Tianlin Duan
email: tianlin.duan@ebc.uu.se@xavier

Zebin Zhang

Mathieu Genete

Céline Poux

Adrien Sicard

Martin Lascoux

Vincent Castric

Xavier Vekemans
email: xavier.vekemans@univ-lille.fr

Dominance between self-incompatibility alleles determines the mating system of Capsella allopolyploids

Keywords: Self-incompatibility, polyploidy, S-locus, SRK, SCR, genetic dominance, Capsella

The shift from outcrossing to self-fertilization is one of the main evolutionary transitions in plants, and has broad effects on evolutionary trajectories. In Brassicaceae, the ability to impede self-fertilization is controlled by two genes, SCR and SRK, tightly linked within the S-locus. A series of small non-coding RNAs also encoded within the S-locus regulates the transcriptional activity of SCR alleles, resulting in a linear dominance hierarchy between them. Brassicaceae allopolyploid species are often self-compatible (SC) even when one of their parents is self-incompatible, but the causes of the loss of self-incompatibility (SI) in polyploid lineages have generally remained elusive. We used a series of synthetic hybrids obtained between self-fertilizing Capsella orientalis and outcrossing C. grandiflora to test whether the breakdown of SI in allopolyploid species, such as C. bursa-pastoris, could be explained by the dominance interactions between S-haplotypes inherited from the parental lineages. After establishing a database of reference Sallele sequences, we used RNA-sequencing data from young inflorescences to measure allele-specific expression of the SCR and SRK genes in diploid and tetraploid synthetic hybrids. We then compared the observed expression of SCR alleles with the predicted dominance relationship between S-haplotypes in pollen and with the seed set from autonomous self-fertilization in the synthetic hybrids. Our results formally establish that upon hybridization, the immediate effect on the mating system depends on the relative dominance between S-alleles inherited from the parental species. They illustrate that a detailed understanding of the genetic architecture of the control of SI is essential to predict the patterns of association between the mating system and changes in ploidy.

.

Introduction

Mating systems have far-reaching effects on plant evolution [START_REF] Wright | Evolutionary consequences of self-fertilization in plants[END_REF]. For instance, shifts from outcrossing to self-fertilization are expected to reduce the effective rate of recombination and genetic polymorphism [START_REF] Glémin | Impact of mating systems on patterns of sequence polymorphism in flowering plants[END_REF]), while at the same time providing reproductive assurance when mates are scarce (Jain, 1976). The establishment of polyploid populations is an iconic example of these effects.

Whole-genome duplication (WGD) is prevalent in plants [START_REF] Soltis | Polyploidy and genome evolution in plants[END_REF], and polyploid species are overrepresented in the Arctic flora [START_REF] Brochmann | Polyploidy in arctic plants[END_REF]) as well as in invasive [START_REF] Pandit | Ploidy influences rarity and invasiveness in plants[END_REF]) and domesticated plants [START_REF] Salman-Minkov | Whole-genome duplication as a key factor in crop domestication[END_REF]. Moreover, ancient WGD events on phylogenies seem to be associated with drastic environmental changes [START_REF] Vanneste | Analysis of 41 plant genomes supports a wave of successful genome duplications in association with the Cretaceous-Paleogene boundary[END_REF]. Therefore, WGD has often been hypothesized to allow faster adaptation and niche differentiation in changing environments (Selmecki et al.2015;[START_REF] Baniaga | Polyploid plants have faster rates of multivariate niche differentiation than their diploid relatives[END_REF]. However, beside these potential long-term advantages, newly formed polyploid genotypes are also expected to suffer from the immediate lack of gametes of the same cytotype and from the lower fitness of interploidy hybrids. This phenomenon, known as "minority cytotype exclusion" [START_REF] Levin | Minority cytotype exclusion in local plant populations[END_REF][START_REF] Husband | Constraints on polyploid evolution: a test of the minority cytotype exclusion principle[END_REF], is expected to drastically hinder the success of newly formed polyploid lineages.

Self-fertilization should greatly increase the establishment success of polyploid populations by allowing them to avoid minority cytotype exclusion [START_REF] Fowler | Critical factors in the establishment of allopolyploids[END_REF]. However, empirical surveys on the association between polyploidy and self-fertilization either confirmed the positive association [START_REF] Barringer | Polyploidy and self-fertilization in flowering plants[END_REF][START_REF] Robertson | Comparative evidence for the correlated evolution of polyploidy and self-compatibility in Solanaceae[END_REF], or found no association [START_REF] Mable | Polyploidy and self-compatibility: is there an association?[END_REF]. This suggests that the current models fail to incorporate important details of the interaction between polyploidy and self-fertilization, such as e.g. mechanisms controlling the mating system or the confounding effects of hybridization. An intriguing observation is that allopolyploid lineages (in which WGD occurred in association with hybridization) often exhibit low outcrossing rates, whereas autopolyploid lineages (where "only" WGD occurred) often exhibit predominant outcrossing or mixed mating systems [START_REF] Husband | Mating consequences of polyploid evolution in flowering plants: Current trends and insights from synthetic polyploids[END_REF]).

The mating system of Brassicaceae species is controlled by a sporophytic self-incompatibility system, in which self-pollen recognition is caused by the allele-specific interaction between a pollen coat ligand protein (encoded by the SCR gene) and a stigma transmembrane receptor kinase (encoded by the SRK gene, [START_REF] Takayama | Direct ligand-receptor complex interaction controls Brassica self-incompatibility[END_REF]. The two genes are tightly linked within a small genomic region called the Slocus, where a large number of S-alleles (also called S-haplotypes) typically segregate in selfincompatible species. S-haplotypes form a complex dominance hierarchy in anthers, whereby small noncoding RNA (sRNA) generated by dominant S-haplotypes transcriptionally silence the SCR gene of recessive S-haplotypes [START_REF] Tarutani | Trans-acting small RNA determines dominance relationships in Brassica self-incompatibility[END_REF][START_REF] Durand | Dominance hierarchy arising from the evolution of a complex small RNA regulatory network[END_REF]. Hence, while a large fraction of individuals are heterozygous at the S-locus, in most cases transcripts from only one of the two SCR alleles are present [START_REF] Kakizaki | Linear Dominance Relationship among Four Class-II S Haplotypes in Pollen is Determined by the Expression of SP11 in Brassica Self-Incompatibility[END_REF][START_REF] Burghgraeve | Base-Pairing Requirements for Small RNA-Mediated Gene Silencing of Recessive Self-Incompatibility Alleles in Arabidopsis halleri[END_REF], resulting in phenotypic dominance.

Diversification of S-haplotypes in Brassicaceae is very ancient, as indicated by the very high level of nucleotide divergence among S-haplotype sequences and extensive trans-specific and even trans-generic sharing among related taxa [START_REF] Castric | Plant self-incompatibility in natural populations: a critical assessment of recent theoretical and empirical advances[END_REF]. In Arabidopsis and Capsella, S-haplotypes are classified into four main dominance classes, related to their phylogenetic relationships with class I being the most recessive and class IV being the most dominant [START_REF] Prigoda | Phenotypic and Genotypic Expression of Selfincompatibility Haplotypes in Arabidopsis lyrata Suggests Unique Origin of Alleles in Different Dominance Classes[END_REF]Durand et al.2014;[START_REF] Bachmann | Genetic basis and timing of a major mating system shift in Capsella[END_REF]. Several allopolyploid species of the Brassicaceae family originated from the hybridization between a self-incompatible (SI) and a self-compatible (SC) parental species, including A. suecica (with A.thaliana as SC parent and A. arenosa as SI parent; Novikova et al. 2017), A. kamchatica (with A. lyrata as SC parent and A. halleri as SI parent;[START_REF] Shimizu-Inatsugi | The allopolyploid Arabidopsis kamchatica originated from multiple individuals of Arabidopsis lyrata and Arabidopsis halleri[END_REF][START_REF] Kolesnikova | Genome of selfing Siberian Arabidopsis lyrata explains establishment of allopolyploid Arabidopsis kamchatica[END_REF], and C. bursa-pastoris (with C. orientalis as SC parent and C. grandiflora as SI parent; [START_REF] Douglas | Hybrid origins and the earliest stages of diploidization in the highly successful recent polyploid Capsella bursa-pastoris[END_REF]. These three species have a recent allopolyploid origin, and all share the common feature of being self-compatible. The reason why these allopolyploid species originating from SIxSC are SC rather than SI is intriguing. An interesting possibility could be that the dominance interactions between Shaplotypes could have caused the instantaneous breakdown of SI in these species if the (non-functional) S-haplotype contributed by the SC species had retained the ability to suppress the expression of the (functional) SCR alleles contributed by the SI species in hybrid offspring (reviewed in [START_REF] Novikova | Ancestral self-compatibility facilitates the establishment of allopolyploids in Brassicaceae[END_REF]. Consistent with this hypothesis, the allotetraploid A. suecica, A. kamchatica, and C. bursa-pastoris all share the same nonfunctional S-allele as their respective SC parental species [START_REF] Novikova | Genome sequencing reveals the origin of the allotetraploid arabidopsis suecica[END_REF][START_REF] Bachmann | Genetic basis and timing of a major mating system shift in Capsella[END_REF][START_REF] Bachmann | On the origin of the widespread self-compatible allotetraploid Capsella bursa-pastoris (Brassicaceae)[END_REF][START_REF] Kolesnikova | Genome of selfing Siberian Arabidopsis lyrata explains establishment of allopolyploid Arabidopsis kamchatica[END_REF]). In addition, at least some resynthesized A. suecicaor C. bursa-pastoris-like allotetraploids are immediately SC after hybridization [START_REF] Novikova | Genome sequencing reveals the origin of the allotetraploid arabidopsis suecica[END_REF][START_REF] Bachmann | On the origin of the widespread self-compatible allotetraploid Capsella bursa-pastoris (Brassicaceae)[END_REF][START_REF] Duan | Expression pattern of resynthesized allotetraploid Capsella is determined by hybridization, not whole-genome duplication[END_REF], also supporting that the loss of SI could be an instant outcome of possessing one (relatively dominant) non-functional S-haplotype. A key prediction from this scenario is that the mating system of the resulting hybrid should vary according to the dominance of the Shaplotype contributed by the SI parent relative to that of the SC parent. However, formal proof of this hypothetical process, i.e. the establishment of a direct causal link between the relative dominance of the S-haplotypes, the expression of SCR alleles in anthers, and the loss of SI in allopolyploids, is still lacking.

Here, we used an experimental approach based on a series of synthetic allopolyploid individuals obtained between the selfer C. orientalis and the outcrosser C. grandiflora [START_REF] Duan | Expression pattern of resynthesized allotetraploid Capsella is determined by hybridization, not whole-genome duplication[END_REF] to test whether the breakdown of SI observed in allopolyploid species in Brassicaceae, such as C. bursa-pastoris, could be explained by the dominance interaction between S-haplotypes in anthers. First, we used published genomic and transcriptomic resequencing data to assess a methodology to infer S-locus genotypes and pollen S-locus phenotypes in Capsella. Then we used RNA-sequencing (RNA-seq) data from young inflorescences to measure allele-specific expression of the SCR and SRK genes in synthetic diploid and tetraploid C. orientalis x C. grandiflora hybrids, as an approximation of the early stages of natural C. bursa-pastoris. Finally, we compared the observed expression of SCR alleles with the seed set from autonomous self-pollination in the synthetic hybrids. Altogether, our results demonstrate that the relative dominance of S-alleles inherited from the parental species is a key determinant of the mating system of nascent polyploid lineages.

Results

A reliable methodology to determine S-locus genotypes and phenotypes in Capsella.

First, we produced a comprehensive set of S-allele reference sequences in Capsella. For this, we genotyped individuals at the SRK gene using the NGSgenotyp pipeline [START_REF] Genete | Genotyping and De Novo Discovery of Allelic Variants at the Brassicaceae Self-Incompatibility Locus from Short-Read Sequencing Data[END_REF] [START_REF] Guo | Recent speciation of Capsella rubella from Capsella grandiflora, associated with loss of self-incompatibility and an extreme bottleneck[END_REF][START_REF] Neuffer | S-alleles and mating system in natural populations of Capsella grandiflora (Brassicaceae) and its congeneric relatives[END_REF]; see Supplementary Information).

We obtained a fully resolved S-locus genotype for 177 individuals (Table S1), and identified 74 different S-alleles, including 25 previously unknown C. grandiflora S-alleles (Table S2). For most of them we were able to obtain full sequences of the exon 1 of SRK (available at https://www.doi.org/10.6084/m9.figshare.22567558). Among those new S-alleles we identified an apparently functional SRK allele, noted H4047, that shares 99% identity with the SRK pseudogene (for which the coding sequence is interrupted at position 949 of exon 1) found at the S-locus in subgenome A of C. bursa-pastoris by [START_REF] Bachmann | On the origin of the widespread self-compatible allotetraploid Capsella bursa-pastoris (Brassicaceae)[END_REF]. A more complete description of this set of S-alleles is given in the Supplementary Information. Beside these 74 SRK alleles, we also identified five sequences clustering with SRK alleles (H0002, H0003, H0011, H0012 and H0013 in Table S1, named CgrSRK01, CgrSRK06, CgrSRK09, CgrSRK51 and CgrSRK63 in [START_REF] Paetsch | Evolution of the self-incompatibility system in the Brassicaceae: identification of S-locus receptor kinase (SRK) in self-incompatible Capsella grandiflora[END_REF][START_REF] Neuffer | S-alleles and mating system in natural populations of Capsella grandiflora (Brassicaceae) and its congeneric relatives[END_REF], see Table S2) that we considered as paralogous sequences unlinked to the S-locus, as already documented in [START_REF] Schierup | Identification and Characterization of a Polymorphic Receptor Kinase Gene Linked to the Self-Incompatibility Locus of Arabidopsis lyrata[END_REF] and Prigoda et al. (2005, see Supplementary Information).

Second, we validated the use of transcriptomic data to determine S-locus genotypes using the NGSgenotyp pipeline. We applied the approach described above to genotype the S-locus of four C. grandiflora, four C. orientalis, and 16 C. bursa-pastoris individuals, using published genome resequencing data as well as RNA-seq data obtained separately from leaf, root or flower bud tissues from the exact same individuals [START_REF] Kryvokhyzha | Towards the new normal: Transcriptomic convergence and genomic legacy of the two subgenomes of an allopolyploid weed (Capsella bursa-pastoris)[END_REF]. Strictly identical S-locus genotypes were inferred based on genomic DNA and RNA-seq data from flower bud tissues (Table S3), as expected by the codominant expression of SRK in pistils reported in Brassicaceae [START_REF] Hatakeyama | The S receptor kinase gene determines dominance relationships in stigma expression of self-incompatibility in Brassica[END_REF][START_REF] Burghgraeve | Base-Pairing Requirements for Small RNA-Mediated Gene Silencing of Recessive Self-Incompatibility Alleles in Arabidopsis halleri[END_REF], hence demonstrating that RNA-seq data from flower buds can be used to reliably genotype the S-locus in Capsella. We note that all four individuals of C. grandiflora were heterozygous at the Slocus, and all C. orientalis individuals were homozygous for the allele called H4004n (we use the notation "n" to indicate that this allele is non-functional), in agreement with [START_REF] Bachmann | Genetic basis and timing of a major mating system shift in Capsella[END_REF][START_REF] Bachmann | On the origin of the widespread self-compatible allotetraploid Capsella bursa-pastoris (Brassicaceae)[END_REF] note that these authors refer to this allele as CoS12). In agreement with [START_REF] Bachmann | On the origin of the widespread self-compatible allotetraploid Capsella bursa-pastoris (Brassicaceae)[END_REF], most allotetraploid C. bursa-pastoris individuals had two copies of the H4004n allele derived from the non-functional C. orientalis parental allele and all individuals had two copies of the non-functional H4047n allele, derived from the functional C. grandiflora parental allele H4047 (Table S3). However, we found that a second allele, H2002, already known in C. grandiflora, appeared to be segregating at the Slocus in the C. orientalis subgenome (present in two copies in accession DUB-RUS9 and in one copy, together with H4004n, in LAB-RUS-4). As expected, we generally observed no or very low expression of SRK in leaves or roots (Table S3).

Third, we evaluated whether S-locus phenotypes in pollen could be assessed based on RNA-seq data. We obtained RNA-seq data from flower buds for seven diploid C. grandiflora individuals used as parents in the production of synthetic polyploids and hybrids with C. orientalis (see below), as well as for six synthetic autotetraploids (respectively Cg2 and Cg4 in Fig. 1). We obtained the S-locus genotypes (at SRK) of these individuals using the NGSgenotyp pipeline, as described above. Five C. grandiflora Salleles were found to segregate in this experimental material, again with approximately balanced transcript levels between both SRK alleles in heterozygotes (Fig. 2, Table S4). Then, we obtained full SCR transcript sequences for each of the five S-alleles by applying the de novo assembly module of the NGSgenotyp pipeline (available at https://www.doi.org/10.6084/m9.figshare.22567558), based on a reference database of known SCR sequences from A. halleri and A. lyrata (see Supplementary Material).

We used these new reference sequences to compare patterns of allele-specific expression for SRK and SCR (Table S4). In agreement with the results of [START_REF] Burghgraeve | Base-Pairing Requirements for Small RNA-Mediated Gene Silencing of Recessive Self-Incompatibility Alleles in Arabidopsis halleri[END_REF] in A. halleri, allele-specific expression was much more asymmetric between SCR alleles than between SRK alleles. Indeed, in all heterozygous individuals, one of the two alleles contributed over 99% of the total SCR transcript levels, with a single exception (individual Cg4-1-3), where the relative expression was still as high as 93% (Table S4). This very strong allelic imbalance was also found in tetraploid individuals, suggesting that a dominant allele is capable of repressing several co-occurring alleles at once. The identity of the predominantly expressed allele was fully concordant with expectations based on the predicted classes of dominance between S-alleles (Table S4). Even in cases where two S-alleles of the same dominance class co-occurred (e.g. H2008 andH2022 in individual Cg2-12-3, or H4015 andH4035 in Cg4-1-3), the asymmetry of the transcript levels remained very strong, hence enabling us to determine the dominance hierarchy among the five alleles, as follows: H4035 > H4015 > H2008 > H2022 > H1001. We also analyzed one diploid and one tetraploid C. orientalis individual (Table S4) and observed a single S-allele in pistil and pollen, H4004n, whose SCR sequence was fully identical to that reported by [START_REF] Bachmann | Genetic basis and timing of a major mating system shift in Capsella[END_REF]. These results also confirm that both SRK and SCR are still expressed in C. orientalis, as reported by [START_REF] Bachmann | Genetic basis and timing of a major mating system shift in Capsella[END_REF], even though they are non-functional. 2,S5). Overall, five different S-alleles were segregating among these 53 individuals: the non-functional C. orientalis H4004n allele along with four C. grandiflora alleles (all alleles described above except H1001). Assuming that H4004n has retained the ability to transcriptionally repress S-alleles of a lower dominance class, as suggested by [START_REF] Bachmann | On the origin of the widespread self-compatible allotetraploid Capsella bursa-pastoris (Brassicaceae)[END_REF], we used two different approaches to predict the SI phenotype of hybrids: (1) based on the S-locus genotype inferred from SRK data, a hybrid individual was predicted to be SC if it carried the C. orientalis H4004n allele and none of the S-alleles derived from C. grandiflora that are predicted to be more dominant than H4004n (i.e. H4035 and H4015), and to be SI otherwise (i.e. carrying H2008 and/or H2022); (2) based on the SCR relative expression data, a hybrid individual was predicted to be SC if the relative SCR read depth of allele H4004n was higher than 0.5, and to be SI otherwise. As shown in Table 2, predictions of the SI phenotypes based on the two methods were highly congruent. Overall, 18 of the 27 homoploid hybrids (F individuals), all 7 tetraploidized hybrids (Sh individuals), and 10 of the 19 hybrids between tetraploid C. grandiflora and C. orientalis (Sd individuals) were predicted to be functionally self-compatible. Indeed, in these individuals the presence of H4004n SCR transcripts was associated with efficient transcriptional repression of H2008 and H2022, as expected given their predicted classes of dominance. We observed only two exceptions to this general pattern (individuals F-5-6 and F-10-4, where H2022 had a higher level of expression than H4004n). Conversely, presence of H4015 and/or H4035 was associated with transcriptional repression of H4004n (Table 1), confirming recessivity of the latter with respect to these other class IV S-alleles.

The S-locus genotypes and phenotypes of diploid and synthetic tetraploid hybrids

Inferred S-locus phenotypes predict the mating system of hybrids

We then compared the autonomous seed set of diploid and tetraploid hybrids to test whether the S-locus genotypes and the observed transcriptional dominance of SCR alleles can explain which C. orientalis × C. grandiflora hybrids are SC and which are SI. Seed production under autonomous selfing was used as an indicator of self-compatibility and was classified into three categories to represent individuals with almost no seed (<10 seeds), few seeds (10-300 seeds) and a large number of seeds (> 300 seeds).

We found that both the SRK-predicted dominance of the H4004n allele and the relative expression level of its SCR allele are strong predictors of the ability of autonomous seed production, with only a few exceptions (Fig. 3). The SRK-based prediction of self-compatibility was strongly associated with seed production categories (Fisher's exact test, p-value < 0.001), with most individuals predicted to be selfincompatible producing no or few seeds under autonomous pollination (<300 seeds). In contrast, individuals that were predicted to be self-compatible usually produced more than 300 seeds. Similarly, the relative expression level of the H4004n SCR allele significantly differed among the three seed production categories (Kruskal-Wallis H test, p-value < 0.001). Individuals with a higher proportion of the H4004n SCR allele expression (e.g., larger than 0.25) usually had autonomous seed production above 300, and individuals with a lower proportion of H4004n expression usually had no or few seeds.

Discussion

Assessing dominance relationships between S-alleles using RNA-seq data

At the S-locus in Brassicaceae, the genotype-to-phenotype map is complicated by the widespread existence of dominance/recessivity interactions between S-alleles. Determination of these dominance relationships ultimately relies on phenotypic assays based on controlled pollinations. Recently, [START_REF] Burghgraeve | Base-Pairing Requirements for Small RNA-Mediated Gene Silencing of Recessive Self-Incompatibility Alleles in Arabidopsis halleri[END_REF] demonstrated that phenotypic dominance in pollen can be predicted with high accuracy from the simple comparison of transcript abundances, using quantitative RT-PCR of SCR transcripts. However, allele-specific qPCR primers need to be designed and optimized for every single Shaplotype whose expression is to be quantified, which is not practical when large numbers of Shaplotypes segregate. Here, we show that RNA-seq data from flower buds can be used to reliably infer dominance relationships between S-alleles in pollen. These data are relatively simpler to obtain, as they do not require any specific optimization step beyond a generic RNA-seq library construction and can thus be generalized more readily than the qRT-PCR approach of [START_REF] Burghgraeve | Base-Pairing Requirements for Small RNA-Mediated Gene Silencing of Recessive Self-Incompatibility Alleles in Arabidopsis halleri[END_REF], or the laborintensive phenotypic assessment of dominance. A limitation of this new approach, however, is that it can only quantify transcripts of SCR alleles whose nucleotide sequence is known a priori, which is only the case for a subset of the numerous S-haplotypes typically found in SI species, including C. grandiflora. A potential caveat to this new approach is that it relies on comparing mapping densities of (Illumina) sequencing reads on the nucleotide sequence of SCR alleles, which have relatively short coding sequences, thus making accurate mapping a potential challenge. The high levels of nucleotide divergence among SCR alleles is expected to (at least partially) compensate for this limitation, and accordingly, we found that for the five S-alleles considered in our crossing design, cross-mapping of individual sequencing reads among alleles is negligible, which makes the method highly reliable.

When applying the method to diploid C. grandiflora individuals, we found that the putatively dominant S-haplotype represented >99% of the global level of SCR transcripts in all individuals (Table S5), corresponding to nearly complete dominance at the transcriptional level, in line with [START_REF] Burghgraeve | Base-Pairing Requirements for Small RNA-Mediated Gene Silencing of Recessive Self-Incompatibility Alleles in Arabidopsis halleri[END_REF]. The relative dominance levels we inferred among the Capsella S-haplotypes we studied here were also fully concordant with the dominance interactions previously measured by controlled crosses for the trans-specifically shared S-haplotypes in A. lyrata and A. halleri [START_REF] Prigoda | Phenotypic and Genotypic Expression of Selfincompatibility Haplotypes in Arabidopsis lyrata Suggests Unique Origin of Alleles in Different Dominance Classes[END_REF][START_REF] Llaurens | Does frequencydependent selection with complex dominance interactions accurately predict allelic frequencies at the self-incompatibility locus in Arabidopsis halleri? Evolution[END_REF][START_REF] Durand | Dominance hierarchy arising from the evolution of a complex small RNA regulatory network[END_REF]. Specifically, we confirmed that both S-haplotypes from class IV were dominant over both S-haplotypes from class II, which were themselves dominant over the single class I Shaplotype (Table S5). Another original result from our analysis is that a single SCR allele was majoritarily expressed in each tetraploid C. grandiflora individual, suggesting that the transcriptional silencing machinery controlling dominance remains effective in a tetraploid context, in line with the phenotypic patterns of dominance between S-haplotypes observed in tetraploid individuals of A. lyrata [START_REF] Mable | Inheritance and dominance of self-incompatibility alleles in polyploid Arabidopsis lyrata[END_REF]. We note that the exact number of gene copies of each allele observed in tetraploid individuals in our study remains uncertain because S-locus genotypes were determined using RNA-seq data instead of genomic resequencing [START_REF] Genete | Genotyping and De Novo Discovery of Allelic Variants at the Brassicaceae Self-Incompatibility Locus from Short-Read Sequencing Data[END_REF]. Hence, precise genotyping of tetraploid individuals based on genomic DNA will be needed to quantify the extent of this phenomenon.

The phenotypic effect of the observed dominance relationship among S-alleles was tested by measuring seed production in the diploid and tetraploid C. orientalis x C. grandiflora hybrids. Our measurement of seed production under autonomous selfing is not a formal test of SC, and several confounding effects could blur the link between the dominance relationship of SCR alleles and seed production. In particular, newly formed interspecific hybrids are expected to have lower fitness due to interactions between divergent genomes [START_REF] Fishman | When Two Rights Make a Wrong: The Evolutionary Genetics of Plant Hybrid Incompatibilities[END_REF], therefore individuals with no seed or few seeds could also result from hybrid incompatibility rather than SI. Furthermore, the hybrid individuals were not strictly separated in the growth chamber during flowering time, so for individuals that generated a small number of seeds, we cannot rule out the possibility of pollen contamination from other Capsella plants.

Nonetheless, we still found a strong association between relative expression level of the non-functional H4004n SCR allele and autonomous seed production, which provides evidence that the machinery controlling dominance relationships between SCR alleles is a major determinant of the mating system of the hybrids we obtained.

S-haplotypes dominance mediates the effect of WGD on the mating system of allopolyploids in the

Brassicaceae

Associations between WGD and an autogamous mating system have been largely debated in plant biology, but a general consensus is still lacking [START_REF] Mable | Polyploidy and self-compatibility: is there an association?[END_REF][START_REF] Barringer | Polyploidy and self-fertilization in flowering plants[END_REF][START_REF] Husband | Mating consequences of polyploid evolution in flowering plants: Current trends and insights from synthetic polyploids[END_REF][START_REF] Robertson | Comparative evidence for the correlated evolution of polyploidy and self-compatibility in Solanaceae[END_REF]. Part of the confusion stems from the fact that broad-scale studies focused on taxa comprising a mixture of different SI systems (e.g. self-recognition-based sporophytic SI, self-recognition or non-self recognition-based gametophytic SI), and different types of polyploidy (autopolyploidy, allopolyploidy, or a combination of both, i.e. segmental allopolyploidy). Moreover, the mechanistic causes of the loss of SI in polyploid lineages have generally remained elusive.

In the specific case of allopolyploidy in Brassicaceae, our results formally establish that the mating system immediately upon hybridization depends on the relative dominance of the non-functional allele inherited from the selfer C. orientalis as compared to that of the functional S-allele(s) inherited from the outcrosser C. grandiflora, in line with the model proposed by [START_REF] Novikova | Ancestral self-compatibility facilitates the establishment of allopolyploids in Brassicaceae[END_REF]. This observation raises at least three intriguing questions. First, the non-functional S-allele needs to still retain the capacity to remain dominant. This is the consequence of the particular genetic architecture of dominance between S-haplotypes, where dominance modifiers (small non-coding RNAs) are decoupled from the gene they regulate [START_REF] Billiard | Evidence for Fisher's dominance theory: How many "special cases[END_REF]. This particular genetic architecture of dominance might be less rare than it was long thought to be [START_REF] Billiard | The integrative biology of genetic dominance[END_REF]. Second, the variation we observed relies on the fact that the non-functional S-haplotype that was fixed in C. orientalis has an intermediate level of dominance.

If it had been the most recessive, then all allotetraploid individuals would by definition have inherited a more dominant S-haplotype from C. grandiflora, and would thus remain SI. In contrast, if C. orientalis had fixed the most dominant S-haplotypes, then all hybrids would have turned SC. This raises the question of why these SC species have fixed S-haplotypes at intermediate levels of dominance, while in SI species (including C. grandiflora) the most recessive S-allele typically has by far the highest frequency [START_REF] Billiard | A General Model to Explore Complex Dominance Patterns in Plant Sporophytic Self-Incompatibility Systems[END_REF]). Some factors affecting the fixation probability of SC mutations have been studied by [START_REF] Tsuchimatsu | Effects of pollen availability and the mutation bias on the fixation of mutations disabling the male specificity of self-incompatibility[END_REF], but to the best of our knowledge, the effect of dominance on the fixation probability of SC mutations in a sporophytic SI system remains to be investigated formally. A third interesting question is why most known examples of recent allotetraploids in Brassicaceae involve hybridization between a selfer and an outcrosser [START_REF] Novikova | Ancestral self-compatibility facilitates the establishment of allopolyploids in Brassicaceae[END_REF]. A more general survey of recent allopolyploids would be needed to determine the generality of this pattern, but one tempting hypothesis is that, given an appropriate combination of S-haplotypes, this kind of crosses might instantaneously generate SC allotetraploid individuals avoiding minority cytotype exclusion [START_REF] Novikova | Ancestral self-compatibility facilitates the establishment of allopolyploids in Brassicaceae[END_REF]. Determining whether differences in the intensity of genetic conflicts between the outcrosser and the selfer genomes (in particular over development of the endosperm, [START_REF] Rebernig | Non-reciprocal Interspecies Hybridization Barriers in the Capsella Genus Are Established in the Endosperm[END_REF] can oppose this selective advantage would be an interesting next step. Finally, a parallel can be drawn with the more general process of Haldane's sieve, in which advantageous alleles tend to be fixed more readily in natural populations when they are dominant because they are directly exposed to natural selection [START_REF] Haldane | A mathematical theory of natural and artificial selection, Part I[END_REF][START_REF] Haldane | A Mathematical Theory of Natural and Artificial Selection, Part V: Selection and Mutation[END_REF]. Here in contrast, the selective advantage would go to hybrid lineages that have inherited from their SI parent a S-haplotype more recessive than the S-allele that was fixed in the selfing lineage.

A similar mechanism was proposed by [START_REF] Bachmann | On the origin of the widespread self-compatible allotetraploid Capsella bursa-pastoris (Brassicaceae)[END_REF] for the evolution of selfing in the allotetraploid C. bursa-pastoris from C. orientalis and C. grandiflora parents. Intriguingly, while C. bursa-pastoris is a strong selfer, the S-haplotype it inherited from C. grandiflora (H4047) belongs to class IV, and thus would a priori be expected to be at least as dominant as the S-haplotype it inherited from C. orientalis (H4004n). Also, we identified one C. bursa-pastoris individual lacking the H4004n allele but carrying a (more recessive) class II allele (H2002, Table S3), presumably inherited from C. orientalis.

Hence, other mechanisms might be needed to explain the loss of SI in the early development of the C. bursa-pastoris lineage. It should be noted, however, that dominance interactions between the class IV Salleles have been firmly established at the phenotypic level for a small number of S-alleles only, so the possibility remains that some class IV S-alleles (in this particular case, H4047) could actually be more recessive than H4004n. This is suggested by the observation of a putative target of the small non-coding RNA produced by the C. orientalis S-haplotype in close proximity to the H4047n SCR pseudogene within the A subgenome of C. bursa-pastoris [START_REF] Bachmann | On the origin of the widespread self-compatible allotetraploid Capsella bursa-pastoris (Brassicaceae)[END_REF]). As we found that both alleles (H4004 and H4047) are segregating in the Cg-9 population, it would be interesting to obtain living material carrying those alleles and perform controlled crosses to test their relative dominance.

Material and methods

Methodological approach to type S-alleles in Capsella experimental material based on RNA-seq data

To build an extended dataset of reference sequences of SRK from the self-incompatible species Capsella grandiflora, we genotyped 180 individuals of the Cg-9 population of C. grandiflora from Monodendri, Greece [START_REF] Josephs | Association mapping reveals the role of purifying selection in the maintenance of genomic variation in gene expression[END_REF] at the SRK gene with the NGSgenotyp pipeline [START_REF] Genete | Genotyping and De Novo Discovery of Allelic Variants at the Brassicaceae Self-Incompatibility Locus from Short-Read Sequencing Data[END_REF]) using raw Illumina reads available from Sequence Read Archive (SRA, Table S1). For the SRK reference database, we used available sequences of SRK from A. lyrata and A. halleri [START_REF] Takou | Maintenance of Adaptive Dynamics and No Detectable Load in a Range-Edge Outcrossing Plant Population[END_REF][START_REF] Genete | Genotyping and De Novo Discovery of Allelic Variants at the Brassicaceae Self-Incompatibility Locus from Short-Read Sequencing Data[END_REF], and 62 partial sequences from Capsella grandiflora [START_REF] Guo | Recent speciation of Capsella rubella from Capsella grandiflora, associated with loss of self-incompatibility and an extreme bottleneck[END_REF][START_REF] Neuffer | S-alleles and mating system in natural populations of Capsella grandiflora (Brassicaceae) and its congeneric relatives[END_REF]; see Table S2). Briefly, this pipeline filters raw reads with a dictionary of k-mers extracted from the reference database, then uses Bowtie2 to align filtered reads against each reference sequence from the database and produces summary statistics with Samtools (v1.4; [START_REF] Danecek | Twelve years of SAMtools and BCFtools[END_REF] allowing it to identify S-alleles present in each individual. The pipeline NGSgenotyp also contains a de novo assembly approach module which produces full sequences of the S-domain of SRK for alleles present as partial sequences in the database as well as for newly identified S-alleles.

We then compared the results of the S-alleles genotyping approach obtained from either genomic DNA or RNA-seq data from flower buds, leaf and root tissues. For this we used published data from [START_REF] Kryvokhyzha | Towards the new normal: Transcriptomic convergence and genomic legacy of the two subgenomes of an allopolyploid weed (Capsella bursa-pastoris)[END_REF] on four individuals each of C. grandiflora and C. orientalis, and on 16 individuals of C. bursa-pastoris. We applied the NGSgenotyp pipeline separately on each dataset, using the SRK reference database expanded with the Capsella S-allele sequences obtained above. Our analysis showed that S-allele typing based on RNA-seq data from flower buds gave identical results than those obtained from genomic DNA, so in the rest of the analyses we only used flower buds RNA-seq data.

Creating and sequencing the transcriptome of synthetic hybrids and polyploids and assessing their mating system

To test the dominance relationship among SI alleles and its phenotypic consequences we used diploid and tetraploid hybrids of C. orientalis x C. grandiflora, using the synthetic hybrids generated by Duan et al.

(2023; Fig. 1 andTable 2). We measured fruit set under autonomous selfing to test whether selfcompatibility of these hybrids can be predicted by the dominance of SCR alleles, by comparing the SCR alleles identified in transcriptomes of inflorescences with fruit-set from spontaneous self-pollination in 27 diploid and 26 tetraploid hybrids.

In short, the diploid and tetraploid hybrids were generated from one inbred line of C. orientalis (URAL-RUS5), and seeds that were collected from two wild C. grandiflora individuals of the same population (85.1 and 85.24). Specifically, all the synthetic hybrids were descendants of three C. grandiflora individuals (85.1-5, 85.24-1, and 85.24-5), two of which (85.24-1 and 85.24-5) had the same maternal plant. Diploid hybrids (F) were obtained by crossing C. orientalis with C. grandiflora. Tetraploid hybrids (allotetraploids) were generated in two ways: in the first case the two diploid species were first crossed, and whole-genome duplication (WGD) was induced on the first generation of diploid hybrids with colchicine solution, resulting in "hybridization-first" synthetic allotetraploids (Sh); in the second case, WGD was induced in both diploid species, then the synthetic autotetraploids were crossed to obtain "WGD-first" allotetraploids (Sd). In addition, several diploid hybrids were suspected to have spontaneous WGD without colchicine treatment based on observations on organ size and the shape of trichomes, including individual F-3-5 which was used in the present study. In all interspecific crosses, diploid or tetraploid C. orientalis served as the maternal plant. The second generation of diploid hybrids, Shallotetraploids and Sd-allotetraploids as well as the diploid and tetraploid parental species were then grown together in a growth chamber. Each of the three hybrid groups was represented by six lines (independent hybridization events), and each line was represented by six individuals. The six individuals of the same line were full siblings from self-fertilization. An overview of the mating scheme used to create the different resynthesized hybrids and polyploids is given in Fig. 1.

The ability to generate seeds with only autonomous selfing was recorded for all hybrid individuals as a categorical factor. The hybrid individuals were classified into three categories based on the total number of seeds: 1) having almost no seed (< 10 seeds), 2) having few seeds (10-300 seeds), and 3) having plenty of seeds (> 300 seeds). As the hybrid individuals were not strictly separated in the growth chamber during flowering, the cutoff of ten seeds was applied to reduce noise from occasional pollen contamination from other Capsella plants. The seed set data of two individuals were removed from the dataset, because they were severely affected by disease during flowering time.

The first group of RNA-seq data was from [START_REF] Duan | Expression pattern of resynthesized allotetraploid Capsella is determined by hybridization, not whole-genome duplication[END_REF]. Total RNA was extracted from the inflorescence of 6 diploid hybrids and 14 allotetraploids, using a cetyl-trimethyl-ammonium-bromide (CTAB) based method. Sequencing libraries were prepared with Illumina TruSeq Stranded mRNA (poly-A selection) kit, and sequenced on three NovaSeq 6000 S4 lanes with 150-bp paired-end reads (SNP&SEQ Technology Platform in Uppsala). One sequencing library was prepared and sequenced for each diploid sample, and two libraries were prepared for each tetraploid sample. On average 38.6 and 77.3 million read pairs were generated for the diploid and tetraploid samples, respectively.

To obtain a larger sample size, we performed a second group of RNA-seq on inflorescences of 33 additional hybrid individuals, including 20 diploid hybrids and 13 allotetraploids. Inflorescence samples of this second group were from the same experiment as those from the first group, and were collected at the same time, and stored at -80°C before sequencing. Total RNA was extracted using an RNeasy Plant Mini Kit (Qiagen). The library preparation and sequencing platform were the same as the first group, except that one library was prepared for each individual, regardless of ploidy level. Sequencing of the second group of inflorescence samples yielded an average library size of 97.4 million RNA-seq reads.

Determination of S-locus genotype and phenotype in the Capsella hybrids and synthetic polyploids and confrontation with mating system phenotype assessments

In Brassicaceae, the self-incompatibility phenotype in pollen depends on complex dominance relationships among S-alleles, achieved through modifier genetic elements consisting in small RNAs encoded by precursors lying at the S-locus of dominant alleles and targeting the SCR gene of recessive alleles [START_REF] Tarutani | Trans-acting small RNA determines dominance relationships in Brassica self-incompatibility[END_REF][START_REF] Durand | Dominance hierarchy arising from the evolution of a complex small RNA regulatory network[END_REF]. Dominance in pollen is thus regulated at the transcriptional level, and is associated with very strong inhibition of mRNA production of recessive alleles [START_REF] Burghgraeve | Base-Pairing Requirements for Small RNA-Mediated Gene Silencing of Recessive Self-Incompatibility Alleles in Arabidopsis halleri[END_REF], which could potentially be revealed by analyzing RNA-seq data. Hence, we tested this approach by performing S-allele typing with the NGSgenotyp pipeline using separately a SRK reference database, to determine the S-locus genotype of individuals (because SRK alleles are always co-expressed in the style, [START_REF] Hatakeyama | The S receptor kinase gene determines dominance relationships in stigma expression of self-incompatibility in Brassica[END_REF], and a SCR reference database, to determine which Sallele is majoritarily expressed in pollen (and thus putatively dominant). We applied this approach to RNA-seq data from 8 diploid individuals (7 C. grandiflora + 1 C. orientalis) and 7 tetraploids (6 C. grandiflora + 1 C. orientalis) used as parents in the hybrid experiments [START_REF] Duan | Expression pattern of resynthesized allotetraploid Capsella is determined by hybridization, not whole-genome duplication[END_REF], with the same SRK reference database as above enlarged with newly obtained C. grandiflora allele sequences, and for SCR with a reference database of sequences from A. halleri and A. lyrata (Genbank sequences). For SCR, because of the higher sequence divergence among S-alleles than for SRK, we modified the NGSgenotyp parameters by reducing the k-mer size used for filtering to a value of 15 (the default size used for SRK was 20). This allowed us, with the de novo assembly module of NGSgenotyp, to obtain full coding sequences of SCR for all C. grandiflora alleles present in the hybrids. In order to quantify the relative expression of SCR alleles, we mapped individual RNA-seq data against each reference SCR sequence with the genotyp module of NGSgenotyp and used the mean read depth delivered by Samtools (v1.4; [START_REF] Danecek | Twelve years of SAMtools and BCFtools[END_REF] to compute the ratio of the mean read depth of the predominantly expressed allele (i.e., the putative dominant allele) to the sum of the read depths of all alleles present. We applied the same approach in synthetic tetraploid C. grandiflora individuals, but for SRK data we could only report the number and identity of alleles present, and thus it was not possible to precisely genotype individuals, i.e.

to determine the number of gene copies of any given allele identified (when the total number of alleles detected in an individual was lower than 4, which was the case for all tetraploid individuals).

Once the proposed approach was validated on C. grandiflora diploid and tetraploid individuals, we applied it to the experimental diploid and tetraploid hybrids. We identified two categories of hybrid individuals, in terms of pollen SI phenotype, depending on relative dominance levels of the inherited C. grandiflora, and Tyler Kent and Stephen Wright for a preliminary analysis of S-alleles diversity in Cg-9 population. Finally, we thank Barbara Mable for her critical comments on the manuscript.

Figures and Tables

  Diploid and tetraploid hybrids were produced between C. grandiflora and C. orientalis (Fig. 1; Duan et al. 2023), using seeds from two wild individuals of C. grandiflora and one inbred line of C. orientalis. Diploid hybrids were generated by crossing C. orientalis with C. grandiflora, while allotetraploids were created either by inducing genome doubling in diploid hybrids with colchicine treatment ("Sh" allotetraploids), or by crossing colchicine-induced autotetraploid C. orientalis with autotetraploid C. grandiflora ("Sd" allotetraploids). In all crosses, diploid or tetraploid C. orientalis was used as the maternal plant, mimicking the formation of natural C. bursa-pastoris (Hurka et al. 2012). Then we analyzed RNA-seq data from flower buds of 27 diploid hybrids (F), seven tetraploidized hybrids (Sh) as well as 19 C. orientalis x C. grandiflora autotetraploid hybrids (Sd; Fig. 1, Tables

  grandiflora and C. orientalis parental alleles: individuals with predominant expression of the C. orientalis allele, which is a non-functional S-allele; and individuals with predominant expression of one of the C. grandiflora alleles. Dominance of the non-functional C. orientalis allele is expected to cause a breakdown of self-incompatibility because it will impede recognition and rejection of self-pollen, and thus we assigned an expected self-compatible phenotype to those individuals, and an expected self-incompatible phenotype to individuals with a dominant SCR allele from C. grandiflora. Then we compared seed production data under autonomous selfing with the prediction of self-compatible/self-incompatible phenotype based on SCR dominance. The association between SRK-predicted self-compatibility and seed production categories was tested with Fisher's exact test in R software environment version 3.6.3 (R Core Team, 2020). The relative expression level of the H4004n SCR allele (mean read depth of H4004n/sum of the mean read depth of all other S-alleles of the individual) in the three seed production categories were compared by the Kruskal-Wallis H test. The work on SI in the Lille group is supported by the Agence Nationale de la Recherche (TE-MoMa project, grant number ANR-18-CE02-0020-01), the Région Hauts-de-France and the Ministère de l'Enseignement Supérieur et de la Recherche (CPER Climibio and CPER Ecrin grants), and the European Fund for Regional Economic Development. The study in Uppsala was supported by grant 2019-00806 from the Swedish Research Council to ML and Nilsson-Ehle research grant from the Royal Physiographic Society in Lund to TD. ML also acknowledges support from the Erik Philip-Sörensen Foundation. TD was supported by the Sven and Lilly Lawski Foundation. The computation and data handling were provided by the Swedish National Infrastructure for Computing (SNIC) at Uppmax, partially funded by the Swedish Research Council through grant agreement no 2018-05973. We also thank Elsa Sundkvist for her help with seed counting, Jesper Besgaard for early sharing of unpublished SRK sequences from C.

Fig. 1

 1 Fig. 1 Crosses and induced whole-genome duplication (WGD) for generating diploid (F) and two types of tetraploid hybrids (Sd and Sh) from diploid Capsella orientalis (Co2) and Capsella grandiflora (Cg2; Duan et al. 2023). Autotetraploid Capsella orientalis (Co4) and Capsella grandiflora (Cg4) were created as intermediate steps. The generation numbers of the synthetic polyploids or hybrids are indicated by subscripts, and plant materials used in the present study are highlighted in bold.

Fig. 2

 2 Fig. 2 Phylogeny of SRK alleles identified in parents or hybrids between Capsella grandiflora (Cgr) and C. orientalis (Co). Phylogenetic reconstruction using maximum likelihood under the TPM3uf+I+G4 model implemented by PhyML (see Supplementary Information). We also included sequences identified in C. bursa-pastoris (Cbp), and trans-specific alleles from Arabidopsis halleri (Ah), A. lyrata (Al) or A. arenosa (Aar) when available. Functional specificities and dominance classes of the S-alleles are indicated. The outgroup is represented by an ARK3 sequence from Capsella rubella (Cru). Bootstrap values are represented by dots on the nodes: black: BP = 100, grey: 95<BP<99, white: BP=89, no dot: BP < 80.

Fig. 3

 3 Fig. 3 Relationships of the SRK-predicted dominance of the H4004n SCR allele, the observed relative expression level of the H4004n SCR allele and seed production in the diploid and tetraploid C. orientalis C. grandiflora.

  Fig. 1

Fig. 3 Table 1 .

 31 Fig. 3

  on publicly available short-read resequencing data of 180 C. grandiflora individuals from Monodendri, Greece (the Cg-9 population in[START_REF] Josephs | Association mapping reveals the role of purifying selection in the maintenance of genomic variation in gene expression[END_REF], starting from a database of SRK sequences from A. lyrata and A.

	halleri and 62 partial C. grandiflora SRK sequences previously obtained by Sanger sequencing by Jesper
	Bechsgaard and Mikkel Schierup

Acknowledgements

Data availability

The RNA-sequencing data of the additional Capsella hybrids generated by this article are available in Sequence Read Archive (SRA) of National Center for Biotechnology Information (NCBI), and can be accessed with BioProject number PRJNA946929. All new Capsella SRK and SCR sequences obtained by de novo assembly with the NGSgenotyp pipeline are posted at https://www.doi.org/10.6084/m9.figshare.22567558.

Author contributions

Conception and coordination of the study: TD, ML, VC, XV Creation of experimental material: TD, AS RNASEQ data: TD, ZZ Bioinformatics: TD, MG Data analysis: TD, XV, VC, CP, MG Writing of the manuscript: TD, XV, VC, ML All authors reviewed the manuscript.

Conflict of interest

The authors declare no conflict of interest.


HSIs can be represented as 2D arrays after rearranging the two pixel modes, then: -Pixels = Samples -Spectral bands = Features These arrays are processed using matrix techniques.

Tensor Representation:

By separating the features, we are able to extract meaningful patterns as well as observe some inter-feature relationships. Feature Extraction is done thanks to CP Decomposition (CPD). -Even though the image is vectorized, spatial information is not entirely lost thanks to using small neighborhoods for morphological filtering, which also ensures that the rank stays low.

-In order to extract the features, we use CPD with nonnegative constraints to ensure physical interpretability.

-Each column triplet {𝐚 :,𝐫 , 𝐛 :,𝐫 , 𝐜 :,𝐫 } of features represents a full material's spatial highlight, spectral reflectance, and morphological pattern.

-CPD is unique under only mild conditions.

Feature

  Extraction of Multimodal Images Based on Tensor Decomposition and Mathematical Morphology (poster) Mohamad Jouni, Mauro Dalla Mura, Pierre Comon To cite this version: Mohamad Jouni, Mauro Dalla Mura, Pierre Comon. Feature Extraction of Multimodal Images Based on Tensor Decomposition and Mathematical Morphology (poster). 2020 UGA & UT Workshop, Dec 2020, Online, Japan. . hal-04306074 Blind Source Separation (looking at 4 column triplets of features): -We use AO-ADMM to minimize the cost function: 𝓣 -(𝐀, 𝐁, 𝐂) 𝐅 + 𝐫 𝐀 + 𝐫 𝐁 + 𝐫(𝐂) by solving for each matrix in an alternating ADMM way. The function 𝐫 𝐇 represents the constraints applied only at 𝐇 => CPD is flexible with modal constraints. Feature Extraction of Multimodal Images Based on Tensor Decomposition and Mathematical Morphology Mohamad Jouni 1 , Mauro Dalla Mura 1,2 , Pierre Comon 1 1 Univ. Grenoble Alpes, Grenoble INP, Gipsa-Lab, 38000 Grenoble, France 2 Tokyo Tech World Research Hub Initiative (WRHI), School of Computing, Tokyo Institute of Technology, Tokyo, Japan  Contribution: We extract spatial morphological features from a hyper-spectral image (HSI), represent the data as multi-modal arrays, then process them using tensor decomposition to improve the quality of feature extraction.1. HSIs naturally provide us with spectral features of the objects of the scene in order to understand its physical composition, but spectral features are not enough. 2. We extract spatial neighborhood features through Morphology, but the state-of-the-art either mixes the two feature modalities or treats them independently => Data are lost! 3. Solution: Tensor analysis jointly accounts to the full data block and maintains the interfeature relationships. This is explored in classification and blind separation of image data.

-

  state of the art, CPD allows us to blindly & uniquely unmix the HSI data without loss. References 𝐀 looks like a matrix of samples and features, so by passing it to a classifier: It is possible to explain the features as a scatter plot in a multi-dimensional feature space. (Relations with ELMM for spectral unmixing and variability.) -Comon, P. (2014). Tensors: a brief introduction. IEEE Signal Processing Magazine, 31(3), 44-53. -Jouni, M., Mura, M. D., & Comon, P. (2020). Hyperspectral Image Classification Based on Mathematical Morphology and Tensor Decomposition. Mathematical Morphology-Theory and Applications, 4(1), 1-30.