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Figure 15: Several final segmentations. Final segmentations obtained with our method. S. Nuvoli et al. / Automatic Surface Segmentation for Seamless Fabrication Using 4-axis Milling Machines

Figure 15 Several final segmentations. Final segmentations obtained with our method. S. Nuvoli et al. / Automatic Surface Segmentation for Seamless Fabrication Using 4-axis Milling Machines

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Figure 1: Our processing pipeline in a nutshell (left to right): we first compute the rotation axis on the input mesh; we then partition the mesh in millable height-field portions (including the top and bottom ones) taking also into account the information of the saliency map; we compute a milling sequence and fabricate the object using the 4-axis milling machine; we clean up the result to obtain the final real object.
Figure 1: Our processing pipeline in a nutshell (left to right): we first compute the rotation axis on the input mesh; we then partition the mesh in millable height-field portions (including the top and bottom ones) taking also into account the information of the saliency map; we compute a milling sequence and fabricate the object using the 4-axis milling machine; we clean up the result to obtain the final real object.
Figure 3: Fabrication process outline. We first carve the set of height-fields on the side of the model from the raw block of material (a); then, we unpin the block from the rotation tool, and we mill the two height-fields (top and bottom) (b). Notice that we use a mold (colored with lighter grey) to keep the model correctly in place for milling the second between top and bottom. Our fabrication approach assumes that at least one of the two extremes of the stock is a flat base; this is a reasonable requirement since stocks used for 4-axis machining are generally cylinders. The top and bottom regions’ manufacturing is simple 3-axis milling passes: the first one is performed by fastening the block on its flat base. In contrast, for the second, a mold is used to properly place the block and to guarantee a proper alignment in the process. Note that, without these two phases, we would not be able to carve a portion of the block. Fig. 3 illustrates the entire process.
Figure 3: Fabrication process outline. We first carve the set of height-fields on the side of the model from the raw block of material (a); then, we unpin the block from the rotation tool, and we mill the two height-fields (top and bottom) (b). Notice that we use a mold (colored with lighter grey) to keep the model correctly in place for milling the second between top and bottom. Our fabrication approach assumes that at least one of the two extremes of the stock is a flat base; this is a reasonable requirement since stocks used for 4-axis machining are generally cylinders. The top and bottom regions’ manufacturing is simple 3-axis milling passes: the first one is performed by fastening the block on its flat base. In contrast, for the second, a mold is used to properly place the block and to guarantee a proper alignment in the process. Note that, without these two phases, we would not be able to carve a portion of the block. Fig. 3 illustrates the entire process.
Figure 2: Our fabrication setup. The milling tool on the left has three degrees of freedom in the operational field of the machine; the rotating shaft adds the fourth one. A 4-axis CNC machine extends the classical 3-axis CNC device with an extra degree of freedom corresponding to the object’s rota- tion around an axis. In a typical manufacturing setup, as illustrated in Fig. 2, the carved volume (called stock) is fixed to a rotating shaft through a solid flat base. In this kind of setup, manufacturing the two extremes of the object is not trivial since both extremes of the stock have to be robustly fixed to the rotating shaft during the whole fabrication, as shown in Fig. 2. For security reasons and to ensure fabrication stability, the drill bit should not approach the shafts. Thus, the fabrication of a 3D object starting from a solid block of material must involve at least three phases: while we can fabricate the large majority of a shape by exploiting the different milling directions provided by the rotation axis, two additional fabrication phases are necessary to carve the top and bottom regions.
Figure 2: Our fabrication setup. The milling tool on the left has three degrees of freedom in the operational field of the machine; the rotating shaft adds the fourth one. A 4-axis CNC machine extends the classical 3-axis CNC device with an extra degree of freedom corresponding to the object’s rota- tion around an axis. In a typical manufacturing setup, as illustrated in Fig. 2, the carved volume (called stock) is fixed to a rotating shaft through a solid flat base. In this kind of setup, manufacturing the two extremes of the object is not trivial since both extremes of the stock have to be robustly fixed to the rotating shaft during the whole fabrication, as shown in Fig. 2. For security reasons and to ensure fabrication stability, the drill bit should not approach the shafts. Thus, the fabrication of a 3D object starting from a solid block of material must involve at least three phases: while we can fabricate the large majority of a shape by exploiting the different milling directions provided by the rotation axis, two additional fabrication phases are necessary to carve the top and bottom regions.
Figure 4: Effect of axis selection. The choice of the axis in (a) generates height-fields of varying depth and size, thus complicating tool-path creation, the axis shown in (b) produces height-fields of similar size and depth, even if their number is reduced by one; moreover, (b) has better top and bottom regions to three-phase machining. Axis (a) is the principal inertia axis, while axis (b) is the axis selected by the optimization method of Sec.3.2.2. S. Nuvoli et al. / Automatic Surface Segmentation for Seamless Fabrication Using 4-axis Milling Machines
Figure 4: Effect of axis selection. The choice of the axis in (a) generates height-fields of varying depth and size, thus complicating tool-path creation, the axis shown in (b) produces height-fields of similar size and depth, even if their number is reduced by one; moreover, (b) has better top and bottom regions to three-phase machining. Axis (a) is the principal inertia axis, while axis (b) is the axis selected by the optimization method of Sec.3.2.2. S. Nuvoli et al. / Automatic Surface Segmentation for Seamless Fabrication Using 4-axis Milling Machines
Figure 5: Filtering and decomposition into regions given an axis. (a) Orig- inal model. (b) Smoothed model with top height field in blue, bottom height field in red, and side region in gray, S. Nuvoli et al. / Automatic Surface Segmentation for Seamless Fabrication Using 4-axis Milling Machines
Figure 5: Filtering and decomposition into regions given an axis. (a) Orig- inal model. (b) Smoothed model with top height field in blue, bottom height field in red, and side region in gray, S. Nuvoli et al. / Automatic Surface Segmentation for Seamless Fabrication Using 4-axis Milling Machines
Figure 6: Saliency field computation. We show how we generate mesh saliency for the Moai model. The salience values are between 0 (red), and 1 (blue). We also support user-driven saliency determination by letting the user interactively select/unselect with a brushing tool the areas of interests (Fig. 6d). This makes it possible, in particular, to have users mark semantically essential regions.
Figure 6: Saliency field computation. We show how we generate mesh saliency for the Moai model. The salience values are between 0 (red), and 1 (blue). We also support user-driven saliency determination by letting the user interactively select/unselect with a brushing tool the areas of interests (Fig. 6d). This makes it possible, in particular, to have users mark semantically essential regions.
Figure 7: Effect of the saliency term on the segmentation. From left to right: segmentation without considering saliency, final saliency values, seg- mentation with saliency. Once we have obtained the decomposition for each chart, we rein- troduce the high-frequency details lost during pre-filtering (Sec. 3.1)
Figure 7: Effect of the saliency term on the segmentation. From left to right: segmentation without considering saliency, final saliency values, seg- mentation with saliency. Once we have obtained the decomposition for each chart, we rein- troduce the high-frequency details lost during pre-filtering (Sec. 3.1)
S. Nuvoli et al. / Automatic Surface Segmentation for Seamless Fabrication Using 4-axis Milling Machines
S. Nuvoli et al. / Automatic Surface Segmentation for Seamless Fabrication Using 4-axis Milling Machines
Figure 9: Occlusion effects. Black faces represent triangles no longer visi- ble from the spindle after restoring the high frequencies. by using Laplacian surface reconstruction framework [Sor06]. As in Muntoni et al. [MLS*18], we constrain the reconstruction with height-field constraints that ensure fabricability, without taking into account that high-frequency detail reintroduction could cause some triangles to be no longer visible due to slight occlusions (Fig. 9). Again, we sacrifice the exact fidelity of the shape to simplify the pro- cess. The tool-path generation software will automatically produce the closest fabricable shape.
Figure 9: Occlusion effects. Black faces represent triangles no longer visi- ble from the spindle after restoring the high frequencies. by using Laplacian surface reconstruction framework [Sor06]. As in Muntoni et al. [MLS*18], we constrain the reconstruction with height-field constraints that ensure fabricability, without taking into account that high-frequency detail reintroduction could cause some triangles to be no longer visible due to slight occlusions (Fig. 9). Again, we sacrifice the exact fidelity of the shape to simplify the pro- cess. The tool-path generation software will automatically produce the closest fabricable shape.
Figure 10: 2D representation of how we generate the surface (called walls) for embedding the chart surface into the stock. (a) Limitations of a surface that is parallel with respect with the fabrication direction. (b) Surface that form a given angle guarantees fabricability and manufacturing quality. Note that, in our examples, we used a cylinder as stock to maxi- mize the final result size and minimize scrap material, but any other shape is allowed.
Figure 10: 2D representation of how we generate the surface (called walls) for embedding the chart surface into the stock. (a) Limitations of a surface that is parallel with respect with the fabrication direction. (b) Surface that form a given angle guarantees fabricability and manufacturing quality. Note that, in our examples, we used a cylinder as stock to maxi- mize the final result size and minimize scrap material, but any other shape is allowed.
Figure 11: Collision between walls and other charts. (a) The oblique walls of the chart Cy collide with a portion of the surface of another chart C, (highlighted in red). In this case, the red area would be unintentionally removed when performing the milling of Cz and, therefore, fabricated from a wrong direction. (b) Milling C, before Cz solves the issue, because the remaining volume to be removed (lighter blue) does not contain the actual wall of Co.
Figure 11: Collision between walls and other charts. (a) The oblique walls of the chart Cy collide with a portion of the surface of another chart C, (highlighted in red). In this case, the red area would be unintentionally removed when performing the milling of Cz and, therefore, fabricated from a wrong direction. (b) Milling C, before Cz solves the issue, because the remaining volume to be removed (lighter blue) does not contain the actual wall of Co.
Figure 12: Set of meshes generated by our method. On top, the stocks (with the part that will be milled in a lighter color), and on bottom the resulting shape. At each step, the resulting shape will be rotated and used as stock for the next step. The ordering of the charts is computed in order to minimize the area fabricated from a wrong direction. Our results leave intentionally small portions of waste material that can be removed manually after the fabrication. These small portions lie between adjacent charts, and the precision of the milling machine determines their thickness. To save fabrication time, we remove these portions manually at the end of the milling process. Note that we could avoid them by increasing the angle parameter used to generate the chart walls.
Figure 12: Set of meshes generated by our method. On top, the stocks (with the part that will be milled in a lighter color), and on bottom the resulting shape. At each step, the resulting shape will be rotated and used as stock for the next step. The ordering of the charts is computed in order to minimize the area fabricated from a wrong direction. Our results leave intentionally small portions of waste material that can be removed manually after the fabrication. These small portions lie between adjacent charts, and the precision of the milling machine determines their thickness. To save fabrication time, we remove these portions manually at the end of the milling process. Note that we could avoid them by increasing the angle parameter used to generate the chart walls.
Figure 13: Fabrication sequence. The first row shows the initial stock (a) and target shapes for side region (b, c, d, e). Each target shape is rotated properly and used as stock for the next step. The second row shows the resulting mesh after 4-axis fabrication and the target shapes for the bottom (g) and top (h) regions.
Figure 13: Fabrication sequence. The first row shows the initial stock (a) and target shapes for side region (b, c, d, e). Each target shape is rotated properly and used as stock for the next step. The second row shows the resulting mesh after 4-axis fabrication and the target shapes for the bottom (g) and top (h) regions.
S. Nuvoli et al. / Automatic Surface Segmentation for Seamless Fabrication Using 4-axis Milling Machines Figure 14: Fabricated models. For the four fabricated models we show here (left to right): the initial shape; its saliency map; the two ends (in red and blue) displayed on the smoothed model; the decomposition of the model in height-fields; the fabricated object (in PVC for the kitten and the Buddha models, and wood for the batman and David models).
S. Nuvoli et al. / Automatic Surface Segmentation for Seamless Fabrication Using 4-axis Milling Machines Figure 14: Fabricated models. For the four fabricated models we show here (left to right): the initial shape; its saliency map; the two ends (in red and blue) displayed on the smoothed model; the decomposition of the model in height-fields; the fabricated object (in PVC for the kitten and the Buddha models, and wood for the batman and David models).
Table 1: Model Statistics. The first column shows the number of faces (NF), the next four columns (S, saliency; O optimal orientation; A face associ- ation; M fabrication sequence and mesh generation) list the timing of the computation, in the last column we report the number of obtained charts (NC). Times are in seconds. © 2021 The Author(s) Computer Graphics Forum © 2021 The Eurographics Association and John Wiley & Sons Ltc
Table 1: Model Statistics. The first column shows the number of faces (NF), the next four columns (S, saliency; O optimal orientation; A face associ- ation; M fabrication sequence and mesh generation) list the timing of the computation, in the last column we report the number of obtained charts (NC). Times are in seconds. © 2021 The Author(s) Computer Graphics Forum © 2021 The Eurographics Association and John Wiley & Sons Ltc
Table 2: Model dimensions. We report, for the four fabricated models, the size in millimeters of the target model and the initial stock. Figure 16: Finishing phase. Side regions finished with a Imm ball cutter.
Table 2: Model dimensions. We report, for the four fabricated models, the size in millimeters of the target model and the initial stock. Figure 16: Finishing phase. Side regions finished with a Imm ball cutter.
Figure 17: Fabrication phases.. Manufacturing the first of the two ends of the model (a), then the second end of the model (c) using its mold (b). Theoretically valid segmentations may not be practically feasible.
Figure 17: Fabrication phases.. Manufacturing the first of the two ends of the model (a), then the second end of the model (c) using its mold (b). Theoretically valid segmentations may not be practically feasible.
Figure 18: Fabricated model before and after the final sanding pass. In (a) the model has still part of the stock that, initially, was surrounding it; in (b) only the portion that is to be sanded is present. in (c), the final polished model, using the mold as a base S. Nuvoli et al. / Automatic Surface Segmentation for Seamless Fabrication Using 4-axis Milling Machines
Figure 18: Fabricated model before and after the final sanding pass. In (a) the model has still part of the stock that, initially, was surrounding it; in (b) only the portion that is to be sanded is present. in (c), the final polished model, using the mold as a base S. Nuvoli et al. / Automatic Surface Segmentation for Seamless Fabrication Using 4-axis Milling Machines
Figure 20: Practical limitations. The fabrication of this shape has two problems: many small non-reachable charts and two little opposite extremes. We could solve the first by sacrificing fidelity on the non-visible surface. Still, the second issue could lead to instability during the object’s manufacturing. The shape shown in Fig. 20 does not have two substantial portions of the surface that we can pick as opposite extremes. Our approach finds two small regions (the bottom of one leg, in red, and the trunk tip, in blue) that, depending on the material used for the manufac- turing and the final object’s size, could easily lead to instability fabrication. Side regions must be wide enough to sustain the entire object weight during the whole manufacturing process. We could ad- dress this problem by just adding a term in the optimization problem described in Sec. 3.2.2 that penalizes the formation of small top and bottom areas. Alternatively, we can discard the axes that generate top and bottom regions smaller than a certain threshold. However, we cannot guarantee to have a suitable solution for the general case. In general, for this kind of shape, single-block fabrication on 4-axis machines is challenging, if not impossible, and it would be ideal for splitting the model into multiple components that can be fabricated separately and assembled afterward.
Figure 20: Practical limitations. The fabrication of this shape has two problems: many small non-reachable charts and two little opposite extremes. We could solve the first by sacrificing fidelity on the non-visible surface. Still, the second issue could lead to instability during the object’s manufacturing. The shape shown in Fig. 20 does not have two substantial portions of the surface that we can pick as opposite extremes. Our approach finds two small regions (the bottom of one leg, in red, and the trunk tip, in blue) that, depending on the material used for the manufac- turing and the final object’s size, could easily lead to instability fabrication. Side regions must be wide enough to sustain the entire object weight during the whole manufacturing process. We could ad- dress this problem by just adding a term in the optimization problem described in Sec. 3.2.2 that penalizes the formation of small top and bottom areas. Alternatively, we can discard the axes that generate top and bottom regions smaller than a certain threshold. However, we cannot guarantee to have a suitable solution for the general case. In general, for this kind of shape, single-block fabrication on 4-axis machines is challenging, if not impossible, and it would be ideal for splitting the model into multiple components that can be fabricated separately and assembled afterward.
Figure 19: Failure case due to intrinsic limitations of 4-axis milling. In the depicted cup-shaped surfaces, faces in black are non-visible from any possible direction.
Figure 19: Failure case due to intrinsic limitations of 4-axis milling. In the depicted cup-shaped surfaces, faces in black are non-visible from any possible direction.
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