Definition 1. Measuring distances on the image space. Infinitesimal transformations. The transformation from j to jh is therefore divided in two complementary processes. Both transformations are the main ingredients of any morphing process between two images. Differentiable structure. Remark 1. Infinitesimal transformation spaces. Geometric transformation. Photometric transformation. Differentiable curves and tangent space. Definition 2 C 1 curves in JW. Let I be an interval in R.
Definition 3. Definition 4. Riemannian structure. Definition 5. Proposition 1. Moreover, p is linear from Tj JW to W. This issue is addressed in Proposition 2, the proof of which is provided in Appendix A.
Proposition 2. Groups of diffeomorphisms. We quote these results in the following theorem. The fact that GB is a group is proved in [30]. Further results on these groups and on AT can be found in Appendix C.
This is also related to developments in optimal design [28]. Geodesics on JW. Minimizing geodesics. The space of C 1 curves is not well suited to deal with proofs of the existence of curves of minimal length between two images j0 and j1 , i. We introduce below the more tractable space of curves with square integrable speed.
The proof of this proposition is postponed to Appendix A. Proposition 3. We can now introduce the space H 1 [0, 1], JW of regular curves. Definition 6. Proposition 4. The complete proof follows by usual density arguments. We carry on with an important result which characterizes regular paths in JW. Theorem 2. The proof is postponed to Appendix B. Theorem 3. Let j0 and j1 be in JW. Definition 7. Characterization of geodesics.
Photometric optimality. Theorem 4. Definition 8. Study of the geodesic equation. Directional derivatives in L2. In this section, we try to clarify the last equa- tion of system 15 , at least in some situations of interest.
Definition 9. Remark 3. The existence of Dj. Theorem 5. Lemma 7. Lemma 8 Gronwall. Lemma 9. Therefore, the proof boils down to show that the interversion of derivatives underlying the formal argument above can be made rigorous. Lemma We now prove the pointwise convergence of the pth derivative. The last upper bound now tends to 0, by dominated convergence. Appendix D. Theorem Proof of Lemma 2.
Appendix F. Proof of Lemma 4. Proof of Lemma 5. We now can collect the estimates we have obtained to conclude the proof of Lemma 5. Amit, U. Grenander, and M. Piccioni, Structural image restoration through deformable templates, J. Amit and P. Arnold and B. Fourier Grenoble , 1 , pp. Bajcsy and C. Bajcsy and S. Kovacic, Multiresolution elastic matching, Comp.
Vision, Graphics, and Image Proc. Pattern Anal. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Christensen, R. Rabbitt, and M. Image Process. Cootes, C. Taylor, D. Cooper, and J. Graham, Active shape models: Their training and application, Comp.
David and S. Zucker, Potentials, valleys, and dynamic global coverings, Int. Vision, 5 , pp. Dupuis, U. Foias, D. Holm, and E. D, , pp. Geman and S. Grenander, Lectures in Pattern Theory, Appl. Grenander and D. Keenan, Towards automated image understanding, J. Grenander and M. Miller, Representations of knowledge in complex systems with discussion section , J. B, 56 , pp. Marsden and T.
Marsden, T. Ratiu, and A. And the average of deburring time for each workpiece after adjusting the robot deburring speed is shortened by about 3. Article :. DOI: Abstract In this paper, we discuss a geometrical model of a space of deformable images or shapes, in which infinitesimal variations are combinations of elastic deformations warping and of photometric variations.
Keyphrases local geometry deformable template velocity-based image linear approximation infinite dimensional shape manifold elastic deformation photometric variation deformable image large deformation general construction geodesic equation geometrical model robust estimation infinitesimal variation.
0コメント