Fast and robust detection of a known pattern in an image

International audienceMany image processing applications require to detect a known pattern buried under noise. While maximum correlation can be implemented efficiently using fast Fourier transforms, detection criteria that are robust to the presence of outliers are typically slower by several orders of magnitude. We derive the general expression of a robust detection criterion based on the theory of locally optimal detectors. The expression of the criterion is attractive because it offers a fast implementation based on correlations. Application of this criterion to Cauchy likelihood gives good detection performance in the presence of outliers, as shown in our numerical experiments. Special attention is given to proper normalization of the criterion in order to account for truncation at the image borders and noise with a non-stationary dispersion

Copyright Protection of Color Imaging Using Robust-Encoded Watermarking

In this paper we present a robust-encoded watermarking method applied to color images for copyright protection, which presents robustness against several geometric and signal processing distortions. Trade-off between payload, robustness and imperceptibility is a very important aspect which has to be considered when a watermark algorithm is designed. In our proposed scheme, previously to be embedded into the image, the watermark signal is encoded using a convolutional encoder, which can perform forward error correction achieving better robustness performance. Then, the embedding process is carried out through the discrete cosine transform domain (DCT) of an image using the image normalization technique to accomplish robustness against geometric and signal processing distortions. The embedded watermark coded bits are extracted and decoded using the Viterbi algorithm. In order to determine the presence or absence of the watermark into the image we compute the bit error rate (BER) between the recovered and the original watermark data sequence. The quality of the watermarked image is measured using the well-known indices: Peak Signal to Noise Ratio (PSNR), Visual Information Fidelity (VIF) and Structural Similarity Index (SSIM). The color difference between the watermarked and original images is obtained by using the Normalized Color Difference (NCD) measure. The experimental results show that the proposed method provides good performance in terms of imperceptibility and robustness. The comparison among the proposed and previously reported methods based on different techniques is also provided

The Moduli of Singular Curves on K3 Surfaces

In this article we consider moduli properties of singular curves on K3 surfaces. Let $\mathcal{B}_g$ denote the stack of primitively polarized K3 surfaces $(X,L)$ of genus $g$ and let $\mathcal{T}^n_{g,k} \to \mathcal{B}_g$ be the stack parametrizing tuples $[(f: C \to X, L)]$ with $f$ an unramified morphism which is birational onto its image, $C$ a smooth curve of genus $p(g,k)-n$ and $f_*C \in |kL|$. We show that the forgetful morphism $$\eta \; : \; \mathcal{T}^n_{g,k} \to \mathcal{M}_{p(g,k)-n}$$ is generically finite on one component, for all but finitely many values of $p(g,k)-n$. We further study the Brill--Noether theory of those curves parametrized by the image of $\eta$, and find a Wahl-type obstruction for a smooth curve with an unordered marking to have a nodal model on a K3 surface in such a way that the marking is the divisor over the nodes.Comment: Final version. To appear in J. Math. Pures. App

Severi Varieties and Brill-Noether theory of curves on abelian surfaces

Severi varieties and Brill-Noether theory of curves on K3 surfaces are well understood. Yet, quite little is known for curves on abelian surfaces. Given a general abelian surface $S$ with polarization $L$ of type $(1,n)$, we prove nonemptiness and regularity of the Severi variety parametrizing $\delta$-nodal curves in the linear system $|L|$ for $0\leq \delta\leq n-1=p-2$ (here $p$ is the arithmetic genus of any curve in $|L|$). We also show that a general genus $g$ curve having as nodal model a hyperplane section of some $(1,n)$-polarized abelian surface admits only finitely many such models up to translation; moreover, any such model lies on finitely many $(1,n)$-polarized abelian surfaces. Under certain assumptions, a conjecture of Dedieu and Sernesi is proved concerning the possibility of deforming a genus $g$ curve in $S$ equigenerically to a nodal curve. The rest of the paper deals with the Brill-Noether theory of curves in $|L|$. It turns out that a general curve in $|L|$ is Brill-Noether general. However, as soon as the Brill-Noether number is negative and some other inequalities are satisfied, the locus $|L|^r_d$ of smooth curves in $|L|$ possessing a $g^r_d$ is nonempty and has a component of the expected dimension. As an application, we obtain the existence of a component of the Brill-Noether locus $\mathcal{M}^r_{p,d}$ having the expected codimension in the moduli space of curves $\mathcal{M}_p$. For $r=1$, the results are generalized to nodal curves.Comment: 29 pages, 3 figures. Comments are welcome. 2nd version: added some references in Rem. 7.1

Uniform families of minimal rational curves on Fano manifolds

It is a well-known fact that families of minimal rational curves on rational homogeneous manifolds of Picard number one are uniform, in the sense that the tangent bundle to the manifold has the same splitting type on each curve of the family. In this note we prove that certain --stronger-- uniformity conditions on a family of minimal rational curves on a Fano manifold of Picard number one allow to prove that the manifold is homogeneous

Time-causal and time-recursive spatio-temporal receptive fields

We present an improved model and theory for time-causal and time-recursive spatio-temporal receptive fields, based on a combination of Gaussian receptive fields over the spatial domain and first-order integrators or equivalently truncated exponential filters coupled in cascade over the temporal domain. Compared to previous spatio-temporal scale-space formulations in terms of non-enhancement of local extrema or scale invariance, these receptive fields are based on different scale-space axiomatics over time by ensuring non-creation of new local extrema or zero-crossings with increasing temporal scale. Specifically, extensions are presented about (i) parameterizing the intermediate temporal scale levels, (ii) analysing the resulting temporal dynamics, (iii) transferring the theory to a discrete implementation, (iv) computing scale-normalized spatio-temporal derivative expressions for spatio-temporal feature detection and (v) computational modelling of receptive fields in the lateral geniculate nucleus (LGN) and the primary visual cortex (V1) in biological vision. We show that by distributing the intermediate temporal scale levels according to a logarithmic distribution, we obtain much faster temporal response properties (shorter temporal delays) compared to a uniform distribution. Specifically, these kernels converge very rapidly to a limit kernel possessing true self-similar scale-invariant properties over temporal scales, thereby allowing for true scale invariance over variations in the temporal scale, although the underlying temporal scale-space representation is based on a discretized temporal scale parameter. We show how scale-normalized temporal derivatives can be defined for these time-causal scale-space kernels and how the composed theory can be used for computing basic types of scale-normalized spatio-temporal derivative expressions in a computationally efficient manner.Comment: 39 pages, 12 figures, 5 tables in Journal of Mathematical Imaging and Vision, published online Dec 201

Tame class field theory for arithmetic schemes

We extend the unramified class field theory for arithmetic schemes of K. Kato and S. Saito to the tame case. Let $X$ be a regular proper arithmetic scheme and let $D$ be a divisor on $X$ whose vertical irreducible components are normal schemes. Theorem: There exists a natural reciprocity isomorphism \rec_{X,D}: \CH_0(X,D) \liso \tilde \pi_1^t(X,D)^\ab\. Both groups are finite. This paper corrects and generalizes my paper "Relative K-theory and class field theory for arithmetic surfaces" (math.NT/0204330