.

More globally, the conventional ΛCDM cosmology used in WMAP and Planck analyses leads to a conceptually unclear situation for the ratio between relative velocities and distances at cosmic scales given by the Lundmark-Lemaître-Hubble (LLH) constant (see [START_REF] Gonzalez-Mestres | Pre-Big Bang, fundamental Physics and noncyclic cosmologies[END_REF]19] and references [START_REF] Steer | History: Who discovered Universe expansion?[END_REF] to [START_REF] Hubble | A relation between distance and radial velocity among extra-galactic nebulae[END_REF]). In the ΛCDM approach, the value of the LLH constant from Friedmann equations depends crucially on a set of poorly identified phenomenological parameters. On the contrary, if H is the LLH constant and t the age of the Universe, the product H t can be identically equal to 1 in other cosmological geometries already before introducing matter, energy and gravitation [START_REF] Gonzalez-Mestres | Pre-Big Bang, fundamental Physics and noncyclic cosmologies[END_REF][START_REF] Gonzalez-Mestres | Planck data, spinorial space-time and asymptotic Universe, mparc 13-33[END_REF]. No cosmological constant is then needed, while the cosmological constant is a source of trouble in standard cosmology.

Towards a new cosmology?

Even assuming the existence of a realistic physical concept similar to the cosmological constant, in alternative approaches to quantum field theory [START_REF] Gonzalez-Mestres | Pre-Big Bang, fundamental Physics and noncyclic cosmologies[END_REF][START_REF] Gonzalez-Mestres | Lorentz symmetry violation, dark matter and dark energy[END_REF] one can expect this phenomenon to be generated only in the presence of standard matter. The ultimate vacuum may be made of more fundamental objects not obeying QFT, with a ground-state structure that does not incorporate standard bosons (including their zero modes) as permanent condensates. In this case, the contribution of the new cosmological constant to Friedmann-like equations will decrease like the matter density as the Universe expands, and the relation H = t -1 will be preserved in suitable space-time patterns as a limit at large t [START_REF] Gonzalez-Mestres | Pre-Big Bang, fundamental Physics and noncyclic cosmologies[END_REF][START_REF] Gonzalez-Mestres | Planck data, spinorial space-time and asymptotic Universe, mparc 13-33[END_REF] except if new (small) corrections to the effective geometry must be taken into account.

It is not difficult to conceive patterns escaping the usual cosmological constraints. Starting from the standard Friedmann equations [START_REF] Enqvist | Cosmological inflation[END_REF][START_REF] Gorbunov | Introduction to the Theory of the Early Universe[END_REF], we can consider the relation :

H 2 = 8π G ρ/3 -k R -2 c 2 + Λ c 2 /3 (1)
where H = a -1 s da s /dt is the LLH constant, a s the scale factor, G the gravitational constant, ρ the energy density associated to standard matter, c the speed of light, k R -2 the curvature parameter, R the present curvature distance scale of the Universe (the curvature radius, and possibly the radius of the Universe, for k = 1) and Λ the cosmological constant.

Then, taking ρ = 0, Λ = 0 and a -1 s da s /dt = R -1 dR/dt leads for k = -1 (negative curvature, hyperbolic space) to the identity H t = 1 with the simple relation dR/dt = c. But such a result does not appear to be compatible with cosmological observations, as the value R = c t would be too small for a realistic fit to data. The problem can potentially be solved [START_REF] Gonzalez-Mestres | Planck data, spinorial space-time and asymptotic Universe, mparc 13-33[END_REF] replacing c by a much larger constant c ′ , but this implies abandoning the standard Friedmann formulation based on a direct extrapolation from general relativity and on the role of c as a universal critical speed.

Starting from a positive space curvature, the spinorial space-time (SST) geometry we introduced in 1996-97 [START_REF] Gonzalez-Mestres | Physical and Cosmological Implications of a Possible Class of Particles Able to Travel Faster than Light[END_REF][START_REF] Gonzalez-Mestres | Space, Time and Superluminal Particles[END_REF] yields a more clear result concerning the possible geometric origin of the relation H = t -1 where the cosmic time t is now given by the modulus of a cosmic SU(2) spinor (or a function of this modulus). The cosmic space at time t is described by the associated hypersphere with a natural positive space curvature and no critical speed at that stage. The law H = t -1 holds then automatically in the absence of matter and cosmological constant [START_REF] Gonzalez-Mestres | Pre-Big Bang, fundamental Physics and noncyclic cosmologies[END_REF][START_REF] Gonzalez-Mestres | Cosmic rays and tests of fundamental principles[END_REF], just as in the previous negative curvature example. The associated Friedmann-like equation equivalent of (1) is then:

H 2 = -dH/dt (2)
where -dH/dt = t -2 is a positive curvature term from the SST geometry, with a sign contrary to that expected from the standard Friedmann equations and much larger in modulus due to the explicit identity between the cosmic time and the radius of the Universe. Thus, in the SU(2) spinorial space-time the inverse square of the age of the Universe plays the role of the Friedmann-like curvature term replacing from the beginning the expressionk R -2 c 2 in the equation equivalent to [START_REF]Wilkinson Microwave Anisotropy Probe (WMAP[END_REF]. In this way, the ratio c R -1 is directly replaced by an independent inverse time scale and thesign multiplying the space curvature term is removed. The curvature term is now positive for k = 1. As already stressed, no space units, critical speed(s), matter, relativity and gravitation have been introduced yet. Standard matter and a cosmological constant are not required in order to get a sensible value of H. Indeed, SST appears to give rise to a new cosmology.

Another specific property of the spinorial space-time is the existence of a privileged space direction for each point of the cosmic space-time [START_REF] Gonzalez-Mestres | Pre-Big Bang, fundamental Physics and noncyclic cosmologies[END_REF][START_REF] Gonzalez-Mestres | Cosmic rays and tests of fundamental principles[END_REF]. This is a direct consequence of the use of two complex coordinates, instead of the usual four real ones, to describe the cosmic space-time position. Cosmic SU(2) transformations correspond to space translations, and the privileged space direction is associated to the SU(2) generator that leaves unchanged the space-time position spinor [START_REF] Gonzalez-Mestres | Cosmic rays and tests of fundamental principles[END_REF][START_REF] Gonzalez-Mestres | Space, Time and Superluminal Particles[END_REF]. Space translations following this generator leave the cosmic space-time position spinor invariant up to a complex phase. Thus, the possible existence of a privileged space direction in WMAP and Planck data on Cosmic Microwave Background (CMB) should not be considered as a strange phenomenon even if further data and analyses are required to confirm and interpret such a result.

Together with parity violation [START_REF] Gonzalez-Mestres | Spinorial space-time and Friedmann-like equations (I), mparc[END_REF][START_REF] Gonzalez-Mestres | Spinorial space-time and privileged space direction (I), mparc[END_REF], this basic property of the cosmic spinorial space-time, again previous to the introduction of standard matter and relativity, can indeed be at the origin of the asymmetry in the average temperatures on opposite hemispheres of the sky (therefore defining a privileged space direction) possibly confirmed by Planck [START_REF]Planck 2013 results. XXIII. Isotropy and statistics of the CMB[END_REF] after previous WMAP observations. The fact that such a natural effect may look surprising from a conventional point of view suggests that the fundamental principles of Cosmology possibly remain to be explored and explicitly formulated.

The spinorial space-time (SST)

Although the use of spinors in Particle Physics to describe fermions is very old [START_REF] Dirac | The Quantum Theory of the Electron[END_REF][START_REF] Pauli | Zur Quantenmechanik des magnetischen Elektrons[END_REF], and the most fundamental presently known standard matter fields have spin 1/2, the basic description of space-time has remained based on the conventional four real numbers describing integer angular momenta: a time dimension and three space dimensions. However, particles with half-integer spin are not, strictly speaking, representations of the real SO(3) rotation group. In [START_REF] Gonzalez-Mestres | Physical and Cosmological Implications of a Possible Class of Particles Able to Travel Faster than Light[END_REF][START_REF] Gonzalez-Mestres | Space, Time and Superluminal Particles[END_REF], we suggested to replace this standard real four-dimensional space-time by the complex two-dimensional one actually felt by spin-1/2 particles. The standard rotation group SO(3) for space dimensions is then replaced by the spinorial SU(2) allowing spin-1/2 particles to be representations of the actual rotation group. Such a natural choice turns out to have important cosmological implications [START_REF] Gonzalez-Mestres | Planck data, spinorial space-time and asymptotic Universe, mparc 13-33[END_REF][START_REF] Gonzalez-Mestres | Spinorial space-time and privileged space direction (I), mparc[END_REF].

A privileged space direction

In our original formulation of the spinorial space-time (SST), it was necessary [START_REF] Gonzalez-Mestres | Space, Time and Superluminal Particles[END_REF] to elucidate how the usual four real space-time coordinates would be related to the two complex coordinates of the SU(2) spinor ξ. We then were led to first consider: i) the SU(2) invariant | ξ | 2 = ξ † ξ where the dagger stands for hermitic conjugate; ii) the vector z = ξ † σξ where σ is the vector formed by the Pauli matrices. The role of both expressions had to be carefully determined.

In [START_REF] Gonzalez-Mestres | Physical and Cosmological Implications of a Possible Class of Particles Able to Travel Faster than Light[END_REF], we had already proposed to interpret t = | ξ | as the cosmic time. Then, one has: z = t 2 where z is the modulus of z . Actually, other definitions of t in terms of z (i.e. t = z) are also possible and lead to the same physical results provided the definition of space coordinates is adapted to that of the cosmic time. The point ξ = 0 corresponds to the cosmic time origin in a naturally expanding Universe were the speed of light is not an intrinsic fundamental constant and inflation is no longer needed. Because of the pre-existing spinorial geometry, the S 3 hypersphere at constant t incorporates an additional structure with a privileged space direction associated to each comoving observer.

It immediately turned out [START_REF] Gonzalez-Mestres | Space, Time and Superluminal Particles[END_REF] that it was not possible to interpret z as providing the space coordinates: one coordinate, corresponding to an overall phase of the cosmic spinor, was missed by both t and z . Therefore, a different description of space was necessary on the S 3 hypersphere of constant t . The vector z = ξ † σξ turned out to be the tangent direction of a whole cosmic trajectory naturally associated to the cosmic spinor ξ . Such a direction corresponds locally to the σ matrix that leaves ξ invariant. Thus, the privileged space direction immediately emerged when trying to define space coordinates in [START_REF] Gonzalez-Mestres | Space, Time and Superluminal Particles[END_REF], as a natural consequence of the additional geometric structure carried by the S 3 hypersphere of constant t obtained from the SST. The additional geometric structure was just the expression of the original complex coordinates. This point was further developed in the 2011 Post Scriptum to [START_REF] Gonzalez-Mestres | Cosmic rays and tests of fundamental principles[END_REF], and reminded in [START_REF] Gonzalez-Mestres | Pre-Big Bang, fundamental Physics and noncyclic cosmologies[END_REF] before the Planck announcement [START_REF]Planck 2013 results. XXIII. Isotropy and statistics of the CMB[END_REF]. The SST prediction of a privileged space direction for each comoving observer may have been confirmed by Planck data. As reminded in [START_REF] Gonzalez-Mestres | Spinorial space-time and Friedmann-like equations (I), mparc[END_REF][START_REF] Gonzalez-Mestres | Spinorial space-time and privileged space direction (I), mparc[END_REF], parity violation is also expected to play a crucial role.

Space in the constant-time hypersphere

The standard real space coordinates are then generated locally in the SST [START_REF] Gonzalez-Mestres | Space, Time and Superluminal Particles[END_REF]. The SU(2) transformations of a cosmic spinor ξ correspond to curved space translations leaving unchanged the space-time origin ξ = 0 . Relative space coordinates are defined in this way through the parameters of the relevant cosmic SU(2) rotation. Conventional space rotations of these coordinates around a fixed space-time point ξ 0 are then SU(2) transformations acting on the SU(2) group itself. The distance between two points on the space hypersphere at t = t 0 is given by t 0 times the angle θ separating the two points in a circle of radius t 0 . As a 2π SU(2) rotation changes the sign of the spinor ξ 0 , the value θ = 2π corresponds to the antipodal space-time positionξ 0 .

On the S 3 ξ = t 0 hypersphere and taking ξ = ξ 0 as the space origin (the observer position), a point ξ can be described [START_REF] Gonzalez-Mestres | Space, Time and Superluminal Particles[END_REF] as :

ξ = U ξ 0 ( 3
)
where U is a SU(2) transformation. Writing:

U = exp (i/2 t -1 0 σ. x) ≡ U( x) (4)
the vector x , with 0 ≤ x (modulus of x) ≤ 2π t 0 , can be naturally interpreted as the position vector at constant time t 0 . The privileged space direction just defined corresponds to the set of values of x such that U( x) ξ 0 = exp (i φ) ξ 0 where exp (i φ) is a complex phase factor. Such a definition is invariant under SU(2) transformations and remains stable under comoving time evolution. Using the equivalent definition (local tangent direction) provided by the vector | ξ 0 | -2 ξ † 0 σ ξ 0 , the value of | ξ | -2 ξ † σ ξ is common to all points ξ lying on the privileged space trajectory and on its comoving equivalents.

Under a SU(2) transformation V turning σ. x into V σ. x V -1 (the vector representation), the operator U defined in (4) transforms into VUV -1 (space rotation around ξ 0 ). The vector v obtained from the equation V = exp (i/2 σ. v) defines the rotation axis and angle. Acting on position vectors, the SU(2) transformations are actually SO(3) rotations. x provides the space coordinates of ξ with respect to ξ 0 and transforms like a SO(3) real vector [START_REF] Gonzalez-Mestres | Space, Time and Superluminal Particles[END_REF].

Cosmology

Cosmologically comoving frames naturally appear in the SST and correspond to straight lines through the cosmic origin ξ = 0. They provide local privileged rest frames. At the stage considered here, the spinorial space-time does not require the introduction of any critical speed. The cosmic time is actually the only physical dimension and the distance between two comoving observers on the S 3 hypersphere is proportional to t times the angular separation as seen from the space-time origin ξ = 0. This automatically leads to: i) a constant relative speed between two comoving observers given by their angular separation; ii) a universal ratio between relative velocities and distances among comoving observers for a given value of t (the LLH law defining the LLH constant H).

Then, the H t = 1 law automatically follows in the absence of standard matter and of a cosmological constant [START_REF] Gonzalez-Mestres | Pre-Big Bang, fundamental Physics and noncyclic cosmologies[END_REF][START_REF] Gonzalez-Mestres | Cosmic rays and tests of fundamental principles[END_REF]. Such a strong result is close to Planck data that suggest H ≃ 67.3 km/s/Mpc and t ≃ 13.82 Gyr leading to H t ≃ 0.95.

The H t = 1 relation can thus be a suitable limit for the evolution of the Universe, as the standard matter density is expected to decrease with the Universe expansion faster than the SST curvature term [START_REF] Gonzalez-Mestres | Planck data, spinorial space-time and asymptotic Universe, mparc 13-33[END_REF][START_REF] Gonzalez-Mestres | Spinorial space-time and Friedmann-like equations (I), mparc[END_REF]. Since no critical speed has yet been introduced, the Universe corresponding to the cosmic SST can be naturally assumed to be much larger than our visible Universe.

As discussed above, another important cosmological implication of the SST geometry is the existence of a privileged space direction for each commoving observer. Standard cosmology using only real space-time coordinates cannot exhibit such a geometric property, as the possible memory of an underlying spinorial space-time structure is lost from the beginning by such an approach. It is therefore crucial to elucidate if the asymmetry between opposite hemispheres of the sky observed by Planck [START_REF]Planck 2013 results. XXIII. Isotropy and statistics of the CMB[END_REF] has a primordial origin or corresponds to a more accidental phenomenon.

Standard relativity and SST Cosmology

An essential characteristics of the SST is the fact that it naturally incorporates a cosmic space curvature previous to the introduction of matter and relativity. Furthermore, no global Lorentz symmetry appears to be required as local preferred reference frames exist in standard Cosmology.

Then, conventional relativity is not necessarily a global cosmic law and may instead just correspond to the space-time structure felt locally by standard matter at low energy [START_REF] Gonzalez-Mestres | Cosmological Implications of a Possible Class of Particles Able to Travel Faster than Light[END_REF][START_REF] Gonzalez-Mestres | Vacuum Structure, Lorentz Symmetry and Superluminal Particles[END_REF]. The SST structure would also be dominant at very small distances and manifest itself in particular in the internal structure of spin-1/2 particles generating a spinorial wave function [START_REF] Gonzalez-Mestres | Cosmic rays and tests of fundamental principles[END_REF].

If the SST geometry is the expression of some form of preonic vacuum, standard matter can nucleate during a specific period of the history of the Universe corresponding to a new form of the conventional Big Bang. Thus, the spinorial space-time appears as a natural framework for pre-Big Bang patterns. It may then happen that standard matter does not nucleate everywhere in the Universe, but only in some regions, and that vacuum is not really homogeneous at very large cosmic scale. If so, the whole Universe would not be really homogeneous and isotropic as usually assumed. Finding possible signatures of such a situation would be a major cosmological challenge.

The introduction of a spinorial space-time implies by itself an unavoidable modification of the cosmological equations derived from general relativity, as the curvature tensor does no longer vanish in the absence of standard matter and of a cosmological constant. Therefore, the Einstein field equation [START_REF] Einstein | Die Feldgleichungen der Gravitation[END_REF][START_REF] Einstein | Die Grundlage der allgemeinen Relativitätstheorie[END_REF]:

R µν -1/2 g µν + g µν Λ = 8π G c -4 T µν (5)
(R µν = Ricci curvature tensor, R = scalar curvature, g µν = metric tensor, T µν = stress-energy tensor), usually considered as the starting point of modern cosmology, can no longer remain unchanged. A new term should be added to this equation accounting for the pre-existing global cosmic curvature of the SST. One can then modify (5) writing:

R µν -1/2 g µν + g ′ µν S + g µν Λ = 8π G c -4 T µν (6)
where the term g ′ µν S describes the SST curvature term. Taking a local frame where g ′ µν S is diagonal with a time-like direction and three space-like directions, the structure of g ′ µν S is fundamentally different from standard relativity. In particular, the speed of light c does not govern the ratio between space and time components. Then, a very small SST space curvature is compatible with a dominant time-like contribution from the same g ′ µν S tensor to the relevant Friedmann-like equation.

A modified Friedmann equation

Such an approach does not contradict experiment. The SDSS-III Collaboration [START_REF]SDSS-III Collaboration[END_REF][START_REF]BOSS Measures the Universe to One-Percent Accuracy[END_REF] has recently published [START_REF] Anderson | The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: Baryon Acoustic Oscillations in the Data Release 10 and 11 galaxy samples[END_REF][START_REF] Tojeiro | The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: galaxy clustering measurements in the low redshift sample of Data Release[END_REF] more stringent bounds on the effective cosmic space curvature. But the space-like part of the tensor g ′ µν S can be consistent with these bounds without questioning the leading role of its time-like component in a modified Friedmann equation as suggested by [START_REF] Gonzalez-Mestres | [END_REF].

Assuming that the local comoving frames from standard cosmology are close enough to those generated by the SST cosmic geometry, a new version of the first Friedmann equation can be:

H 2 = 8π G ρ/3 -k R -2 c 2 + t -2 + K + Λ c 2 /3 ( 7
)
where K is a correction term accounting in particular for : -a possible small difference between the comoving frames of standard cosmology and those (preexisting) obtained from the underlying SST cosmic geometry; -a reaction of the nucleated standard matter to the pre-existing expansion of the Universe led by the SST geometry [START_REF] Gonzalez-Mestres | Pre-Big Bang, fundamental Physics and noncyclic cosmologies[END_REF][START_REF] Gonzalez-Mestres | Planck data, spinorial space-time and asymptotic Universe, mparc 13-33[END_REF];

-vacuum inhomogeneities at cosmic scale and other non-standard effects.

As the present value of H t is close to 1, one can reasonably assume K to be small. But the situation may have been different at earlier stages of the Universe. Similar corrections will also be present in a modified second Freedmann equation where the possible difference (tending to disappear at large t) between SST and standard comoving frames can produce a significant effect.

As previously emphasized, in such a description of the Universe evolution Λ does not stand for the conventional cosmological constant from standard QFT but for a possible new concept [START_REF] Gonzalez-Mestres | Pre-Big Bang, fundamental Physics and noncyclic cosmologies[END_REF][START_REF] Gonzalez-Mestres | High-energy cosmic rays and tests of basic principles of Physics[END_REF] associated to pre-BigBang cosmologies [START_REF] Gonzalez-Mestres | Pre-Big Bang, vacuum and noncyclic cosmologies[END_REF][START_REF] Gonzalez-Mestres | Planck, cosmic rays and unconventional cosmologies, contribution to the Planck 2011 Conference[END_REF] and to new fundamental Physics [START_REF] Gonzalez-Mestres | Superbradyons and some possible dark matter signatures[END_REF][START_REF] Gonzalez-Mestres | Lorentz symmetry violation, dark matter and dark energy[END_REF].

SST and space curvature

Equation [START_REF]Planck 2013 results. I. Overview of products and scientific results[END_REF] has been formulated without considering a possible variation of the sign of the term t -2 on the right-hand side. As previously seen, he sign and the value of this term have been obtained [START_REF] Gonzalez-Mestres | Pre-Big Bang, fundamental Physics and noncyclic cosmologies[END_REF][START_REF] Gonzalez-Mestres | Cosmic rays and tests of fundamental principles[END_REF] from the S 3 spatial hypersphere naturally associated to the spinorial space-time at constant cosmic time t . But the question can be raised whether the SST may lead to a different value and sign for the contribution of the curvature term to the first modified Friedmann equation.

At the stage of the calculation leading to this curvature term, only a cosmic time scale exists and plays simultaneously the role of an effective distance scale with the spherical space structure naturally suggested by the SST geometry. No distinct distance scale or critical speed has yet been introduced. Thus, it is in principle possible to redefine the spatial distance at cosmic scale as seen by a comoving observer at a point ξ 0 on the ξ = t hypersphere.

Hyperbolic coordinates in the SST

In particular, an effective hyperbolic curvature can replace the initial spherical one with a suitable redefinition of the observed distances. Contrary to the hyperspherical configuration common to all comoving observers, such a hyperbolic structure would have a privileged comoving observer. But in all cases, the spinorial structure of space-time is more fundamental and will remain unchanged by this new definition of distances on the S 3 hypersphere.

If D is the cosmic distance on the space hypersphere corresponding to the angular distance θ times t [START_REF] Gonzalez-Mestres | Pre-Big Bang, fundamental Physics and noncyclic cosmologies[END_REF][START_REF] Gonzalez-Mestres | Cosmic rays and tests of fundamental principles[END_REF] and leading to the LLH law [START_REF] Gonzalez-Mestres | Planck data, spinorial space-time and asymptotic Universe, mparc 13-33[END_REF][START_REF] Gonzalez-Mestres | Cosmic rays and tests of fundamental principles[END_REF], a possible alternative to this natural definition would be to send to infinity, from the point of view of the spatial distance on the S 3 hypersphere, the antipodal pointξ 0 (θ = 2π) as seen by the comoving observer based at ξ 0 . This transformation turns D = θ t into a new definition of distance, D ′ , such as for instance:

D 2 = 4 π 2 | ξ 0 | 2 D ′ 2 (4 π 2 | ξ 0 | 2 + D ′ 2 ) -1 (8)
D ′ 2 = 4 π 2 | ξ 0 | 2 D 2 (4 π 2 | ξ 0 | 2 -D 2 ) -1 (9)
leading to :

D ′ 2 = 4 π 2 | ξ 0 | 2 θ 2 (4 π 2 -θ 2 ) -1 (10)
D = 2 π | ξ 0 | (θ = 2 π) corresponds to D ′ = ∞
and the spherical curvature can thus become a hyperbolic one. The LLH law and the H t = 1 equation remain unchanged by this redefinition of distance where D ′ is proportional to t.

The standard orthogonal coordinates x = t sin (θ/2) and y = t cos (θ/2) , where θ is the angular distance to ξ 0 , can then be replaced by suitable new coordinates x ′ and y ′ such that:

(dD ′ ) 2 = (dx ′ ) 2 + (dz ′ ) 2 (11)
A hyperbolic geometry can be built writing:

x ′ = α | ξ 0 | [exp(w) -exp(-w)] (12)
y ′ = α | ξ 0 | [exp(w) + exp(-w)] (13)
y ′2 -x ′2 = 4 α 2 | ξ 0 | 2 ( 14
)
where α is a constant and w a new variable. One has then:

(dD ′ /dw) 2 = 2 α 2 | ξ 0 | 2 [exp(2w) + exp(-2w)] (15)
allowing also to relate w and θ. With w = 0 for θ = 0 and taking small values of both variables, one gets: (dθ/dw) 2 ≃ 4 α 2 [START_REF] Gonzalez-Mestres | Vacuum Structure, Lorentz Symmetry and Superluminal Particles[END_REF] so that dθ/dw ≃ 1 for α = 1/2 . This is just an example of a possible hyperbolic structure generated from the spinorial space-time and using | ξ 0 | as the natural distance scale. Another possibility in order to generate a hyperbolic curvature would be to send to infinity, from the point of view of distance as seen by the observer, a whole circle of the spatial hypersphere writing:

D ′ 2 = 4 π 2 | ξ 0 | 2 D 2 ( 4 λ π 2 | ξ 0 | 2 -D 2 ) -1 (17)
where 0 < λ < 1 .

Then, one gets:

-

D ′ = ∞ for D = 2π λ 1/2 | ξ 0 | -an imaginary D ′ for larger values of D.
In particular, no signal traveling in space from comoving points at D > 2π λ 1/2 | ξ 0 | can ever reach the comoving observer at ξ 0 . However, this hidden region of the Universe remains connected to the observer through the spinorial coordinates in potential causality-violating paths.

International Conference on New Frontiers in Physics 2013

Some consequences

As the new definitions of the spatial distance just considered remain equal to | ξ 0 | times a function of θ, the relation H = t -1 is preserved together with the spinorial space-time structure.

Thus, contrary to conventional patterns, the spinorial space-time can account for both spherical and hyperbolic space curvatures with the same (dominant) positive-definite curvature term in the Friedmann-like equation and the H = t -1 law in the absence of matter and of a cosmological constant. In particular, the curvature term t -2 in (7) turns out to be related to the primordial space-time geometry and not to the space curvature directly observed.

Similarly, as the underlying SST structure is not modified by the coordinate transformation, nothing in the transition from spherical to hyperbolic curvature appears to remove the existence of a privileged space direction for each comoving observer. Comoving observers will remain defined by the original SST geometry and the change in the definition of distances will not modify the LLH law.

A potentially significant difference between the spherical and hyperbolic views of space with the underlying spinorial space-time geometry is that the spherical description with positive curvature is generated as a direct consequence of the mathematical spinorial structure and is the same for all comoving observers. This is not the case of the hyperbolic structure that is specific to each observer even in the choice of the (antipodal) point sent to infinity.

Such unconventional properties of the Cosmology based on the spinorial space-time clearly indicate the presence of a more fundamental geometry than that usually considered in theories obtained from standard relativity. This ultimate geometry would play a crucial role in both particle structure and the origin of our Universe. Further work in this direction is obviously required.

Conclusion

The indications from Planck and WMAP suggesting the existence of a privileged space direction as predicted by the spinorial space-time, together with the automatic implementation of the LLH law and of the H t = 1 relation in the SST approach irrespectively of the apparent sign of the space curvature, are encouraging properties from a phenomenological point of view.

Further confirmation of the observational evidence [START_REF]Planck 2013 results. XXIII. Isotropy and statistics of the CMB[END_REF] for a privileged space direction seems necessary, as well as a careful study of the structure of this phenomenon. It also remains to be checked in practice if the new (SST) curvature term in the modified first Friedmann equation ( 7) can successfully replace the conventional cosmological constant in a global phenomenological description of the Universe. Alternative cosmological models should be built and compared with observation.

In any case, the fact that the SST approach can account for both spherical and hyperbolic descriptions of the observed space with the same (leading) cosmological weight of the space curvature in [START_REF]Planck 2013 results. I. Overview of products and scientific results[END_REF] and the same law relating H and t is a new and promising feature. These basic properties turn out to be directly related to the structure of the new ultimate space-time geometry provided by the SST.

Another encouraging feature of the SST approach is that it provides the potential space-time expression of a new fundamental vacuum structure beyond standard QFT, possibly associated to new ultimate constituents of matter and allowing to avoid the cosmological constant problem [START_REF] Gonzalez-Mestres | Pre-Big Bang, fundamental Physics and noncyclic cosmologies[END_REF][START_REF] Gonzalez-Mestres | Pre-Big Bang, vacuum and noncyclic cosmologies[END_REF]. Explicit vacuum and particle structure, and pre-Big Bang, patterns along these lines remain to be explicitly built and compared to available data.

The study of ultra-high energy cosmic rays can provide complementary relevant information [START_REF] Gonzalez-Mestres | High-energy cosmic rays and tests of basic principles of Physics[END_REF][START_REF] Gonzalez-Mestres | Ultra-high energy physics and standard basic principles[END_REF]. Possible superbradyon searches are also an important topic. In particular, cosmic superbradyons can be part of the dark matter [START_REF] Gonzalez-Mestres | Superbradyons and some possible dark matter signatures[END_REF] or decay emitting high-energy cosmic rays [START_REF] Gonzalez-Mestres | Physical and Cosmological Implications of a Possible Class of Particles Able to Travel Faster than Light[END_REF].

Post Scriptum (April 24, 2014)

After this paper was sent (on February 5) to the editors, the BICEP2 collaboration [START_REF]BICEP2 Collaboration, BICEP2 II: Experiment and Three-Year Data Set[END_REF] reported [START_REF]BICEP2 I: Detection Of B-mode Polarization at Degree Angular Scales[END_REF] a B-mode polarization of the cosmic microwave background (CMB) radiation.

BICEP2 data are often interpreted as a strong direct evidence for cosmic inflation and primordial gravitational waves. It is claimed that the detected B-modes of CMB cannot be generated primordially by density perturbations and that only gravitational waves originating from the inflationary expansion of the Universe can have produced such a signature. But, as pointed out in [START_REF] Gonzalez-Mestres | CMB B-modes, spinorial space-time and Pre-Big Bang (I), mparc[END_REF], this reasoning is tacitly based on the standard Big Bang approach where inflation is in any case required to get a potentially consistent description of the Universe. Pre-Big Bang models can easily escape these constraints and do not require any inflationary scenario to produce CMB B-modes.

In particular, the spinorial space-time discussed here naturally generated a privileged space direction for each comoving observer. Then, contrary to standard cosmological models where only CMB E-modes associated to gradients can be present except for the B-modes due to inflationary gravitational waves, a cosmology based on the spinorial space-time easily generates B-modes through rotations around the privileged space direction and vector products by this direction. This is just a conservative example of the potentialities of unconventional space-time geometries in pre-Big Bang cosmologies. Pre-Big Bang cosmologies [START_REF] Gonzalez-Mestres | Pre-Big Bang, fundamental Physics and noncyclic cosmologies[END_REF][START_REF] Gonzalez-Mestres | Cosmological Implications of a Possible Class of Particles Able to Travel Faster than Light[END_REF] can describe the origin of the Universe and its largescale structure more naturally than ad hoc models based on cosmic inflation.

Thus, the BICEP2 result, if confirmed, can potentially provide an experimental evidence for the spinorial space-time we suggested in 1996-97 [START_REF] Gonzalez-Mestres | Physical and Cosmological Implications of a Possible Class of Particles Able to Travel Faster than Light[END_REF][START_REF] Gonzalez-Mestres | Space, Time and Superluminal Particles[END_REF].

It must be noticed, however, that the present experimental situation remains unclear and that it is not yet excluded that the observed signal be due to galactic dust [START_REF] Liu | Fingerprints of Galactic Loop I on the Cosmic Microwave Background[END_REF]. Further experimental work is therefore required before any theoretical assertion can be made on the grounds of such data.

Another recent claim is that Fermi data provide evidence for a possible annihilation of 31-40 GeV dark matter particles in the region surrounding the galactic center [START_REF] Daylan | The Characterization of the Gamma-Ray Signal from the Central Milky Way: A Compelling Case for Annihilating Dark Matter[END_REF]. Alternative interpretations of Fermi data have also been dealt with [START_REF] Demianski | Beyond the ΛCDM cosmology: complex composition of dark matter[END_REF][START_REF] Boehm | A weighty interpretation of the Galactic Centre excess[END_REF]. As already stressed, the possibility that remnant superbradyons be the dark matter has been considered in our papers since 1995 [START_REF] Gonzalez-Mestres | Superbradyons and some possible dark matter signatures[END_REF][START_REF] Gonzalez-Mestres | Physical and Cosmological Implications of a Possible Class of Particles Able to Travel Faster than Light[END_REF].

The possible existence of galactic dark matter would not, in any case, provide an evidence for standard cosmology and inflation. Pre-Big Bang dynamics with new physics can generate several kinds of dark matter. In pre-Big Bang patterns, superbradyons and the SST can be related to the ultimate structure of matter and to a more fundamental space-time beyond standard relativity, as already felt in part by standard fermions. In particular, this new early cosmology would be directly based on the ultimate internal structure of quarks, leptons and gauge bosons [START_REF] Gonzalez-Mestres | Pre-Big Bang, fundamental Physics and noncyclic cosmologies[END_REF][START_REF] Gonzalez-Mestres | Cosmic rays and tests of fundamental principles[END_REF].