EUSIPCO 2009

Transcript

1. Rank transformation and Manifold Learning for Multivariate Mathematical Morphology Olivier 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
2. 1 Introduction 2 Rank Transformation 3 Vector Quantization 4 Codebook Reordering by Manifold Learning 5 Results Olivier Lezoray (Universit´ e de Caen) EUSIPCO 2009 August 2009 2 / 22
3. 1 Introduction 2 Rank Transformation 3 Vector Quantization 4 Codebook Reordering by Manifold Learning 5 Results Olivier Lezoray (Universit´ e de Caen) EUSIPCO 2009 August 2009 3 / 22
4. Introduction Mathematical Morphology (MM) relies on the use of a 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
5. 1 Introduction 2 Rank Transformation 3 Vector Quantization 4 Codebook Reordering by Manifold Learning 5 Results Olivier Lezoray (Universit´ e de Caen) EUSIPCO 2009 August 2009 5 / 22
6. Ordering of vector data A multivariate image can be represented 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
7. Rank transform Definition A rank transformation r : Rp → 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
8. 1 Introduction 2 Rank Transformation 3 Vector Quantization 4 Codebook Reordering by Manifold Learning 5 Results Olivier Lezoray (Universit´ e de Caen) EUSIPCO 2009 August 2009 8 / 22
9. Vector Quantization (VQ): Principle Usual approaches to mathematical morphology do 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
10. Vector Quantization: Algorithm LBG is a VQ algorithm that produces 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
11. 1 Introduction 2 Rank Transformation 3 Vector Quantization 4 Codebook Reordering by Manifold Learning 5 Results Olivier Lezoray (Universit´ e de Caen) EUSIPCO 2009 August 2009 11 / 22
12. VQ codebook reordering A codebook produced by VQ is not 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
13. Manifold Learning The codebook is a sub-manifold of the complete 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
14. Initial Quantized Rank QV L QV Rank L Lexicographic Rank Lexicographic L Lexicographic Lexicographic IHSL IHSL Rank α-trimmed L α-trimmed Rank Bit-Mixing L Bit-Mixing Lexicographic Lexicographic Rank Laplacian L Laplacian Rank Diffusion L Diffusion Eigenmaps Eigenmaps Maps Maps Olivier Lezoray (Universit´ e de Caen) EUSIPCO 2009 August 2009 14 / 22
15. 1 Introduction 2 Rank Transformation 3 Vector Quantization 4 Codebook Reordering by Manifold Learning 5 Results Olivier Lezoray (Universit´ e de Caen) EUSIPCO 2009 August 2009 15 / 22
16. Rank Transform comparison Principle Comparing rank transforms is equivalent to 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
17. Results synthetic bpp 0 2 4 6 8 10 DM LE LX ATLX VQ BM GMST MJ PW LXHSL natural1 bpp 0 2 4 6 8 natural2 bpp 0 2 4 6 8 Average bpp 0 2 4 6 8 Figure: Loss-less compression results, in bits per pixel, applied to the rank images after applying the following re-ordering methods: Laplacian Eigenmaps (LM), Diffusion Maps (DM), Lexicographic (LX), Lexicographic IHSL (LXHSL), BitMixing (BM), Pairwise (PW), Graph Minimum Spanning Tree (GMST), α-trimmed Lexicographic (ATLX), Majority (MJ), VQ (Vector Quantization). Olivier Lezoray (Universit´ e de Caen) EUSIPCO 2009 August 2009 17 / 22
18. Results Original Rank by Diffusion Maps Codebook (512 colors) Dilation (11 × 11) Erosion (11 × 11) Closing (11 × 11) Olivier Lezoray (Universit´ e de Caen) EUSIPCO 2009 August 2009 18 / 22
19. Opening (11 × 11) Opening by reconstruction Alternate Sequential Filter (11 × 11) 3 iterations with a square Gradient (3 × 3) Watershed Olivier Lezoray (Universit´ e de Caen) EUSIPCO 2009 August 2009 19 / 22
20. R´ esultats Channel 1 Channel 10 Rank Image ASF (7 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
21. Conclusion Contribution A general approach for building a complete lattice 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
22. Thanks for your attention. Publications available at: http://www.info.unicaen.fr/∼lezoray Olivier Lezoray (Universit´ e de Caen) EUSIPCO 2009 August 2009 22 / 22