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Title: Local Editing of Cross-Surface Mappings with Iterative Least Squares Conformal Maps

Paper content:
In this paper, we propose a novel approach to improve a given surface mapping through local refinement.
The approach
receives an established mapping between two surfaces and follows four phases:
(i) inspection of the mapping and creation of a sparse
set of landmarks in mismatching regions;
(ii) segmentation with a low-distortion region-growing process based on flattening the
segmented parts;
(iii) optimization of the deformation of segmented parts to align the landmarks in the planar parameterization domain;
and
(iv) aggregation of the mappings from segments to update the surface mapping.
In addition, we propose a new method to deform the
mesh in order to meet constraints (in our case, the landmark alignment of phase
(iii)).
We incrementally adjust the cotangent weights for
the constraints and apply the deformation in a fashion that guarantees that the deformed mesh will be free of flipped faces and will have
low conformal distortion.
Our new deformation approach, Iterative Least Squares Conformal Mapping (ILSCM), outperforms other
low-distortion deformation methods.
The approach is general, and we tested it by improving the mappings from different existing surface
mapping methods.
We also tested its effectiveness by editing the mappings for a variety of 3D objects.
C OMPUTING a cross-surface mapping between two surfaces (cross-parameterization) is a fundamental problem in digital geometric processing.
A wide range of methods have been developed to find such mappings [2] , [3] , [4] , but no single method results in a perfect mapping in every case.
Quite often, the mapping results may be good overall, but some specific, sometimes subtle, semantic features, such as articulations and facial features, may remain misaligned, as illustrated in Figure 1 .
These imperfections of the final result are often unacceptable in a production setting where the artist needs a high degree of control over the final result, and will often sacrifice automation of a method for higher control.
Typically, improving results using surface mapping methods requires the user to iteratively insert some landmarks and solve for the mapping globally.
However, since the imperfections are typically localized to a specific region, a local solution that does not change the mapping globally would be preferred in order to ensure that the method does not introduce artifacts elsewhere on the map.
This paper proposes a surface mapping editing approach providing local and precise control over the map adjustments.
The process begins with the inspection of an existing vertex-topoint surface mapping between two meshes.
In regions where the mapping exhibits some discrepancy, the user sets landmarks positioned at corresponding locations on both meshes.
For each such region, we extract a patch on both meshes in order to localize the changes in the mapping, and we flatten them on a common planar domain.
The mapping is improved based on a 2D deformation optimization that steers the landmarks toward correspondence while limiting distortion and having theoretical guarantees to maintain the local injectivity of the map.
We developed a new 2D deformation approach denoted Iterative Least Squares Conformal Maps (ILSCM), which iteratively minimizes a conformal energy, each iteration ensuring that flips do not occur, and in practice, ensuring progress toward satisfying the constraints.
We chose to work with a conformal energy as we want to be able to improve mappings where the deformation between the pair of meshes is not isometric.
Our editing approach can successfully align the mapping around landmarks without any degradation of the overall mapping.
The local surface maps are extracted from their respective deformed segments and parameterization domains, and are then combined to form an improved global surface mapping.
Our approach solves an important practical problem and offers three novel scientific contributions.
The first is a practical approach for local surface map editing, which we show, using both qualitative and quantitative metrics, provides better results than other stateof-the-art methods.
The second involves a compact segmentation which results in a compromise between a low-distortion flattening and a low-distortion deformation when aligning the landmarks.
The third is a new deformation approach, ILSCM, which preserves conformal energy better than other state-of-the-art methods, and that has theoretical guarantees preventing the introduction of foldovers.
Our approach carves a new path in between the more classical shape-preserving methods, which often lose local injectivity, and the more current methods, which formulate the injectivity constraint as part of the optimization.
These latter approaches typically do not have a bound on the shape-preserving error.
In our approach, we are minimizing only the shape-preserving term (i.e., LSCM energy) and iteratively improving the user constraints while maintaining a locally injective map in each iteration.
We achieve this by carefully controlling the λ parameter in Eq. 1.
At one extreme, if λ is very large (i.e., infinity), the formulation is equivalent to the LSCM formulation.
If λ is very small, it takes many iterations for the user constraints to be satisfied, or in some cases, the user constraints may ultimately not be satisfied.
Our iterative scheme relies on two important observations.
If λ is 0, the solution is the same as the initial configuration.
Therefore, if we start in a locally injective configuration, the final result will be a locally injective configuration.
If the initial configuration is locally injective, there always exists a λ (however small) that will result in a locally injective configuration, where the user constraints are closer to the target.
This scheme will converge to a locally injective configuration.
Consequently, we iteratively repeat the optimization to fight against flipped faces, but convergence cannot be guaranteed.
It is always possible to design a landmark configuration in which the constraints cannot be met without flipped faces.
This is true for the other deformation methods as well.
Appendix B demonstrates different failure cases using different deformation methods.
In our experiments, the constraints are satisfied (up to numerical precision), even for extreme deformations.
In our results, we improved mappings which were initially computed from a variety of methods [1] , [3] , [4] , [10] , [21] , [26] .
Even if these initial mappings minimize different deformation energies, the fact that we rely on the LSCM conformal energy to edit them did not prevent our approach to improve the mappings.
One must keep in mind that the goal of the editing is not to strictly minimize a deformation energy, but to align important semantic features of the objects and maintain injectivity.
We analyzed our results to verify the degree to which the deformation deteriorates the shape of the triangles.
We checked 13 of the results found in this paper, and we considered that a detrimental deformation is one in which the angle becomes more than 20 times narrower after deformation.
Eleven cases had no such triangles, while the two other cases had two and three, respectively.
The worst triangle in our 13 test cases was 24 times narrower than before deformation.
Any deformation method is prone to result in thin triangles, so we compared our approach to LIM, SLIM, and KP-Newton for six examples.
When looking at the worst triangle found in the deformed meshes, ILSCM performed best for four of the test cases, while KP-Newton performed best for two of the test cases.
SLIM and LIM were systematically in third and fourth place behind ILSCM and KP-Newton.
Furthermore, our results were better than LIM, SLIM, and KP-Newton in terms of shape preservation and final triangulation, as can be seen in Fig. 12 and in the video.
We ran our experiments on a 3.40 GHz Intel Core-i7-4770 CPU with 12 GB of memory.
The presented approach was implemented with MATLAB, taking advantage of its sparse matrices and linear solvers.
Table 1 shows computation times for the segmentation and the deformation (including mapping extraction) phases.
Since our deformation phase is an iterative method, the time to edit a mapping depends on the size of the mismatching regions and iterations.
We have presented a novel approach for improving surface mappings locally.
Our approach is based on a low-distortion region-growing segmentation followed by an independent planar parameterization of each segment.
The mapping is then optimized based on an alignment of the user-prescribed landmarks in the parameterization space of each segment.
Our joint planar parameterization deformation for the segments is robust, and results in low distortion.
Our new iterative LSCM approach can be reused in several contexts where a deformation with low distortion is required.
From a practical perspective, our approach has several
(a) Mesh A
(b) Mesh B, initial mapping [10]
(c) Mesh B, WA [10]
(d) Mesh B, our edited mapping
(e) Mesh A
(f) Mesh B, initial mapping [10]
(g) Mesh B, WA [10]
(h) Mesh B, our edited mapping advantages.
It can be used to improve the mapping resulting from
(a) Mesh A skeleton
(b) Mesh B, initial skeleton [26]
(c) Mesh B, edited skeleton Fig. 24 .
When using the mapping to retarget attributes, in this case the skeleton, an incorrect mapping will lead to problems, here putting the thumb joint outside of the mesh.
By locally editing the mapping, it is easy to fix such issues.
any surface mapping method.
It also provides a great deal of control, allowing the user to restrict editing to a specific region and to add as few or as many landmarks as necessary to achieve a desired result.
Our local editing leads to interesting questions which open many avenues for future work.
One such prospective area is higherlevel landmarks such as lines.
This will lead to challenges in terms of easing the interactive placement of these lines on both meshes, but will provide a better set of constraints for the deformation.
Another avenue would be to extend the scope to editing deformation transfer.
This will combine deformation with editing and enable the user to control animation retargeting.

assistant: text: We propose a novel approach to improve a given cross-surface mapping through local refinement with a new iterative method to deform the mesh in order to meet user constraints.