Kluwer - Handbook of Biomedical Image Analysis Vol
.1.pdfShape From Shading Models |
295 |
(4)yx (k, l) = φ(yx)(x, y)φk(,yxl )(x, y)dx dy = (2)(k) (2)(l),
x(3)(k, l) = φ(xx)(x, y)φk(x,l)(x, y)dx dy = D(l) (3)(k),
(3)y (k, l) = φ(y)(x, y)φk(,yyl )(x, y)dx dy = D(k) (3)(l),
x(2)(k, l) = φ(x)(x, y)φk(x,l)(x, y)dx dy = D(l) (2)(k),
(2)y (k, l) = φ(y)(x, y)φk(,yl)(x, y)dx dy = D(k) (2)(l),
x(1)(k, l) = φ(x)(x, y)φk,l (x, y)dx dy = D(l) (1)(k),
(1)y (k, l) = φ(y)(x, y)φk,l (x, y)dx dy = D(k) (1)(l),
where
(1)(k) = |
|
φ(x)(x)φ(x − k)dx, |
(2)(k) = |
φ(x)(x)φ(x)(x − k)dx, |
|
(3)(k) = |
|
φ(xx)(x)φ(x)(x − k)dx, |
(4)(k) = |
|
φ(xx)(x)φ(xx)(x − k)dx |
are 1D connection coefficients and D(0) = 1, D(n) = 0, n = 1. Note that since the 2D basis here is constructed from the tensor product of 1D basis, these 2D connection coefficients can be computed by using 1D coefficients. We also notice that these connection coefficients are independent of the input images; therefore, they only need to be computed once.
The energy function is then linearized by taking the linear term in its Taylor expansion at ( p, q). The next step is to solve the optimization problem associated with the linearized energy function by iterations. Let δpi, j , δqi, j , and δzi, j be the small variation of pi, j , qi, j , and zi, j , respectively, and set
|
∂δW |
|
|
∂δW |
∂δW |
|
||||||||||
|
|
= |
|
|
|
|
= |
|
|
|
= 0. |
|
||||
|
∂δpi, j |
|
∂δqi, j |
∂δzi, j |
|
|||||||||||
We obtain |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
∂ R |
|
|
∂ R |
|
||||||
δpi, j = [C1 D22 − C2 |
|
|
(i, j) |
|
|
(i, j)]/D, |
(5.73) |
|||||||||
∂ p |
∂q |
|||||||||||||||
|
|
|
|
|
|
∂ R |
|
|
∂ R |
|
||||||
δqi, j = [C2 D11 − C1 |
|
(i, j) |
|
(i, j)]/D, |
|
|||||||||||
∂ p |
∂q |
|
||||||||||||||
δzi, j = C3/D33,
296 |
|
|
|
|
|
Shen and Yang |
where |
|
|
|
|
|
|
D11 = R2pi, j |
+ 3 (2)(0) + 1, |
(5.74) |
||||
D22 |
2 |
+ 3 |
(2) |
(0) |
+ 1, |
|
= Rqi, j |
|
|
||||
D33 |
= 2 (2)(0) + 2 (4)(0), |
|
||||
|
|
|
|
2 |
2 |
|
D = D11 D22 − Rpi, j Rqi, j
and
C1 = (E − R)Rp − pi, j
2N−2
+Zi−k, j ( (3)(k) + (1)(k)) − (2 pi−k, j + pi, j−k) (2)(k),
k=−2N+2
C2 = (E − R)Rq − qi, j
2N−2
+Zi, j−k( (3)(k) + (1)(k)) − (qi−k, j + 2qi, j−k) (2)(k),
|
k=−2N+2 |
|
|
2N−2 |
|
C3 = − |
|
|
( pi−k, j + qi, j−k)( (3)(k) + (1)(k)) |
|
|
|
k=−2N+2 |
|
+ (Zi−k, j + Zi, j−k)( (2)(k) + (4)(k)). |
(5.75) |
|
Finally, we can write the iterative formula |
|
|
|
pim, j+1 = pim, j + δpi, j , |
(5.76) |
|
qim, j+1 = qim, j + δqi, j , |
|
|
zim, +j 1 = zim, j + δzi, j . |
|
We now summarize this method as the follows:
Step 0. Compute 1D connection coefficients and 2D connection coefficients.
Step 1. Compute the set of coefficients (5.75) and (5.74).
Step 2. Compute the set of variations δpi, j , δqi, j , and δzi, j (5.73).
Step 3. Update the current ( pim, j , qim, j ) and then the current shape reconstruction Zim, j using Eq. (5.76).
5.4.3 Summary
The wavelet-based method we demonstrated in this section is based on the approximation of the objective function in V0. It should be pointed out that it
Shape From Shading Models |
297 |
did not use the multiscale structure possessed by the wavelet bases, nor the Mallat algorithm to speed up the computation. Since the selected wavelet bases are time-limited (therefore it is not band-limited), it may be not the best choice for approximating differential operators.
At this point, we would like to mention the idea of regularization. The shape from shading problems can be regarded as inverse problems since they attempt to recover physical properties of a 3D surface from a 2D image associated with the surface. Therefore, the Tikhonov regularization approach can be applied to this problem. The time-limited filters, such as the difference boxes [22] or the Daubechies wavelets used in Section 5.4.2, do not satisfy one of the conditions requested by the Tikhonov regularization [61]. In contrast with time-limited filters, band-limited filters are commonly used for regularizing differential operators, since the simplest way to avoid harmful noise is to filter out high frequencies that are amplified by differentiation. Meyer wavelet family constitutes an interesting class of such type of band-limited filters. The ill-posedness/ill-conditioness of the SFS model and its connection to the regularization theory have been discussed in [7]. Minimization (5.21) will lead to a smoother solution (the regularization solution). In some cases, the Lagrange multipliers are the “regularizers.” However, the numerical experiments presented in Section 5.3 are treated by choosing those regularizers equal to 1. The nonlinear ill-posed problems are quite difficult and basically no general approaches seem to exist [7]. For the classic theory of regularization, we highly recommend Tikhonov et al. [60].
A 2D basis constructed from the tensor product of 1D wavelet basis is much easier to compute than the nonseparable wavelets. There is also some ongoing research on nonseparable wavelets for use in image processing. For a detailed discussion on nonseparable wavelets, we recommend [37,38,40] and references therein.
The development of a wavelet-based method which reflects the multiscale nature with an effective algorithm, namely, using Mallat algorithm, is still an open problem.
5.5 Concluding Remarks
In this chapter, we have given a super brief introduction of the shape from shading problems. A variety of elementary numerical techniques related to solution
298 |
Shen and Yang |
of this problem is discussed and implemented to show the basic ideas. However, a short chapter like this one has to omit many related topics, which are both important and exciting. In fact, there are many other techniques and advanced developments in the area. Fortunately, most of them are very well documented in the literature. For instance, the following two approaches reflect different flavors:
1.Statistical learning and neural network. [2] introduced a statistical method to solve the SFS model; the principal component analysis (PCA) was used to derive a low-dimensional parameterization of head shape space, and an algorithm was presented for solving shape from shading based on this approach.
2.Fast matching method. The schemes are of use in a variety of applications, including problems of shape from shading. An excellent review about this method is given by its pioneer [54]. Applications related to vision problems can be found in [55] and [53].
We conclude this chapter by pointing out that there is, in general, no proof of the convergence for the numerical methods introduced in Sections 5.3 and 5.4. An interesting example related to this topic can be found in [30].
5.6 Acknowledgements
The authors would like to thank Dr. Gilbert G. Walter for his encouragement and his valuable suggestions which led to significant improvement of this paper. The first author was partially supported by Professor Naoki Saito’s grant ONR YIP N00014-00-1-0469 while completing this paper. She also wishes to thank Dr. Jianbo Gao for introducing her the reference [31].
Shape From Shading Models |
299 |
Bibliography
[1]Ascher, U. M. and Carter, P. M., A multigrid method for shape from shading, SIAM J. Numer. Anal. Vol. 30, No. 1, pp. 102–115, 1993.
[2]Atick, J. J., Griffin, P. A., and Redlich, A. N., Statistical approach to shape from shading: Reconstruction of 3D face surfaces from single 2D images, Neural Comput., Vol. 8, pp. 1321–1340, 1996.
[3]Bakhvalov, N. S., On the convergence of a relaxation method with natural constraints on the elliptic operator, USSR Comput. Math. Phys., Vol. 6, pp. 101–135, 1966.
[4]Barnes, I. and Zhang, K., Instability of the Eikonal equation and shape from shading, M2AN Math. Model. Numer. Anal. Vol. 34, No. 1, pp. 127– 138, 2000.
[5]Beylkin, G., On the representation of operators in bases of compactly supported wavelets, SIAM J. Numer. Anal., Vol. 29, pp. 1716–1740, 1992.
[6]Bichsel, M. and Pentland, A. P., A simple algorithm for shape from shading, IEEE Proc. Comput. Vis. Pattern Recognit., pp. 459–465, 1992.
[7]Bertero, M., Poggio, T. A., and Torre, V., Ill-posed problems in early vision, Proc. IEEE, Vol. 76, No. 8, pp. 869–889, 1988.
[8]Briggs, W. L., Henson, V. E., and McCormick, S. F., A Multigrid Tutorial, 2nd edn., Society for Industrial and Applied Mathematics, 193 pp. c 2000.
[9]Choe, Y. and Kashyap, R. L., 3-D shape from a shading and textural surface image, IEEE Trans. Pattern Anal. Mach. Intell. Vol. 13, No. 9, pp. 907–999, 1999.
[10]Chabrowski, J. and Zhang, K., On shape from shading problem, In: Functional Analysis, Approximation Theory and Numerical Analysis, World Scientific Publishing, River Edge, NJ, pp. 93–105, 1994.
[11]Courant, R. and Hilbert, D., Methods of Mathematical Physics, 1st edn., Vol. 1, Wiley-Interscience, New York, 560 pp., 1989.
300 |
Shen and Yang |
[12]Daubechies, I, Ten Lectures on Wavelets, CBMS-NSF Series in Appl. Math., SIAM Publications, Philadelphia, PA, 1992.
[13]Debnath, L., Integral Transforms and Their Applications, CRC Press, Boca Raton, FL, Vol. xi, 457 pp., c1995.
[14]Deng, Y. and Li, J., Existence and uniqueness in shape from shading, J. Comput. Sci. Technol. Vol. 12, No. 1, pp. 58–64, 1997.
[15]Duffy, D. G., Transform Methods for Solving Partial Differential Equations, CRC Press, Boca Raton, FL, 1994.
[16]Dupuis, P. and Oliensis, J., An optimal control formulation and related numerical methods for a problem in shape reconstruction, Ann. Appl. Probab., Vol. 4, No. 2, pp. 287–346, 1994.
[17]Dupuis, P. and Oliensis, J., Direct method for reconstructing shape from shading, IEEE Computer Society Conference proceedings on CVPR ’92, Comput. Vis. Pattern Recognit., pp. 453-458, 1992.
[18]Durou, J. D. and Piau, D., Ambiguous shape from shading with critical points, J. Math. Imaging Vis., Vol. 12, No. 2, pp. 99–108, 2000.
[19]Evans, G., Blackledge, J. M., and Yardley, P. D., Analytic Methods for Partial Differential Equations, Spinger-Verlag, New York, 1999.
[20]Evans, L., Partial Differential Equations, Series of graduate studies in mathematics, Vol. 19, 662 pp., American Mathematical Society, Providence, RI, 1998.
[21]Ewing, G. M., Calculus of Variations with Applications, 352 pp., Dover Publications, New York, 1985.
[22]Herskovitz, A. and Binford, T. O., On boundary detection, Artificial Intelligence Laboratory Memo 183, Massachussetts Institute of Technology, Cambridge, MA, 1980.
[23]Hackbusch, W., Multi-Grid Methods and Applications, Springer, Berlin, 1985.
[24]Gautschi, W., Numerical Analysis: An Introduction, 506 pp., Birkhauser, Boston, 1997.
Shape From Shading Models |
301 |
[25]Harris, J. W. and Stocker, H., Handbook of Mathematics and Computational Science, Springer-Verlag, Berlin, 1998.
[26]Horn, B. K. P., Shape from shading: A Method of Obtaining the Shape of a Smooth Opaque Object from One View, Ph.D Thesis, Massachussetts Inst. of Technology, 1970.
[27]Horn, B. K. P. and Brooks, M. J., The variational approach to shape from shading, MIT A. T. Memo 813, 1985.
[28]Horn, B. K. P., Robot Vision, MIT Eng. Comput Sci. Ser., McGraw-Hill, New York, 1986.
[29]Horn, B. K. P. and Brooks, M. J. (eds.), Shape from Shading, MIT Press, Cambridge, MA, 1989.
[30]Horn, B. K. P., Szeliski, R. S., and Yuille, A. L., Impossible shaded images, IEEE Trans. Pattern Anal. Mach. Intell., Vol. 15, No. 2, pp. 166–170, 1993.
[31]Hsieh, J., Liao, H., Ko, M., and Fan, K., Wavelet-based shape from shading, Graph. Models Image Process., Vol. 57, No. 4, pp. 343–362, 1995.
[32]Hurt, N., Mathematical methods in shape-from-shading: A review of recent results, Acta Appl. Math. Vol. 23, pp. 163–188, 1991.
[33]Ikeuchi, K. and Horn, B. K. P., Numerical shape from shading and occluding boundaries, Artif. Intell., Vol. 17, pp. 141–184, 1981.
[34]Kozera, R., Uniqueness in shape from shading revisited, J. Math. Imaging Vision Vol. 7, No. 2, pp. 123–138, 1997.
[35]Lee, K. M. and Kuo, C. J., shape from shading with a linear triangular element surface model, IEEE Trans. Patten Anal. Mach. Intell., Vol. 15, No. 8, pp. 815–822, 1993.
[36]Lee, C. H. and Rosenfeld, A., Improved methods of estimating shape from shading using the light source coordinate system, Artif. Intell., Vol. 26, pp. 125–143, 1985.
[37]Lin, E.-B. and Ling, Y., Image compression and denoising via nonseparable wavelet approximation, J. Comput. Appl. Math., Vol. 155, No. 1, pp. 131–152, 2003.
302 |
Shen and Yang |
[38]Lin, E.-B. and Ling, Y., 2-D nonseparable multiscaling function interpolation and approximation with an arbitrary dilation matrix, Commun. Nonlinear Sci. Numer. Simul., Vol. 5, No. 3, pp. 125–133, 2000.
[39]Mallat, S., A Wavelet Tour of Signal Processing, Acdemic Press, New York, 1998.
[40]Mendivil, F. and Piche,´ D., Two algorithms for non-separable wavelet transforms and applications to image compression, In: Fractals: Theory and applications in Engineering, Springer, London, pp. 325–345, 1999.
[41]Oliensis, J., Shape from shading as a partially well-constrained problem, Comput. Vis., Graph., Image Process. Vol. 54, pp. 163–183, 1991.
[42]Oliensis, J., Uniqueness in shape from shading, Int. J. Comput. Vis. Vol. 6,
pp.75–104, 1991.
[43]Neumaier, A., Solving ill-conditioned and singular linear systems: A tutorial on regularization, SIAM Rev., Vol. 40, No. 3 (Sep., 1998), pp. 636–666.
[44]Ortega, J. M. and Rheinboldt, W. C., Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970. Reprinted as Classics in Applied Mathematics, Vol. 30, SIAM, Publications, Philadelphia, PA, 2000.
[45]Peleg, S. and Ron, G., Nonlinear multiresolution: A shape-from-shading example, IEEE Trans. Pattern Anal. Mach. Intell. Vol. 11, No. 2, pp. 198–206, 1989.
[46]Pentland, A. P., Local analysis of the image, IEEE Trans. Pattern Anal. Mach. Recognit., Vol. 6, pp. 170–187, 1984.
[47]Pentland, A. P., Linear shape-from-shading, Int. J. Comput. Vis., Vol. 4,
pp.153–162, 1999.
[48]Pong, T. C., Haralick, R. M., and Shapiro, L. G., Shape from shading using the facet model, Pattern Recognit. Vol. 22, No. 6, pp. 683–695, 1989.
[49]Poggio, T., Torre, V., and Koch, C., Computational vision and regularization theory, Nature, Vol. 317, pp. 314–319, 1985.
Shape From Shading Models |
303 |
[50]Rouy, E. and Tourin, A., A viscosity solutions approach to shape from shading, SIAM J. Numer. Anal., Vol. 29, No. 3, pp. 867–884, 1992.
[51]Saxberg, B. V. H., A modern differential geometric approach to shape from shading, MIT AI Lab, Tech. Rep. 1117, 1989.
[52]Saxberg, B. V. H., Existence and Uniqueness for shape from shading around critical points, Theory Algorithm, Vol. IJRR (11), pp. 202–224, 1992.
[53]Sethian, J. A., Evolution, implementation, and application of level set and fast marching methods for advancing fronts, J. Comput. Phys., Vol. 169, No. 2, pp. 503–555, 2001.
[54]Sethian, J. A., Fast marching methods, SIAM Rev. Vol. 41, No. 2, pp. 199–235 (electronic), 1999.
[55]Sethian, J. A., Level Set Methods and Fast Marching Methods, 2nd edn., Cambridge Monographs on Applied and Computational Mathematics, 3, Cambridge University Press, Cambridge, xx+378 pp., 1999.
[56]Stuben, K. and Trottenber, U., Multigrid methods: Fundamental algorithms, model problem analysis and applications, In: Multigrid Methods: Proceedings of the Conference, Kolm-Porz, Springer-Verlag, Berlin, pp. 1–176, 1982.
[57]Sweldens, W. and Roose, D., Shape from Shading Using Parallel Multigrid Relaxation, Multigrid Methods, III (Bonn, 1990), Internat. Ser. Numer. Math., Vol. 98, Birkhauser,¨ Basel, pp. 353–364, 1991.
[58]Terzopoulos, D., Image analysis using multigrid relaxation methods, IEEE Trans. Pattern Anal. Mach. Intell., Vol. 8, pp. 129–139, 1986.
[59]Tikhomirov, V. M., Stories about Maxima and Minima. Translated from the Russian by Abe Shenitzer, American Mathematical Society, Providence, RI, 1990.
[60]Tikhonov, A. N. and Arsenin, V. A., Solutions of Ill-Posed Problems, Winston & Sons, Washington, D.C., 1977.
304 |
Shen and Yang |
[61]Tikohonov, A. N. and Samariskii, A. A., Equations of Mathematical Physics, Macmillan, London, 765 pp., 1963.
[62]Trottenberg, U., Schuller, A., and Oosterlee, C., Multigrid, 1st edn., Academic Press, New York, 2000.
[63]Tsai, P. S. and Shah, M., Shape form shading using linear approximation, Image Vis. Comput., Vol. 12, No. 8, pp. 487–498, 1994.
[64]Ulich, G., Provably convergent methods for the linear and nonlinear shape from shading problem, J. Math. Imaging Vis., Vol. 9, No. 1, pp. 69– 82, 1998.
[65]Wei, T. and Klette, R., Theoretical analysis of finite difference algorithms for linear shape from shading, In: Proceedings of Computer Analysis of Images and Patterns: 9th International Conference, CAIP 2001 Warsaw, Poland, September 5–7, 2001, W. Skarbek (Ed.), Lecture Notes in Computer Science, Vol. 2124, Springer-Verlag, Heidelberg, pp. 638–645, 2001.
[66]Walter, G. G. and Shen, X., Wavelets and Other Orthogonal System, 2nd edn., CRC Press, Boca Raton, FL, 2000.
[67]Wells, R. O. and Zhou, X., Wavelet interpolation and approximate solutions of elliptic partial differential equations, Technical Report, Computational Mathematics Laboratory, Rice University, 1993.
[68]Zhang, R., Tsai, P.-S., Cryer, J. E., and Shah, M., Shape from shading: A survey, IEEE Trans. Pattern Anal. Mach. Intell., Vol. 21, No. 8, pp. 690–706, 1999.
[69] Zhao, W., Chellappa, R., Rosenfeld, A., and Phillips, P. J.,
Face recognition: A literature survey. Available at http://citeseer.
nj.nec.com/zhao00face.html, 2000.
[70]Zheng, Q. and Chellappa, R., Estimation of illuminant direction, albedo, and shape from shading, IEEE Trans. Pattern Anal. Mach. Intell., Vol. 13, No. 7, pp. 680–702, 1991.