1. ISMM, October 13 th, 2007 Vaz Et. Al Michael S. Vaz a, Atilla P. Kiraly b, Russell M. Mersereau c a Barco – Medical Imaging Division, Beaverton OR, USA b Siemens Corporate Research, Princeton NJ, USA c Georgia Institute of Technology, Atlanta GA, USA Multi-level decomposition of Euclidean spheres

2. ISMM, October 13 th, 2007 Vaz Et. Al Overview Motivation Method –2D Example –3D Example Results & Analysis Conclusion & Future work Questions (also, at any time during the presentation)

3. ISMM, October 13 th, 2007 Vaz Et. Al Motivation Mathematical morphology (MM) is used for a wide variety of applications. Some examples: –Object recognition –Image segmentation –Industrial inspection In medical imaging, MM has such uses as: –Brain segmentation from MR images –Airway/vessel segmentation from CT images Medical imaging –Anisotropic voxalization –Need for scale-robust algorithms –High resolution imaging  raw image resolution is increasing as acquisition technology increases –Increased image resolution can increase computational load of analysis very rapidly –Dilation/erosion optimized for high resolution large structuring elements (SE) is very useful

4. ISMM, October 13 th, 2007 Vaz Et. Al Our motivating application Perfusion visualization for pulmonary embolism

5. ISMM, October 13 th, 2007 Vaz Et. Al We need fast MM for Euclidean spheres Lung segmentation algorithm uses mathematical morphology (MM) Multiple Euclidean sphere structuring elements (SE) Approximations to the ball SE are not tolerated 512 x 512 x 512 CT volume was takes ~20 minutes to process

6. ISMM, October 13 th, 2007 Vaz Et. Al Even the commercial package falls short As sphere SE size increases Processing time increases rapidly for a brute-force (direct) implementation The commercial package (SDC) is still not good

7. ISMM, October 13 th, 2007 Vaz Et. Al Why do we need yet another method? Many methods have been proposed to accelerate MM Majority of these involve some form of SE decomposition HGW algorithm and Logarithmic decomposition are very fast 1D –Only a subset of useful 2D/3D SE lend themselves to a factorization that allows the use of these fast 1D methods –Doesn’t work for the Euclidean sphere – which we need! Some methods are only applicable to binary images or require a special encoding/pre-processing of the image Some methods cater to specific constraints such as limited region of support and specialized hardware

8. ISMM, October 13 th, 2007 Vaz Et. Al No really! Why? For the Euclidean disk/sphere –Many methods that are said to work for all convex symmetric SE just don’t work (problem with definition of convexity) –Many methods for 2D sacrifice accuracy for efficiency: not acceptable! –Even after the tradeoff, these methods do not lend themselves to the 3D sphere –The method using histograms by Doogenbroeck and Talbot (1996) does the job, but we were not satisfied with implementation constraints (not good for parallelization) SE decomposition as a union of partitions was described by Anelli and Broggi (1998) –Genetic algorithms used to perform the decomposition –Great idea but some of us tend to shy away from genetic algorithms

9. ISMM, October 13 th, 2007 Vaz Et. Al Logarithmic decomposition (LD) –simple, elegant and powerful Want a true 3D decomposition Wish to harness LD in a decomposition of Euclidean sphere SE Inspiration

10. ISMM, October 13 th, 2007 Vaz Et. Al Method: 2D example Decomposition as a union of partitions Each partition is comprised of –Cubic factor C i –Sparse factor S i

11. ISMM, October 13 th, 2007 Vaz Et. Al

12. ISMM, October 13 th, 2007 Vaz Et. Al Computation reuse across partitions

13. ISMM, October 13 th, 2007 Vaz Et. Al Decomposing the SE: algorithm outline 1.Initialization: set CSE (current SE) to be equal to SE 2.Find the largest cube C i with which CSE can be morphologically opened without change 3.Find the corresponding sparse factor S i  At this point we have decomposed a partition from CSE 4.Update CSE with RSE (remaining SE) 5.Go to Step 2 and repeat till RSE is NULL

14. ISMM, October 13 th, 2007 Vaz Et. Al Decomposing the SE (RSE i formula WRONG in paper)

15. ISMM, October 13 th, 2007 Vaz Et. Al Determining S i 1.Once C i is available, obtain test cubic factors CT j dilating C i with T j 2.Now obtain candidate sparse factors ST 1, ST 2, ST 3 RT j = CSE o CT j ST j = (CSE Θ C i ) – (RT j Θ C i ) 3.Test each ST j for sparseness In general spase factors are not edge- connected for 2D or not face- connected for 3D 4.Select the ST j that is sparse and has the largest discrete mass  assign it to S i

16. ISMM, October 13 th, 2007 Vaz Et. Al Determining S i (ST j formula is WRONG in paper)

17. ISMM, October 13 th, 2007 Vaz Et. Al 3D example: radius-5.5 sphere Once decomposition is available, usage is identical to 2D Finding S i is slightly more elaborate –For 2D we have T1, T2, T3 –For 3D we have T1 – T7 SE mass = 739 Total ops for dilation = 49/output voxel 11 ops due to C i 38 ops due to S i

18. ISMM, October 13 th, 2007 Vaz Et. Al 3D decomposition as a union of partitions

19. ISMM, October 13 th, 2007 Vaz Et. Al Cubic and sparse factors of P 1 and P 2

20. ISMM, October 13 th, 2007 Vaz Et. Al Cubic and sparse factors of P 3 and P 4

21. ISMM, October 13 th, 2007 Vaz Et. Al Results Performance for a 512 x 512 x 480 chest CT image, using an Intel Pentium IV Xeon Dual CPU platform with 2GB RAM Clear advantage of proposed method over the commercial SDC morphology toolbox For increasing SE size, processing time increase is minimal for the proposed method

22. ISMM, October 13 th, 2007 Vaz Et. Al True 3D decomposition Clear advantage to true 3D decomposition vs. decomposing as a union of 2D slices Could further reduce comparison ops by factorizing S i –at the expense of more memory –increase read/write ops

23. ISMM, October 13 th, 2007 Vaz Et. Al Conclusions An efficient method for MM using Euclidean disk and sphere SE has been presented (no approximations) –These SE don’t lend themselves to well known factorization methods The proposed method works for all 2D/3D convex symmetric SE –Convexity: A convex discrete domain equal to the set of all voxels that fall inside its Euclidean convex hull –Symmetry: A symmetric discrete domain has a clearly defined center. When its center of mass is translated to the origin of the standard Cartesian coordinate system, the translated domain will be symmetric about the x = 0, y = 0 and z= 0 planes The method works for binary (flat) SE and can be applied to binary or gray-scale images The method is not guaranteed to be optimal

24. ISMM, October 13 th, 2007 Vaz Et. Al Future Work Investigate optimality of the method –Is the cost to find the optimal decomposition justified by the gain? Convex, but not symmetric SE –Perhaps relax the orientation of the cubic factors General SE with no constraint –Perhaps relax the need for C 1 to be a factor of C 2 and so on –Also relax the orientation of the C i –Perhaps this will generalize to C i  “dense factor”, where dense factor can be further decomposed/factorized –We expect the gain-per-discrete mass to be less than for convex/symmetric SE –NP-complete problem  genetic algorithms might become interesting Do you have any suggestions? Thank You For Your Attention! Please Ask Questions