The Geodesic Active Field (GAF) approach from image processing - whose mathematical background is the Riemannian theory of submanifolds, was recently extended by the authors to the Finslerian setting, for certain specific metrics of Randers type.      The present work studies the significantly more flexible Generalized Lagrange (GL) extension,      which allows a versatile adapting of the GAF process to Finslerian, pseudo-Finslerian and      Lagrangian structures.      The mathematically essential GAF mean curvature flow PDEs of three such GL structures      (Randers-Ingarden, Synge-Beil and proper Generalized Lagrange) are explicitly obtained,      discussed, implemented, and their corresponding feature evolution is compared with the      classic results produced by the established original Riemannian GAF model

Balan, Vladimir

Stojanov, Jelena

University of Niš: Facta Universitatis (E-Journals) / Универзитет у Нишу

FACTA UNIVERSITATIS (NISˇ)Ser. Math. Inform. Vol. 30, No 3 (2015), 353–359APPLICATIONS OF THE MEAN CURVATURE FLOW ASSOCIATED TOANISOTROPIC GENERALIZED LAGRANGEMETRICS IN IMAGEPROCESSINGVladimir Balan and Jelena StojanovAbstract. The Geodesic Active Field (GAF) approach from image processing - whosemathematical background is the Riemannian theory of submanifolds - was recently ex-tended by the authors to the Finslerian setting, for certain specific metrics of Randerstype. The presentwork studies the significantlymore flexibleGeneralized Lagrange (GL)extensions, which allow a versatile adapting of the GAF process to Finslerian, pseudo-Finslerian and Lagrangian structures. The GAF mean curvature flow PDEs of three suchGL structures (Randers-Ingarden, Synge-Beil, and proper Generalized Lagrange) are ex-plicitly obtained, discussed, implemented, and their corresponding feature evolutionsare compared with the classic results produced by the established original RiemannianGAF model.Keywords: The Geodesic Active Field; Riemannian manifold; Image processing; Gener-alized Lagrange metrics.1. Introduction. Related works.The Beltrami framework and its practical importance in image processing wereextensively described in [5, 11, 14, 15]. Applications to image segmentation, edgedetection, denoising and stereo vision were achieved bymeans of a flow technique,which evolves surfaces associated to digital images. The related surface evolutiondetails are comprehensively presented in [10, 12].A curve evolution technique, called geodesic active contours (GAC) is used inimage segmentation [6]. As a natural extension, the geodesic active field (GAF)framework for image registration was introduced in [16], and further improvedin [17]. The GAF technique is based on minimization of the deformation fieldbetween a pair of images. It evolves a surface corresponding to the deformationfield to achieve the minimality of a weighted Polyakov action [5], by employingthe mean curvature (MC) flow. The weighted Polyakov action minimization in theReceived September 02, 2014; Accepted March 02, 20152010 Mathematics Subject Classification. Primary 53B40; Secondary 53C60, 53C44353354 V. Balan and J. StojanovRiemannian type framework is achieved by the Beltrami PDE flow - an isotropic(directional independent) evolutive equation of a 2-dimensional geometric activeobject.On the other hand, [9] and [13] promote anisotropic extensions of geodesicactive contours. They consider a curve as 1-dimensional geometric active object,and use anisotropic weighted curve length for segmentation and for curve extrac-tion, respectively. The non-weighted anisotropic evolution of image surfaces wasrecently proposed in [1], where an approximate minimizing flow contains only themost significant term from the scaled extremal equation. The present work relieson the general anisotropic Beltrami framework, which was recently developed bythe authors in [2]; it discusses as well discretization and applicable aspects of theobtained theoretical results.2. Beltrami framework and anisotropic extensionThe central object of a Beltrami framework is a surface embedded into a Rieman-nian manifold, and its evolution in accordance with the minimization of a globalscalar property of the surface, expressed as a Poyakov action. Let us consider theembedding mapX : D→M, (dim D = n < dim M = m).(2.1)The embedding produces the submanifold Σ = X(D) of the Riemannian manifold(M, h), usually called the image surface. This is endowed with a Riemannian metric, which is not necessarily induced from h. The preliminary aim is to describethe evolution of the image surface (Σ, ) towards the extreme state of the minimalPolyakov action S(X, , h, f ) =∫L dx1 . . . dxn defined by the Lagrangian densityL = f (X, , h) · μν ∂Xi∂xμ∂Xj∂xνhij√,(2.2)where we denote by f the weight function and by  the determinant of the metrictensor (μν).We shall further briefly denote, for a given function Φ:Φ;α :=∂Φ∂xαΦ,i :=∂Φ∂XiΦ,( iα) :=∂Φ∂Xiα.The Euler-Lagrange equation∂tXr = −121√γhir(L,i − L,( iα);α)(2.3)yields the descent flow, called the Beltrami flow∂Xrt = fτr(X) − n2f,i hir + f,i σμXiσXrμ,(2.4)Anisotropic GL Flow in Image Processing 355where τr(X) is the tension field associated to the embedding,τr(X) = ασXrασ − ρθΓσρθXrσ + σμΓrklXkσXlμ.(2.5)The main benefits of the Beltrami framework and of the resulting flow, are thearbitrary and independent choices of  and h, the possibility of considering thevector value feature, and the geometrical quality of the flow (its reparametrizationinvariance).The various choices of metric  and the weight function f yield diﬀerent flows:mean curvature flow, ∂tXr = Hr:  -induced, f ≡ 1;tension flow, ∂tXr = τr:  = (x) -arbitrary, f ≡ 1;Beltrami flow, ∂tXr = ∂tXr(x):  = (x) -arbitrary;anisotropic flow, ∂tXr = ∂tXr(x, v):  = (x, v) -directionally dependent.In the last case - the anisotropic Beltrami framework - the image metric is asmooth tensor field on the tangent bundle TΣ. In order to distinguish it from theused Riemannian type metrics, we denote this with Greek letters, γ=γμνdxμ ⊗ dxν.If the metric γ is symmetric, regular and of constant signature, then (Σ, γ) is ageneralized Lagrange space [4], and the anisotropic Beltrami flow ensuring theminimal Polyakov action is briefly called theGL flow. Its explicit form is developedin [2].Theorem 2.1. (The anisotropic Beltrami flow) Let γ be a generalized Lagrange metricon the surface embedded via (2.1) into the Riemannian manifold (M, h). The PDE of theanisotropic Beltrami flow which provides the minimality of the non-weighted Polyakovaction on the surface (Σ, γ), is∂tXr= τr(X)+12hir{σμ;α[(γσμ),( iα) + γσμ(ln√γ),( iα)]+σμ[(γσμ),( iα);α+ (γσμ),( iα)(ln√γ);α+(γσμ);α(ln√γ),( iα)+ γσμ 1√γ (√γ),( iα);α − (γσμ),i − γσμ(ln√γ),i] },where μν = hij∂μXi∂νXj is the induced metric on the embedded surface.Without losing the generality, one can consider for the anisotropic metric γ, thedeformed induced oneγμν(x, v) = μν(x) + ϕμν(x, v), x ∈ Σ, v ∈ TxΣ.If the embedded surface is endowedwith a Finsler structure F : TΣ→ R [3, 8], thenthe anisotropic metric is defined by the halved Hessianγμν(x, v) =12∂2F2∂vμ∂vν∣∣∣∣(x,v).356 V. Balan and J. StojanovA particular linearly deformed induced norm is the one of Randers type [3, 8, 7],F(x, v) =√μν(x)vμvν + bμ(x)vμ;in this case, the corresponding Beltrami flow is ([2]),∂Rt (Xr) = τr(X) + φ(μν(x), bμ(x), ||v||).(2.6)As well, for Σ endowed with a generalized Lagrange structure of Synge-Beil type,γμν(x, v) = μν(x) + c(X)vμvν,having the covariant directional vector (vμ) = (μνvν), Theorem 2.1 produces theSynge-Beil type Beltrami flow,∂SBt (Xr) = τr(X) + ψ(μν(x), c(X), ||v||).(2.7)3. The Beltrami framework in image processingThe Beltrami framework commonly used in image processingmainly considersa Monge surface in the 3-dimensional Euclidean space X : D→ M, where (M, h) =(R3, hij = diag(1, 1, β2)),X : (x1, x2) → (x1, x2, I(x1, x2)).The components of the induced metric on the surface are(μν) =⎛⎜⎜⎜⎜⎝1 + β2I2x1β2Ix1Ix2β2Ix1Ix2 1 + β2I2x2⎞⎟⎟⎟⎟⎠ ,where, for brevity, we use the notation Ixμ = ∂I∂xμ .The gradient vector field radI = (1αIxα , 2αIxα ) is an ingredient for constructingthe following Finslerian norm of Randers-Ingarden type ([3]):F(x, v) =√μνvμvν + Ixμvμ,(3.1)where bμ = Ixμ is the term which characterizes the Ingarden structure. Further, theFinslerian-Randers type metric components and the PDE of the evolution flow canbe calculated, and only the third component of the flow, ∂tX3 = ∂tI is nontrivial.This allows to derive the flow ∂Rt I for the Monge surface evolution (2.6).Theorem 2.1 and the Synge-Beil structure given on the Monge surface,γμν(x, v) = μν(x) + vμvν,(3.2)yield the corresponding flow ∂SBt I (2.7).We note that this extends the isotropic Beltrami flow used in [16], which coin-cides with the mean curvature flow, ∂tI = H3.Anisotropic GL Flow in Image Processing 3574. Application and tentative implementationA grayscale image is viewed as a matrix Σ =(I(i, j)), whose elements expressthe greyness of pixels:x = (x1, x2) = (i, j) − matrix position, I(i, j) ∈ {0, 1, . . . , 255},where the tangent vectors point to the neighboring pixels characterized by thedisplacementsv(i, j) = (v1, v2) ∈ {(−1,−1), (−1, 0), (−1, 1), (0,−1), (0, 1), (1,−1), (1, 0), (1, 1)}.The evolution is achieved by the successive shifting of the grayscale imageI(i, j)→ I(i, j) + I(i, j),(4.1)whereI(i, j) discretizes one of the following flow types: Randers-Ingarden, Synge-Beil or mean curvature.We use Matlab programming for implementing the three Beltrami flows. Thecorresponding Beltrami evolutions (4.1) are achieved by successive shifting, whereeach iteration assumes the following steps:- accessing pixel (except of the boundary) to get the corresponding feature value;- determining the shift tangent vector - optionally, in the cases of anisotropic flows;- applying the flow expression on the feature value and optionally on the shifttangent vector to obtain the shift value I;- computing the modified feature value, I→ I + I.The main diﬀerence between the implementation of isotropic and anisotropic evo-lution types is in the dependence - in the latter case - of the metric on the eightdiscretized neighboring Gateaux derivatives at each pixel position. Namely, theanisotropic flows exhibit the extra local dependence of the norm and hence metrictensor on the eight neighbors as well, compared to dependence on position only,present in the isotropic flow. In our research point, the gradient is taken for theanisotropic shift.Due to technical limitations, low resolution imageswere considered for process-ing. The shifting was achieved for β = 2, by the use of the Randers-Ingarden flow∂Rt I given by (2.6)-(3.1), the Synge-Beil flow ∂SBt I from (2.7)-(3.2), and the classicalmean curvature flow∂MCt I =12(Ix1x122 + Ix2x211 − 2Ix1x212) .The applicative aspects of the research currently emphasize the constructionof the shift matrices I = (I(i, j)) and the low resolution is a limiting factor invisualizing immediate enhancement of the starting image as eﬀect of the flowaction.The Randers-Ingarden evolution sample obtained after 15 iterations can be seenin Figure (4.1). The eﬀects of the preliminarily implementations can be tracked bymeans of the shift matrices corresponding to the three flows.358 V. Balan and J. StojanovFig. 4.1: Original flower image (60×60), the 16-th RI iteration output, and the latterflow-shift.5. Conclusions and further developmentsThe obtained preliminary results show that for all the three considered flows,the output diﬀers from the input by a slight increase of contrast between com-pact regions of the image. The anisotropic evolutions need extra computationalresources than the mean curvature one, due to the complexity of the anisotropicflow expressions compared to the MC flow. However, though the directional de-pendence slows down the speed of the enhancement process, it has the merit ofconsiderably enabling its sensitivity. Due to their extended local character, theanisotropic flows provide more information than the Riemannian ones, and aim tosimplify the further intended processing of the output features.Further concerns of the subject will address the optimal adapted selection ofthe direction-dependent anisotropic structure, and will consider a non-constantweight function within the Polyakov energy, which tunes the evolution shift, andaccelerates the image enhancement.REFERENCES1. V. Balan, J. Stojanov, Finslerian extensions of geodesic active fields for digital imageregistration, Proceedings in Applied Mathematics and Mechanics / PAMM, 13, 1,2013, pp. 493–494.2. V. Balan, J. Stojanov, Finslerian-type GAF extensions of the Riemannian frameworkindigital image processing , to appear.3. D. Bao, S.-S. Chern and Z. Shen,An Introduction to Riemann-Finsler Geometry, Grad-uate Texts in Mathematics, Volume 200, Springer-Verlag, 2000.4. I. Bucataru, R. Miron, Finsler-Lagrange geometry. Applications to Dynamical Systems,Ed. Acad. Romane, Bucharest, 2007.5. X. Bresson, P. Vanndergheynst J.P. Thiran, Multiscale Active Contours, Int. Jour. ofComp. Vision 70(3) (2006), 197–211.6. V. Caselles, R. Kimmel, G. Sapiro,Geodesic active contours, Int. Jour. of Comp.Vision22(1) (1997), 61–79.Anisotropic GL Flow in Image Processing 3597. X. Cheng, Z. Shen, Finsler Geometry: An approach via Randers Spaces, Science Press& Springer, 2012.8. S.-S. Chern and Z. Shen, Riemann-Finsler Geometry, World Sci. Publishers, 2005.9. G. Gallego, J.I. Ronda and A. Valdes, Directional geodesic active contours, 19th IEEEInternational Conference on Image Processing (ICIP), September 30 - October 3,Orlando, FL, USA, 2012, pp. 2561–2564.10. Y. Giga, Surface evolution equations: a level set method, HokkaidoUniversity TechnicalReport Series in Mathematics No. 71, 2002.11. R. Kimmel and R. Malladi and N. Sochen, Images as embedded maps and minimalsurfaces: movies, color, texture, and volumetric medical images, Int. Journal of Comp.Vision 39(2) (2000), 111–129.12. R. Kimmel, Numerical Geometry of Images, Springer-Verlag, New York 2004.13. J. Melonakos, E. Pichon, S. Angenent and A. Tannenbaum, Finsler active contours,IEEE Trans. on Pattern Analysis and Machine Intelligence 30(3) (2008), 412–423.14. N. SochenandR.DericheandL.-L. Perez, TheBeltrami flow overmanifolds, TechnicalReport No 4897, Sophia Antipolis Cedex, INRIA, France 2003.15. N. Sochen and R. Kimmel and R. Malladi, A general framework for low level vision,IEEE Trans. on Image Processing 73 (1998), 310–318.16. D. Zosso, X. Bresson and J.-P. Thiran, Geodesic active fields - a geometric frameworkfor image registration, IEEE Trans. on Image Processing 20(5) (2011), 1300–1312.17. D. Zosso, X. Bresson, J.-P. Thiran, Fast Geodesic Active Fields for Image RegistrationBased on Splitting and Augmented Lagrangian Approaches, Image Processing, IEEETransactions on , 23(2) (2014), 673–683.Vladimir BalanFaculty of Applied SciencesDepartment Mathematics-InformaticsUniv. Politehnica of BucharestRomaniavladimir.balan@upb.roJelena StojanovTechnical Faculty ”Mihajlo Pupin”University of Novi Sad23000 Zrenjanin, Serbiastojanov.jelena@gmail.com

APPLICATIONS OF THE MEAN CURVATURE FLOW ASSOCIATED TO ANISOTROPIC GENERALIZED LAGRANGE METRICS IN IMAGE PROCESSING

http://casopisi.junis.ni.ac.rs/index.php/FUMathInf/article/download/431/pdf_18

APPLICATIONS OF THE MEAN CURVATURE FLOW ASSOCIATED TO ANISOTROPIC GENERALIZED LAGRANGE METRICS IN IMAGE PROCESSING

Abstract

Similar works

Full text

Available Versions

University of Niš: Facta Universitatis (E-Journals) / Универзитет у Нишу