Lezoray, Christophe Charrier, Abderrahim Elmoataz olivier.lezoray@unicaen.fr Universit´ e de Caen Basse-Normandie, GREYC UMR CNRS 6072, Caen, France EUSIPCO - Glasgow, August 2009 Olivier Lezoray (Universit´ e de Caen) EUSIPCO 2009 August 2009 1 / 22
complete lattice L. A complete lattice L is a non empty set equipped with an ordering relation, such that every non-empty subset K of L has a lower bound ∧K and an upper bound ∨K. Images are modeled by functions mapping their domain space Ω, into a complete lattice L. This is problematic for multivariate images since there is no natural ordering for vectors. Usually, this is performed with a lexicographic ordering (comparing features in a predeﬁned cascade) that introduces strong dissimetry. Our contributation: a general formalism that enables the morphological processing of any multivariate data. Olivier Lezoray (Universit´ e de Caen) EUSIPCO 2009 August 2009 4 / 22
by the mapping f : Ω ⊂ Zl → Rp where l is the image dimension and p the number of channels. One way to deﬁne an ordering relation between vectors is to use a transform h from Rp to Rq followed by the lexicographic ordering on each dimension of Rq: With h : Rp → Rq, and x → h(x) then ∀(xi , xj ) ∈ Rp × Rp, xi ≤ xj ⇔ h(xi ) ≤ h(xj ) (1) Then, the following equivalences can be considered: (total ordering on Rp)⇔(bijective application h : Rp → R⇔ (space ﬁlling curve in Rp)⇔(rank transformation on Rp). Interest of the rank transform h can be seen as a rank transform: to create a total order for building the complete lattice structure, the images’ values are in fact not important, only the rank position on the lattice structure is relevant. When the complete lattice is created, the MM operators only need to perform comparisons between ranks. Olivier Lezoray (Universit´ e de Caen) EUSIPCO 2009 August 2009 6 / 22
N is a function that associates to a vector x ∈ Rp the value r(x) ∈ N where r(x) is the rank position of x on a lattice Lr . Lr is the lattice associated to the rank transformation r that creates a given total ordering. The complete lattice Lr corresponds to an ordered set of multivariate vectors. From the rank transformation, a rank image can be created by associating its rank to each pixel x ∈ Ω. The rank image is the lattice representation of the multivalued image according to the ordering strategy. The advantage is that this rank image can then be used for classical MM processing (it is a scalar image). Olivier Lezoray (Universit´ e de Caen) EUSIPCO 2009 August 2009 7 / 22
not explicitly construct the complete lattice: they ﬁrst deﬁne a total ordering relation that induces a complete lattice. We take an opposite approach: we explicitly build the complete lattice from a multivariate image. Unfortunately, creating the lattice directly from a multivariate image is computationally unfeasible ⇒ we reduce the amount of data by Vector Quantization. VQ maps a vector x to another vector x among m prototypes that deﬁne a codebook. A codebook D is built from a training set S of size m (m m ). A VQ algorithm has to produce a set D of prototypes x that minimizes the distortion deﬁned by 1 m m i=1 min 1≤j≤m xi − xj 2 (2) Olivier Lezoray (Universit´ e de Caen) EUSIPCO 2009 August 2009 9 / 22
2k prototypes after k iterates. Given a multivariate image f , VQ: Rp → Rp is applied to construct a codebook D : N → Rp and an encoder E : Rp → N. An index image g : Zl → N can be deduced from D and E by applying g(x) = E(f (x)) to each vector f (x) = x of the original image f . The initial image can be reconstructed with loss from the index image and the codebook by D(g(x)) = D(E(f (x))). Initial Quantized Rank QV L QV Olivier Lezoray (Universit´ e de Caen) EUSIPCO 2009 August 2009 10 / 22
necessarily an accurate ordering and the rank image can be totally arbitrary (one can consider any index permutation). One index permutation Another index permutation. Both index images will produce the same quantized image ! Given any vector ordering criterion, a codebook can be reordered: this enables to explicitly build the complete lattice. One can then modify the index image to obtain a rank image. Olivier Lezoray (Universit´ e de Caen) EUSIPCO 2009 August 2009 12 / 22
lattice, Manifold Learning is a good candidate to perform the codebook re-ordering. We consider only on two Graph-based methods for nonlinear dimensionality reduction: Diﬀusion Maps and Laplacian Eigenmaps. Let {x1, x2, · · · , xl } ∈ Rp be l sample vectors. Dimensionality reduction consists in searching for a new representation {y1 , y2 , · · · , yl } with yi ∈ Rl . Given a neighborhood graph G associated to these vectors, one considers its adjacency matrix W with Wij = K(xi , xj ) = e −||xi −xj ||2 σ2 and minimizes 1 2 ij Wij yi − yj 2 = Tr(Y T ∆Y ) (3) with ∆ = D − W for Laplacian Eigenmaps and ∆ = W D for Diﬀusion Maps. This is solved by SVD decomposition. The ﬁrst non trivial eigenvector is retained as a new scalar representation. A sorting of the values in this new representation enables the obtaining of a new ordering of the codebook and a rank image. Olivier Lezoray (Universit´ e de Caen) EUSIPCO 2009 August 2009 13 / 22
compare method that reorder a codebook. This has been studied in literature for color images. To compare codebook reordering methods, the assumption that diﬀerences of neighbouring pixels of rank images should follow a Laplacian distribution is used. This is in accordance with the JPEG-LS image coding standard, which also assumes a Laplacian model for the prediction residuals The compression ability of the rank image is a measure of performance for re-ordering schemes. We have used this principle on three databases of 256 colors quantized images: 6 computer-generated images (’synthetic’), 92 natural images (’natural1’), and 12 popular natural images (’natural2’). Olivier Lezoray (Universit´ e de Caen) EUSIPCO 2009 August 2009 16 / 22
iterations MM gradient Watershed with a square) (3 × 3 square) Figure: MM Processing example of a 20-channels multi-spectral barley grain image with a 1024 codebook. Olivier Lezoray (Universit´ e de Caen) EUSIPCO 2009 August 2009 20 / 22
Vector Quantization: data volume reduction. Manifold Learning: data dimension reduction Combination of both these techniques enables the dynamic construction of a codebook, its reordering and the obtaining of a rank image The rank image is a scalar representation of the original multivariate image Outlooks Extending the approach to the construction of the whole complete lattice and not only a sub-manifold of it. Olivier Lezoray (Universit´ e de Caen) EUSIPCO 2009 August 2009 21 / 22