![5
EIGENVALUES AND FREQUENCIES
The eigenvalues of the Laplacian operator 𝐾 are the frequencies of the mesh. In this part, we’ll
show how the Taubin Smoothing algorithm helps eliminating the high frequencies and
preserving the lower ones.
We first look at our weight function 𝑊 which, as mentioned above, must hold that all its
elements are non-negative and each row sums up to 1. Such matrices are called stochastic
matrices, and since 𝑊 is symmetric, the eigenvalues of 𝑊, denoted as 𝑘𝑖
𝑊
, are real and hold
|𝑘𝑖
𝑊
| ≤ 1
And by 𝐾’s construction, the eigenvalues of 𝐾 hold
0 ≤ 𝑘1
𝐾
≤ 𝑘2
𝐾
≤ ⋯ ≤ 𝑘 𝑛
𝐾
≤ 2
Real, bonded below by 0, and above by 2
In the case where 𝑊 is not symmetric, its eigenvalues might not be real, and the behavior of
the fairing algorithm will depend on their distribution in the complex plane. Although if the
eigenvalues are very close to the real line, we can ignore their imaginary parts and the result of
the algorithm should be essentially the same as the symmetric case above.
By 𝑓(𝐾)’s construction
𝑓(𝑘𝑖) = (1 − 𝜆𝑘𝑖)(1 − 𝜇𝑘𝑖)
The eigenvalues of 𝑓(𝐾)
PASS-BAND REGION
𝑓(𝑘) is a square function, since 𝑓(0) = 1 and 𝜆 + 𝜇 < 0, there is a positive 𝑘 𝑝𝑏 called the pass-
band frequency which holds 𝑓(𝑘 𝑝𝑏) = 1. And then
∀𝑘 ∈ [0, 𝑘 𝑝𝑏], 𝑓(𝑘) 𝑁
≈ 1
Pass-band region is preserved and
∀𝑘 ∈ (𝑘 𝑝𝑏, 2], 𝑓(𝑘) 𝑁
≈ 0 for large enough 𝑁
By demanding 𝑓(𝑘) = 1 we find that
𝑘 𝑝𝑏 =
1
𝜇
+
1
𝜆
To minimize the number of iterations, 𝑁, the smoothing strength factor 𝜆 must be as large as
possible while keeping ∀𝑘 ∈ (𝑘 𝑝𝑏, 2], |𝑓(𝑘)| < 1.](https://image.slidesharecdn.com/6ce5ffee-fb73-46dc-8fd4-ff5f926791e3-161213143445/85/A-Signal-Processing-Approach-To-Fair-Surface-Design-5-320.jpg)
![7
IMPLEMENTATION
LAPLACIAN OPERATOR
getEdgeLengthWeights.m, getNeighorWeights.m, getK.m
We start by defining our Laplacian operator
𝐾 = 𝐼 − 𝑊
Where 𝑊 can be chosen to be either the inverse length of edges (𝑤𝑖𝑗) =
‖𝑝 𝑖−𝑝 𝑗‖
−1
∑ ‖𝑝 𝑖−𝑝ℎ‖−1
ℎ∈𝑖∗
, or the
neighbor(uniform) weights (𝑤𝑖𝑗) =
1
|𝑖∗|
PASS-BAND REGION AND FACTORS
getLambdaMiu.m
The easiest way to achieve control over the smoothing process, is to choose a pass-band
frequency 𝑘 𝑝𝑏 first and deduct the values 𝜆 and 𝜇 from it. To do that we tend to solve the
following non-linear equation system
{
𝑘 𝑝𝑏 =
1
𝜆
+
1
𝜇
𝑓(1) = −𝑓(2)
The system was solved using Matlab’s fsolver with the initial guess 𝜆0 = 0.5 , 𝜇0 = −0.5.
For most meshes, a value of 𝑘 𝑝𝑏 in the region [0.01,0.1] produced best results.
SMOOTHING OPERATOR
getfK.m, applySmoothing.m
Finally, we can define our smoothing operator
𝑓(𝐾) = (𝐼 − 𝜆𝐾)(𝐼 − 𝜇𝐾)
We’ve also added the option to get the Laplacian smoothing operator 𝑓(𝐾) = (𝐼 − 𝜆𝐾)
Applying the operator is simple, the result of the smoothing after 𝑁 iterations is as mentioned
before 𝑣 𝑁
= 𝑓(𝐾) 𝑁
𝑣](https://image.slidesharecdn.com/6ce5ffee-fb73-46dc-8fd4-ff5f926791e3-161213143445/85/A-Signal-Processing-Approach-To-Fair-Surface-Design-7-320.jpg)
A Signal Processing Approach To Fair Surface Design
- 1. 1 A Signal Processing Approach To Fair Surface Design Report
- 2. 2 Overview.........................................................................................................................................3 Introduction....................................................................................................................................3 Theory.............................................................................................................................................4 Laplacian Smoothing Algorithm.......................................................................................... 4 Taubin Smoothing Algorithm.............................................................................................. 4 Eigenvalues and Frequencies.............................................................................................. 5 Pass-band Region................................................................................................................ 5 Constraints.......................................................................................................................... 6 Implementation ..............................................................................................................................7 Laplacian Operator.............................................................................................................. 7 Pass-band Region and Factors............................................................................................ 7 Smoothing Operator........................................................................................................... 7 Hard Interpolation Constraint............................................................................................. 8 Soft Interpolation Constraint.............................................................................................. 8 Hierarchal Constraint.......................................................................................................... 8 Eigen-decomposition Smoothing........................................................................................ 9 Results, Conclusions and Discussions ........................................................................................10 Demo Manual ...............................................................................................................................15
- 3. 3 OVERVIEW Taubin Smoothing is a tool for interactive free-form fair surface design. Taubin reduced the problem of surface smoothing, or fairing, to low-pass filtering by generalizing classical discrete Fourier analysis to two-dimensional discrete surface signals. The suggested smoothing algorithm is a very simple surface signal low-pass filter algorithm that applies to surfaces of arbitrary topology. This algorithm works in linear time and space complexity. By combining surface subdivision and different types of constraints, this algorithm proves to be a very effective technique for surfaces of arbitrary topology. Some of the constraints can be imposed without any modification of the algorithm, while others do require a modification to be made. The constraints presented here are Vertex location constraints, both soft interpolation and hard interpolation as well as Hierarchal constraints. Other constraints suggested in the paper, such as vertex normal constraints, were not implemented due to technical difficulties with Matlab to interact and modify the normals in an quick, comfortable and intuitive way. We will talk about each topic briefly and then present our implementation, results and finally a “user manual” for our demo applet. INTRODUCTION Fourier analysis is a natural tool to solve the problem of signal smoothing. The space of signals – functions defined on certain domain – is decomposed into orthogonal subspaces associated with different frequencies, with the low frequency content of a signal regarded as subjacent data, and the high frequency content as noise. The problem lies in calculating such decompositions, which is computationally expensive the larger the vertex count gets, therefore the algorithm Taubin suggested, approximates the result through an iterative process which eliminates the high frequencies with each iteration. The algorithm is based on the Laplacian Smoothing iterative algorithm which shifts each vertex location, according to local information in that vertex, without changing the connectivity of the mesh. But the simple Laplacian Smoothing process is followed by an undesired effect; shrinkage of the mesh. To solve this, an opposite operation is applied to assist in preserving the overall dimensions of the mesh.
- 4. 4 THEORY LAPLACIAN SMOOTHING ALG. Otherwise known as Gaussian filtering, the purpose of the algorithm is to reduce the rapid changes of the curvature throughout the mesh. The idea is to move a vertex 𝑣𝑖 along a certain vector, that vector is the Discrete Laplacian operator applied on that vertex, denoted by ∆𝑣𝑖. Thus, the new vertex location is: 𝑣𝑖 ′ = 𝑣𝑖 + 𝜆∆𝑣𝑖 Where 0 < 𝜆 < 1 is a factor that will determine the strength of the smoothing, Δ𝑣𝑖 = ∑ 𝑤𝑖𝑗(𝑣𝑗 − 𝑣𝑖)𝑗∈𝑖∗ where 𝑖∗ is the indices of the neighboring vertices of 𝑣𝑖, and 𝑤 is the weight function. (*)𝑤𝑖𝑗 must be positive and each row adds up to 1. Otherwise we won’t have control over the smoothing, i.e. instead of attenuating the frequencies, some will be enhanced. In the simple case of a closed curve and 𝑤𝑖𝑗 = 1 |𝑖∗| , we get: ∆𝑣𝑖 = 1 2 (𝑣𝑖−1 − 𝑣𝑖) + 1 2 (𝑣𝑖+1 − 𝑣𝑖) In matrix form: 𝑣1 = (𝐼 − 𝜆𝐾)𝑣 Where K is the Laplace operator As mentioned before, this step produces shrinkage as Gaussian filter isn’t low-pass filter. TAUBIN SMOOTHING ALG. To define a low-pass filter, an opposite step is introduced into the Laplacian algorithm, and so we define our smoothing operator: 𝑓(𝐾) = (𝐼 − 𝜆𝐾)(𝐼 − 𝜇𝐾) Where −1 < 𝜇 < −𝜆 is a new negative scale factor. And therefore the new smoothing step is as follows: 𝑣1 = 𝑓(𝐾)𝑣 Of course, as before, this step can be iterated 𝑁 times and the result: 𝑣 𝑁 = 𝑓(𝐾) 𝑁 𝑣
- 5. 5 EIGENVALUES AND FREQUENCIES The eigenvalues of the Laplacian operator 𝐾 are the frequencies of the mesh. In this part, we’ll show how the Taubin Smoothing algorithm helps eliminating the high frequencies and preserving the lower ones. We first look at our weight function 𝑊 which, as mentioned above, must hold that all its elements are non-negative and each row sums up to 1. Such matrices are called stochastic matrices, and since 𝑊 is symmetric, the eigenvalues of 𝑊, denoted as 𝑘𝑖 𝑊 , are real and hold |𝑘𝑖 𝑊 | ≤ 1 And by 𝐾’s construction, the eigenvalues of 𝐾 hold 0 ≤ 𝑘1 𝐾 ≤ 𝑘2 𝐾 ≤ ⋯ ≤ 𝑘 𝑛 𝐾 ≤ 2 Real, bonded below by 0, and above by 2 In the case where 𝑊 is not symmetric, its eigenvalues might not be real, and the behavior of the fairing algorithm will depend on their distribution in the complex plane. Although if the eigenvalues are very close to the real line, we can ignore their imaginary parts and the result of the algorithm should be essentially the same as the symmetric case above. By 𝑓(𝐾)’s construction 𝑓(𝑘𝑖) = (1 − 𝜆𝑘𝑖)(1 − 𝜇𝑘𝑖) The eigenvalues of 𝑓(𝐾) PASS-BAND REGION 𝑓(𝑘) is a square function, since 𝑓(0) = 1 and 𝜆 + 𝜇 < 0, there is a positive 𝑘 𝑝𝑏 called the pass- band frequency which holds 𝑓(𝑘 𝑝𝑏) = 1. And then ∀𝑘 ∈ [0, 𝑘 𝑝𝑏], 𝑓(𝑘) 𝑁 ≈ 1 Pass-band region is preserved and ∀𝑘 ∈ (𝑘 𝑝𝑏, 2], 𝑓(𝑘) 𝑁 ≈ 0 for large enough 𝑁 By demanding 𝑓(𝑘) = 1 we find that 𝑘 𝑝𝑏 = 1 𝜇 + 1 𝜆 To minimize the number of iterations, 𝑁, the smoothing strength factor 𝜆 must be as large as possible while keeping ∀𝑘 ∈ (𝑘 𝑝𝑏, 2], |𝑓(𝑘)| < 1.
- 6. 6 CONSTRAINTS Constraints enable us to have more control over the smoothing process, we present three types of constraints, hard interpolation, smooth interpolation and hierarchal constraints. Hard interpolation and hierarchal constraints demand no modification of the algorithm but modify the neighborhood structure of the mesh. Soft interpolation will require a small modification of the algorithm but does not change the neighborhood structure. HARD INTERPOLATION CONSTRAINTS Hard interpolation, or non-smooth interpolation, on a set of vertices can be imposed without any modification of the algorithm. Hard interpolation of a vertex prevents it from shifting from its location during the smoothing process. Solely based on the fact that a vertex 𝑣𝑖 without neighbors holds that ∆𝑣𝑖 = 0, imposing hard interpolation on a vertex 𝑣𝑖 is fairly simple. Simply consider 𝑣 to have no neighbors, then 𝑣𝑖 1 = 𝑣𝑖 + 𝜆Δ𝑣𝑖 = 𝑣𝑖 SOFT INTERPOLATION CONSTRAINTS Soft interpolation, or smooth interpolation, offers a solution to perhaps an issue of hard interpolation where the mesh loses it smoothness around the constrained vertices. It takes a different approach to enforce a vertex’s location. Based on applying a smooth deformation on the faired mesh, to restore the constrained vertices back to their original location. When imposed on a vertex 𝑣𝑖, the new smoothing step is defined as: 𝑣 𝐶 𝑁 = 𝑣 𝑁 + (𝑣𝑖 − 𝑣𝑖 𝑁 )𝐹𝑛𝑖 𝐹𝑖𝑖 −1 Where 𝐹 = 𝑓(𝐾) 𝑁 and 𝐹𝑛𝑖 𝐹𝑖𝑖 −1 is our smooth deformation, (𝐹𝑛𝑖 𝐹𝑖𝑖 −1 )𝑖 = 1 which holds the interpolatory constraint (𝑣 𝐶 𝑁)𝑖 = 𝑣𝑖 HIERARCHAL CONSTRAINTS Same as hard interpolation, hierarchal interpolation does not require a modification of the algorithm but of the neighborhood structure of the mesh. We assign to each vertex 𝑣𝑖 a label 𝑙𝑖 and then define the neighborhood structure as follows: 𝑗 ∈ 𝑖∗ ⟺ (𝑖, 𝑗) ∈ 𝐸 ∧ 𝑙𝑖 ≤ 𝑙𝑗
- 7. 7 IMPLEMENTATION LAPLACIAN OPERATOR getEdgeLengthWeights.m, getNeighorWeights.m, getK.m We start by defining our Laplacian operator 𝐾 = 𝐼 − 𝑊 Where 𝑊 can be chosen to be either the inverse length of edges (𝑤𝑖𝑗) = ‖𝑝 𝑖−𝑝 𝑗‖ −1 ∑ ‖𝑝 𝑖−𝑝ℎ‖−1 ℎ∈𝑖∗ , or the neighbor(uniform) weights (𝑤𝑖𝑗) = 1 |𝑖∗| PASS-BAND REGION AND FACTORS getLambdaMiu.m The easiest way to achieve control over the smoothing process, is to choose a pass-band frequency 𝑘 𝑝𝑏 first and deduct the values 𝜆 and 𝜇 from it. To do that we tend to solve the following non-linear equation system { 𝑘 𝑝𝑏 = 1 𝜆 + 1 𝜇 𝑓(1) = −𝑓(2) The system was solved using Matlab’s fsolver with the initial guess 𝜆0 = 0.5 , 𝜇0 = −0.5. For most meshes, a value of 𝑘 𝑝𝑏 in the region [0.01,0.1] produced best results. SMOOTHING OPERATOR getfK.m, applySmoothing.m Finally, we can define our smoothing operator 𝑓(𝐾) = (𝐼 − 𝜆𝐾)(𝐼 − 𝜇𝐾) We’ve also added the option to get the Laplacian smoothing operator 𝑓(𝐾) = (𝐼 − 𝜆𝐾) Applying the operator is simple, the result of the smoothing after 𝑁 iterations is as mentioned before 𝑣 𝑁 = 𝑓(𝐾) 𝑁 𝑣
- 8. 8 HARD INTERPOLATION applySmoothing.m We can easily impose hard interpolation on a set of vertices by resetting the corresponding rows in the Laplacian 𝐾 to zeros. Then 𝑓(𝐾) must be recalculated and applying smoothing is same as before 𝑣 𝑁 = 𝑓(𝐾) 𝑁 𝑣. SOFT INTERPOLATION applySmoothing.m First, we calculate the difference 𝑣 𝑚 − 𝑣 𝑚 𝑁 where 𝑚 is a set of the constrained vertices indices and 𝑣 𝑚 = 𝑣(𝑚, : ) is the set of constrained vertices. Then we tend to solve the following linear system 𝐹𝑚𝑚 𝑦 𝑚 = (𝑣 𝑚 − 𝑣 𝑚 𝑁 ) Where 𝐹 = 𝑓(𝐾) 𝑁 and 𝐹𝑚 = 𝐹(𝑚, 𝑚) Finally, the smoothed result is 𝑣 𝐶 𝑁 = 𝑣 𝑁 + 𝐹𝑦𝑛 HIERARCHAL CONSTRAINTS hierarchizeAM.m Here, we seek to modify The Adjacency Matrix 𝐴𝑀 of our mesh to fulfil the hierarchy levels specified by the label 𝑙. First, we extract all the indices (𝑟𝑖, 𝑐𝑖) where 𝐴𝑀𝑟 𝑖 𝑐 𝑖 = 1 and rebuild the matrix using Matlab’s sparse with the parameters (𝑟, 𝑐, 𝑙 𝑟 ≤ 𝑙 𝑐) meaning we basically zero out neighboring vertices which don’t fulfil the hierarchy condition. Finally, a recalculation of 𝑓(𝐾) is needed and smoothing proceeds as normal. Note: in the code we are assigning the result to a different matrix called hierarchizedAM, this is so we can later revert back to the original adjacency matrix of the mesh. It is also worth mentioning that this type of constraint is independent of the latter two, they can be combined together to get both effects.
- 9. 9 EIGEN-DECOMPOSITION SMOOTHING applySmoothingEigs.m The straight forward way to eliminate high frequencies from a mesh is calculating the eigen- decomposition of the Laplacian operator: 𝐾 = 𝑉𝐷𝑉 𝑇 Where 𝑉 is the eigenvectors of 𝐾, and 𝐷 is a diagonal matrix where the eigenvalues are on the diagonal Using Matlab’s eigs function, we calculated the first 𝑚 eigenvalues diagonal matrix 𝐷 𝑚 and their corresponding eigenvectors 𝑉𝑚 and then the smoothed result is 𝑣 𝑛−𝑚 = 𝑉𝑚 𝑉𝑚 𝑇 𝑣 Smoothing using this method to receive a significantly smoothed result can be very efficient due to the fact we will need to calculate a small number of eigenvalues and eigenvectors, but computation time increases drastically as 𝑚 increases.
- 10. 10 RESULTS, CONCLUSIONS AND DISCUSSIONS Partially noisy sphere Left to right: Original model, after 10 non-shrinking smoothing steps, after 50 steps, after 200 steps. It can be seen that there’s barely difference between smoothing with 50 steps and 200 in this case, it can be said that the process has converged. Also, even after 200 smoothing steps, the model didn’t shrink nor was its general shape altered, i.e. the right side, where the noise was, still can be noticed to be different than the left side. CT scan Left to right: Original model(Spine - First Lumbar Vertebra), the original model voxelized to emulate a CT scan model, voxelized model smoothed 100 steps. Here, we were able to achieve a beautiful and fine visualization of the bone using Taubin Smoothing. We noticed that through this process of emulating a scan and smoothing it, not only can we achieve a higher detail mesh, but also a clean and organized one as well. Can Voxelization and Taubin Smoothing be used as a remeshing technique? 1. Voxelize the mesh 2. Remove interior polygons created from voxelization 3. Smooth
- 11. 11 Wine Glass Left to right: Original low-poly wine glass, after Subdivision and Laplacian Smoothing, after Subdivision and Taubin Smoothing, Phong Shaded. Taubin Smoothing’s advantages and Laplacian Smoothing’s disadvantages are clear in this example. While attempting to achieve a good looking wine glass by subdividing and smoothing, we saw that the best results were achieved by switching between Laplacian and Taubin Smoothing steps, which in a way felt like sculpting. This has led us to ask if we could incorporate Taubin Smoothing, Laplacian smoothing as well as the aforementioned constraints as a sculpting tool set. The user can select parts of the mesh and apply combinations of the tools to alter the mesh shape. For example, we sculpted this glass out of the same low-poly wine glass from above:
- 12. 12 Face Left to right: Original low-poly face, Smoothed with no constraints, smoothed with hard interpolatory constraints on features(eyes, nose, lips and chin). Same as before we can use different kinds of constraints and smoothing methods to design the mesh as we like. For example, using hard and hierarchal constraints, we were able achieve an interesting looking mesh structure
- 13. 13 Hard Vs. Soft Interpolation Left to right: Original model(Sphere with spikes), Smoothed with Hard Interpolation applied to the tips of the spikes on the sides, Smoothed with Soft Interpolation. Hard interpolation’s disadvantage can be seen here as smoothing process was not useful for the interpolated points, whereas in Soft Interpolation we have achieved both the interpolatory constraint and a smooth mesh around the interpolated points. Hierarchal Constraints Right to left: original model, smoothed, smoothed with hierarchal constraints applied. The vertices on the edges of the model were set to one hierarchy level higher than the rest. We can see that, due to the hierarchal neighborhood structure, vertices of the highest hierarchy level (the edges) were smoothed like a curve and the three triangles were rounded into circles.
- 14. 14 Eigendecomposition smoothing Left to right: Original, 𝑚 = 10, 30, 100, 200, 325 This mesh is made up of 2903 vertices which indicates that there is an equal number of eigenvectors. As can be seen here, at around 𝑚 = 200, noise starts to build up at parts where there’s high curvature values(eyelids, ears and horns). We believe that is due to the instability of Matlab’s eigs function. We have also found that for all models, noise starts to get noticed at 𝑚 ≈ 0.1 ∙ |𝑉| and builds up as 𝑚 increases.
- 15. 15 DEMO MANUAL To run the demo, call the script file Demo.m 1. Taubin: Smoothing method selection. Checked - Taubin. Unchecked – Laplacian. 2. Load OFF/Export OFF/Export OBJ: Load meshes and export meshes as .off or .obj files. 3. Edge Lengths/Uniform: Select weight function. 4. Kpb: Set pass-band frequency 𝑘 𝑝𝑏, Enabled only in Taubin Smoothing mode. 5. Subdivide: Subdivide mesh. 6. Select vertices: Use slider to select vertices(will be colored in red). 7. Add as hierarchy level/Revert AM(*): Increment 𝑙(hierarchy level) of selected vertices. Revert AM – restore original adjacency matrix(reset hierarchy levels) 8. Add as constraints/Smooth Interpolation: Set interpolatory constraints on the selected vertices. Smooth Interpolation – checked for Soft interpolation. Unchecked for Hard interpolation (*Cannot combine soft with hard, only the last selection set is valid!) 9. Smooth once/Auto smooth: Apply one step or automatically apply 100 steps of smoothing. The iteration count is displayed under the buttons. 10. Eigs smooth: Use slider to set the level of smoothing with lowest being no smoothing(𝑚 = |𝑉|).(**) 11. Draw Mean: Visualize Mean Curvature function on the mesh 12. Reset: Reset mesh to original data. Removes also constraints and resets hierarchy levels. (*) Make sure there aren’t any vertices in some hierarchy level with no neighbors in the same or higher level. Otherwise, these loner vertices will start moving toward the origin with each smoothing step because in case of no neighbors, ∆𝑣𝑖 = −𝑣𝑖. (**) Set the slider to max(𝑚 = 0) and press Eigs smooth to get an “inflating” animation of the mesh from 𝑚 = 0 to 𝑚 = 0.1 ∙ |𝑉|. On first call to Eigs smooth, please wait for all eigenvectors to be calculated.