3D volume retouching (“Fulfillment” of missing data)
- The surface reconstruction problem can be expressed as follows:
- Given shapes S1, S2, … , find a blending surface Sb that is a reasonable approximation of the surface satisfying a set of constrains.
- reasonable approximation”
- The original surface and the restored surface must smoothly join along predefined boundary regions of the restoration area.
- To avoid superfluous folding over a blended (reconstructed) area.
Examples - Image Retouching (Local approach)
- We attempt to approach the well-known problem of “fulfillment” of missing data
- We assume that due to continuity in the image the boundary of an existing R,G,B surface can be propagated to complete the missing part
- “Wool”, one additional sloping scratch was added to the test image of (Hirani and Totsuka, 1995 )
- Restored image obtained with our algorithm
Examples - Image Retouching
Image retouching (Image size: 550 x 388)
A synthetically produced black scratch is the reconstruction area ΩR. Adobe Photoshop© GUI interface allows us to select automatically this area, after that a landmark area ΩL as a slight extension of the ΩR area (one or two pixels are sufficient) is automatically calculated
- Processing time (SGI Octane 300 Mhz):
- Data sorting: 0.02 sec
- LL decomposition, forward and back substitution: 0.02 sec
- Reconstruction time: 0.02 sec
- (a) Old photograph (courtesy of Amy Miller of Photo-Medic)
- (b) Result produced with Bertalmio, Sapiro, Caselles, and Ballester's algorithm (SIGGRAPH’2000) based on PDEs. Approximately 7 minutes on a 300 MHz Pentium II PC
- (c) Restored image obtained with our algorithm. Approximately 7 sec on AMD Athlon 1GHz
Surface Retouching
- Point sets obtained from computer vision techniques are often are non-uniform and even contain large missing areas of points
- Another source of polygonal data with missing or damaged areas is partly destroyed ancient artefacts. In such applications, an incomplete surface is the surface exactly as measured, and the main problem is to obtain prospective reconstructions that look reasonable
- In some well-known algorithms, authors make the assumption that points are situated sufficiently evenly. In others, it is assumed that holes have entirely specific shapes
Examples - Surface Retouching
Example 1
- Surface retouching of 300 x 300 elevation points given in a pixel raster
- Ш shape - ΩR area
- Red square - Ω+ area
- Processing time (SGI Octane 300 Mhz ):
- Data sorting - 0.1 sec;
- LLt decomposition, forward and back substitution - 0.22 sec;
- Reconstruction time - 0.06 sec.
Example 2
- Surface retouching of range data (captured by the Minolta VIVID 700 laser scanner, Moai model, courtesy of Y. Ohtake and A. Belyaev of Max-Planck-Institut fur Informatik )
- A spherical concavity can be observed (central area of the model).
- Processing time (AMD Athlon 1000 Mhz, 128 MB RAM, Microsoft Windows 2000):
- Data sorting - 0.001 sec;
- LLt decomposition, forward and back substitution - 0.001 sec;
- Reconstruction time - 0.001 sec.
Example 3
- Surface retouching of a synthetic polygonal model
- Left image. A modified “Stanford Bunny” model (courtesy of Y. Ohtake and A. Belyaev of Max-Planck-Institut fur Informatik)
- Right after surface retouching. Processing time for 930 vertices (AMD Athlon 1000 Mhz, 128 MB RAM, Microsoft Windows 2000) ): 1 sec
Example 4
- Surface retouching of a real polygonal model
- Left image. The “Stoned” model (courtesy of R. Scopigno and M. Calliery of Institute CNUCE). Model size - 88478 points.
- Right image is after surface retouching.
Examples - Interactive Surface Retouching and Sculpturing
- Surface retouching of a real polygonal model
- Left image. The “Sangiovannino ” model (courtesy of R. Scopigno and M. Calliery of Institute CNUCE). Model size - 22686 points.
- Right image is after surface retouching. Model size - 103301 points after retouching.
Problems with existing approaches
- Level set models with a positive-curvature flow are used to create a smooth blend (fillet) between solid objects
- Rather simple applications were considered
- Interpolating contour lines with the help of PDE methods, thin plate or FEM interpolation
- Undesirable synthetic effects such as terracing effect or over smoothing appear
Local reconstruction
- Implementation of the partition of unity for generation of polygons from scattered data of (the fragment of Mount Bandai):
- (a) a curvature analysis. Blue area - the surface variation σ > 0.3;
- (b) result of reconstruction (ray tracing). Number of scattered points - 10000, number of vertices after reconstruction - 90000, processing time - 0.941 sec;
- (c) fragment of the mesh as a wire-frame with color attributes in accordance with calculated heights
Local reconstruction
- Surface reconstruction of a technical data set:
- (a) cloud of points, 4100 scattered points are used
- (b) simplified and smoothed mesh shaded
- (c) fragment of the mesh (7141 vertices ) as wire-frame
Improvement of mesh quality using a statistical approach
- In spite of that the average aspect ratio for the mesh (a) is 1.46, there are many badly shaped triangles, especially on the almost vertical parts of the model
- Thus, elimination/enriching of points is needed to avoid a presence of long dangled triangles
- Fig. (b) illustrates the result of implementation of a novel statistical approach for mesh enhancement and siplification
Examples - Surface Smoothing
- Comparison of CSRBF smoothing and Laplacian smoothing.
- (a) Original noisy sphere “Epcot” model, (770 vertices, 1536 polygons); (b) Smoothed model after 5 iterations based on 11-point interpolation. Processing time: 0.6 s.
- Noisy sphere “Epcot” model after processing with Laplacian smoothing
Examples - Surface Smoothing with CSRBFs
- (a) Original “Stanford Bunny” model (35947 vertices)
- (b) Smoothed model after one iteration based on 11 points interpolation (processing time 4.7 sec)
- (c) Smoothed model after one iteration based on 5 points interpolation
- (d) Smoothed model after 5 iterations based on 5 points interpolation
- (a) Original “ballJoint” model (Cyberware Inc, 34267 vertices)
- (b) Smoothed model after one iteration based on 11 points interpolation (processing time 4.1 sec)
Surface simplification
- It is useful to have various simpler versions of original complex models
- A tremendous number of very sophisticated algorithms have been invented to obtain a simplified model
- The space mapping technique can serve for surface simplification
- Bending energy metric can be exploited as a parameter to select candidate points for edge collapse transformation
- We propose using the bending energy htA-1h as an error/quality cost to select candidates for an edge collapse. The approach is based on the use of displacements of N control points as the difference between the initial and final geometric forms. The central point of a star polygon is considered as a point that can slide to the neighboring points
Examples - Surface simplification with RBFs
- Visual results for the Horse model
- (a) - 96966 polygons (b) - 50% (c) - 30% (d) - 10% (e) - 3%
- (a) The modified “Stanford Bunny” model, simplified according to the Hoppe algorithm (30% of original data, processing time - 158.989 sec),
- (b) Simplified model (30%) by using simple geometric error metric
- (c) Simplified model according to our approach (30%, processing time - 59.737 sec)
Improvement of mesh quality using a statistical approach
- If a mesh is created for FEM applications, it is very important to control the mesh gradation smoothness. Shape elements have a strong influence on discretization errors
- Thus, our main premise in creating a mesh refinement algorithm based on a statistical approach is that a more uniform (homogeneous) mesh in the sense of the distribution of elements in a histogram is more amenable to numerical calculations
- Statistical approach provides some latitude in the choice of point placement, allowing softer “transformations” of polygons to be produced
- (a) Fragment of an initial mesh (The “Horse” model)
- (b) After improvement
Examples - Improvement of mesh quality using a statistical approach
- (a) Fragment of the mesh after simplification (13967 triangles)
- (b) Fragment of the mesh after statistical processing
Simplification based on the use of a statistical approach and the bending energy htA-1h as an error/quality cost
- (a) “Ball-Joint” model. 68530 polygons. (b) After simplification. 7320 polygons. (c) After simplification with preserving of sharp features.
- Wire-frame fragments:
- (a) Simplified mesh of the “BallJoint-25” model.
- (b) After simplification with preserving of sharp features.
Examples. Simplification and improvement of mesh quality using a statistical approach and the bending energy
- (a) Fragment of the initial “Horse” mesh
- (b) Fragment of the mesh after statistical processing
- (c) Fragment of the mesh after simplification (40% of the initial mesh)
Examples. Simplification and improvement of mesh quality using a statistical approach and the bending energy
- (a) Original “horse” model (96966 triangles) and a luminosity histogram,
- (b) Simplified model produced accordingly to the use of the bending energy (10% of the original number of triangles) and the luminosity histogram,
- (c) Simplified model produced accordingly to the use of the bending energy and the statistical approach (9% of the original number of triangles) and the luminosity histogram,
- (d) Simplified model produced accordingly to the use of the bending energy and the statistical approach (3% of the original number of triangles) and the luminosity histogram.
