Functionally Based Virtual Computer Art Alexei Sourin School of Computer Engineering Nanyang Technological University Nanyang Avenue, Singapore 639798 +65-790-4292
[email protected] ABSTRACT This article describes how virtual embossing and wood cutting can be done using the function representation of a shape and tools. The software is implemented as an interactive shape modeler where a functional model of the shape is subsequently modified with offset and set-theoretic operations. For visualization, interactive ray tracing is used. Bounding boxes together with the spatial organization of the functional model provide the required fast function evaluation that is usually a bottleneck for functionally based shape modeling systems. The software runs on a personal computer.
Categories and Subject Descriptors I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling - Curve, surface, solid, and object representations; I.3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism – Raytracing, Virtual reality; I.3.8 [Computer Graphics]: Applications.
Keywords Computer art, embossing, carving, virtual reality, functionally based shape modeling.
1
INTRODUCTION
In general, computer art may not necessarily mean that an exact simulation of the original art is to be performed. With the computer, some absolutely new forms, shapes, and expressive techniques, not available in real life, can be created. Also, to simulate the existing artistic techniques, the computer often does the whole job for the artist. There are projects that propose how to turn a photograph into a watercolor, charcoal, colored pen, oil paint picture, or engraving leaving the user an opportunity of controlling the appearance of the picture by adjusting the control parameters. For example, rendering parametric free-form surfaces in pen and ink is proposed in [24]. Interactive creating pen-andink line style drawings from gray scale images is introduced in [21]. Various artistic effects of watercolor and their automatic simulation are described in [4]. Line art drawing is studied in [5-8,11,19]. Expressive rendering is proposed in [10,12].
In [14], digital facial engraving, which imitates traditional copperplate engraving starting from a digital photo of a person, is introduced. In [3,17], methods of carving and sculpting based on 2D images, where the color of each point defines how high or deep the elevation of the respective 3D point should be, are described. There are also projects and software tools that create embossed-like pictures from the image provided as input data. For example, Ulead PhotoImpact (Ulead Systems, Inc) lets the user make the image appear as if it were stamped or imprinted on a solid surface, and Corel PHOTO-PAINT (Corel Corporation) has the filter that makes the image details appear as ridges and crevices on a flat surface. With these tools, one can manage to transform a photograph into an embossed-like 2D image by varying the photo properties and the command parameters. For all these projects, the user of a program should not necessarily be an artist skilled in the respective art to achieve outstanding results in the computer-assisted art. There is another direction in computer art. This is using a computer as a virtual tool that lets an artist work in the virtual environment exactly like, or at least as close to working in real life as possible, without taking over even for the simplest operations. The work of art is to be created with virtual models of the familiar tools, and familiar real world techniques are to be used. A 3D model rather than just a 2D picture often results in this virtual computer art. The advantages of using a computer are that the shape being created is contained entirely in the virtual space, it can be made from any virtual material, and also there will be no problem with “ordering” as many tools as needed. This is very essential for the arts that require special workshops or other conditions that may be inconvenient to have in modern houses. Various interactive sculpting and carving techniques are proposed in [2,20,23]. Interactive local modifications of the basic voxel shape using a modeled carving tool and set-theoretic operations is proposed in [1,9]. In [13], an interactive modeling CSG technique to form solid objects with curved surfaces as if they were sculpted so that the generated objects look like real wooden sculptures is discussed. In [18], shape modeling tools for interactive carving based in a unified manner on the function representation of the underlying model are proposed. Carving with chisels can also be done interactively with a Wacom graphics tablet with using Wacom PenTools. These tools realistically simulate the depth of penetration of the chisel when a pressure sensitive graphics tablet is used. In this article, one more contribution to making virtual computer art is introduced. In Section 2, making virtual embossing is considered. In Section 3, the considerations are extended to making virtual engraving and wood/linoleum cutting. In Section 4, further issues are discussed and conclusion remarks are made.
2 MAKING VIRTUAL EMBOSSING 2.1 What is embossing? Embossing is the art of decorating metals. In this technique, the ornament is raised from the back of the metal by means of hammers and punches (Figure 1) followed by hammering from the front, which is called chasing. Embossing has been used extensively throughout the history of metalworking. It achieved widespread popularity in Europe during the 16th, 17th, and 18th centuries. One of the places where this art originated from is the Caucasus Mountains. In Georgia and Dagestan, it is traditional folk art counting hundreds of years back (Figure 2). For more embossed pictures, refer to [25].
2.2
This project aims to simulate the process of embossing so that it can be done interactively with virtual tools on a virtual sheet of metal with photo-realistic appearance. The author wanted to develop a software tool that would be able to run on a common PC so that one would be able to do embossing virtually everywhere, even while flying in an airplane or sitting in the back seat of a car. For realistic modeling of embossing, a direct physically based simulation could be implemented but it promised to be computationally expensive for our purposes. Alternatively, a pseudo-physical simulation of this process could be done entirely geometrically if the appropriate model of geometric deformation can be defined. This model is proposed in 2.3, real-time 3D rendering is discussed in 2.4, and immersion issues are addressed in 2.5.
2.3 Figure 1: Tools for embossing.
Figure 2. Georgian embossing. “Horses” by I.Ochiauri. Embossing is performed on a sheet of metal about 0.3-1.0 mm thick. It can be copper, brass, aluminum, silver or any other soft and flexible metal. The design is first drawn on the surface of the metal, and the motifs are outlined with a tracer (Figure 4), which transfers the essential parts of the drawing to the back of the plate. The plate is then embedded face down in a rubber sheet or an asphalt block, and the portions to be raised are hammered down into this yielding foundation (Figure 9). Next the plate is removed and re-embedded with the face uppermost. The hammering is continued, this time forcing the background of the design into the foundation. By a series of these processes of hammering and reembedding, followed finally by chasing from the front, the metal attains its finished appearance. After that, the coloring with chemicals remains to be done. The author used to emboss when he was a schoolboy. In this article, he recalls his hobby, and simulates embossing geometrically with a computer.
Physics versus geometry
When the design is bossed up from the back, and the background is obtained by beating down the adjacent areas, basically the same physical process occurs. A hand-held punch, a tracer, or a face of a hammer deforms the metal surface plastically either in relief or in intaglio (incising beneath the surface of the metal) so that the imprints look like the tool shape. Due to the ductility of the metal, the imprints have slightly sloping boundaries. The amount of slope varies depending on the force of hammering or pressing.
Functionally based modeling
A functionally based approach to shape modeling has been used in this project. Each tool, a sheet of metal, and a final embossed shape are defined with the Function Representation (F-Rep). To learn about the F-Rep, refer to [16,26,27]. With the F-Rep model, a complex scene can be defined as a union of functionally defined objects where, figuratively speaking, the functions are meant as geometric DNAs constituting shapes and their properties. Each individual function in the F-Rep is an inequality f(x,y,z)≥0 that is greater than zero for the points inside the respective shape, equal to zero on the surface of the shape, and less than zero for the points outside the shape. The resulting function is an inequality as well. It is a superposition of other functions representing shapes and operations over them. The virtual embossing program is implemented as an interactive shape modeler where the functional model of the metal is subsequently modified with offset and/or set-theoretic operations. The final object is represented in the data structure as a binary tree where each node is an operation and the leaves are the tools. Using functional approach to virtual embossing lets us have a 3D model of the embossed shape rather than its 2D image. This model can be converted to polygonal mesh or any other representation with any desired precision for the visualization or rapid prototyping purposes. When doing virtual embossing, the tools are selected from the set of tools with predefined shapes and sizes. Custom made tools can be defined by varying geometric parameters of the tool models. The user acts as if doing real embossing. The user chooses the tool, hits or force down the metal, and observes the result. If the result is not acceptable, a multilevel undo operation can be used-a charming bonus comparing with real embossing where one wrong stroke can ruin hours of work. Let’s see how the functionally based virtual embossing works step by step. We’ll try to re-make virtually the picture that the author embossed many years ago (Figure. 3).
the metal plate (Figure 4). Mathematically, each curve drawn on the surface of the plate is interpolated by segments of straight lines, where for each segment the negative offset operation is applied along the normal to the surface. The function in Figure 5 resembles a profile of the deformed metal and therefore has been used for tracing simulation.
P Contour
ey
P1
Figure 3: “Girl with a candle”, embossing.
ex
First, let’s represent a sheet of metal of size 2w×2h×2d as a thin solid plate by intersecting six half-spaces as follows:
f embossed = f ( x, y, z ) = min( x + w, min(w − x, min( y + h, min(h − y, min( z + d , d − z ))))) ≥ 0
P2
(1)
The intersection operation is implemented with the min function, which is quite common in computer graphics. Implementation of the intersection operation with the R-functions can be used as well but it is computationally more expensive.
Figure 6: Implementation of tracing. Thus, for each segment of a straight line joining two points P1 and P2, and for any arbitrary point P (Figure 6), the following is to be done:
P' = P − P1 , c=
P12 ⋅ e x P12
P12 = P2 − P1 , s=
P12 ⋅ e y P12
c − s , T= s c
P" = P' T f offset (P) =
Figure 4: Real and virtual transferring a drawing to the back of the metal.
a3 2 2 if 0 ≤ P"⋅e x ≤ P" P"⋅e y + a 0 otherwise f embossed = f embossed − f offset ≥ 0
(2)
Offset value
where a is a parameter defining the size and the shape of a tracer. The result of application of such an offsetting operation is illustrated in Figure 7.
Distance Figure 5: The offset function for simulating embossing. The plate is rendered with an optional drawing mapped on top of it to facilitate further work. Alternatively, the drawing can be placed on a graphics tablet that is in fact more convenient. Next, the contours of the drawing are outlined by dragging lines with a mouse or a pen which simulates transferring a motif to the back of
For raising contours up from behind, the positive offset is to be used:
f embossed = f embossed + f offset ≥ 0 Note that Eq.2 neglects the endpoints of the segment. To avoid possible artifacts at the connection points, the same function is to be applied to the endpoints thus surrounding the whole segment.
Let’s come back to the “Girl with a candle”. By applying the offset function (Eq. 2) subsequently to all the segments interpolating the drawing, all the contours have been made (Figure 8).
In real embossing, there are two ways for raising up the relief regions. The plate is to be embedded face down, and the portions to be raised up are either hammered down, or forced down by pressing the metal with the hand-held punch. In both methods, the portions closer to the contour raise first so that the middle region will go up following the boundary regions (Figure 9). If a large region is to be raised up, than the middle portion also will be raised later on by the same means until the desired elevation is achieved (Figure 10).
Figure 7: Tracing drawings with the offsetting.
Figure 9: Raising up the design.
Figure 8: Making virtual “Girl with a candle”: the contours have been traced. Figure 10: The result of bossing up the design. Next comes raising up the relief regions. This also can be modeled with the offset operations and/or the set-theoretic operations over the plate and the shapes representing the tools. In our case, for bossing up the portions of the plate by hammering or pressing it from behind with the punches, the offset is the most appropriate method. For each application of the punch to point P1, the following is to be done for any point P:
P' = P − P1 f offset =
pa
3
2
q P' + a 2
(3)
f embossed = f embossed + f offset ≥ 0 where a, p, and q are parameters defining the size of the affected region and the height of embossing.
In virtual embossing, we follow the same directions. Either individual points where the offset (Eq. 3) is to be applied are selected thus simulating strokes of the punch, or the path of pressing is outlined with a mouse or a pen inside the contour and near it (Figure 11). In the second case, either the offset defined by segments (Eq. 2) will be used, or the points defining the offset (Eq. 3) will be calculated on the path by the program so that the smooth surface will be resulting. Figure 12 illustrates the result of application of such an operation. In this example, the drawing is traced first. Then, the internal area is raised up by moving the virtual punch near the boundary and inside it. Next, the adjoining area is slightly lowered down by moving another virtual punch near the boundary and outside it thus simulating chasing (Figure 11). The order of operations corresponds to real embossing. For raising up large regions, this method may not work efficiently because the number of individual functions contributing to the resultant shape can be quite large. For such cases, one single function for raising up large regions has been introduced. The artist outlines a contour (Figure 13a) following the path of a
punch (Figure 9). Fast inside-outside odd parity rule test is executed for each rendered point. For the points inside the outlined contour, a uniform offset will be applied to the function. For the points lying outside the contour but within a certain distance from it, a non-uniform descending offset will be applied (Figure 13b).
Path of the tool
a) Uniform offset
Distance
Offset value
Offset value
Resulting offset
Bossing up Tracing
The path bossing up.
Descending offset
b)
Chasing The path tracing.
Distance
of Figure 13: A diagram of raising up a large relief region. of
Offsetting “in point”. Offsetting defined by segments.
P = [x
z ],
y
P1 = [x1
y1
z1 ]
f embossed ( x, y, z ) ≥ 0 f tool = f sphere ( x, y, z ) = r 2 − (x − x1 ) − 2
Figure 11: A diagram of raising up the relief portion of the design.
( y − y1 )2 − (z − z1 )2 ≥ 0
(4)
f embossed = f embossed + f tool + 2 2 f embossed + f tool +
a1 f 1 + embossed a2
2
f tool + a 3
2
≥0
where a1, a2 and a3 are the parameters defining the amount of blended material. For other punch shapes, the appropriate function f1 is to be used. Figure 12: Raising up the relief portions of the design with the offsetting. The function defining this offset is similar to Eq. 3 but it takes into account the distance of the point from the polygon’s boundary. A non-uniform elevation of the region can also be defined thus simulating various pressures when rubbing the metal from behind. In that case, the offset value is interpolated on the values defined for different parts of the contour. A union with blending similar to [15] can be used as an alternative to the offsetting when fancy shaped punches are applied. For example, when bossing up the metal with a semispherical punch for each application of the punch to point P1, the following is to be done for any point P:
The benefit of the offset operations comparing to the set-theoretic operations is that with the offsetting, the contours can be made on the relief portions of the design as well as on the flat surface that is very essential for the further phases of embossing. Besides that, the offsetting deforms the plate rather than adds or subtracts the material. The offset operations have a disadvantage that must be acknowledged. A negative or positive offset applied to the plate from the front causes a similar offset from the other side of the plate. To avoid these “mirror” offsets, the Z-coordinate of the points is to be taken into account. The offset and the blending parameters must be functions of the geometric size of the punch and the metal plate properties to achieve pseudo-physical simulation. At last, the background is to be made by beating down the metal with differently shaped hammers and punches. To simulate the
stroke of the punch with a semi-spherical tip, the following offset and/or the set-theoretic operations have been used: Offsetting : f embossed = f embossed − f offset ≥ 0, where f offset is from Eq. 3 Subtraction : f embossed = min( f embossed ,− ftool ) ≥ 0, where ftool is from Eq. 4 Subtraction with blending : f embossed = f embossed − ftool − a1 f 1 + embossed a2
2
− ftool + a 3
2
2 f embossed + (− ftool )2 −
≥ 0, where ftool is from Eq. 4
The process of embossing continues if other contours, relief regions, and background patterns are to be made. Take a look at the final virtual “Girl with a candle” in Fig. 14 and compare it with the original one in Fig. 3. About 850 individual offset operations are involved in its functional model.
Figure 14: “Girl with a candle”, virtual embossing.
Figure 15: “Seashells”. Virtual embossing made with the settheoretic operations and its real clone.
In Figure 15, another virtual embossing and its real copy are presented. Here, only the set-theoretic operations with blending (Eq. 4) have been used for bossing up the relief portions of the picture. Though subtraction and addition of material is more suitable for sculpting rather than for embossing which only deforms the material, they also can produce an attractive result in certain cases. In all the above examples, the flat surface defined by Eq. 1 has been used for embossing. The same mathematical models work with any shape of the work-piece. Just the appropriate function defining the required shape is to be used in place of the one in Eq. 1. The program offers several predefined shapes and lets the user import any shape that may be required. The user can rotate the work-piece to the desired location so that any point on its surface can be reached with the tools.
2.4
Achieving interactivity
For visualization, interactive ray tracing has been used. Assuming a typical model may contain several thousands of shapes/operations, the ray tracing of it can take quite a long time since it implies many function evaluations for each ray cast. The most effective acceleration techniques developed to reduce ray tracing’s high computational cost are based on space coherence: bounding box hierarchies and space subdivision. During common ray tracing, a space subdivision algorithm associates objects with the bounding shapes in which they reside, and tests each ray for intersection only with the objects inhabiting those shapes. To make ray tracing interactive, only the region that has been affected by the most recent application of the tool is to be redrawn. This method ensures the required fast rendering time of the affected regions, and thus provides both interactivity and photo-realism that are so important for our purposes. To estimate the size of the affected region and to detect which tool instances are involved, the bounding boxes for the tools, and the spatial organization of the model are used. The size of the bounding box is about the size of the tool for simple set-theoretic operations. It requires more complicated estimation of the size of the affected region when the offset operations and the set-theoretic operations with blending are used, since the result of these operations expands beyond the region of the direct impact of the tool. The projection of the bounding box onto the viewing plane defines the region of the image on the screen that is to be updated. For the punches with spherical and other symmetrical shapes, axesaligned bounding boxes projecting into axes-aligned rectangles can be used. For the segments interpolating contours, the use of the axesaligned rectangles is not feasible since these segments can be long which may require re-drawing of large regions. Instead, the bounding rectangles aligned with each segment are used for updating the image. Calculation of the initial and final coordinates of each scan line for these rectangles is done using the Bresenham’s line scan conversion algorithm. The coordinates of the ray tracing scan lines are calculated by increasing/decreasing by an integer value the respective coordinate of each pixel representing the segment. The use of the Bresenham’s line scan conversion algorithm ensures the required fast processing time. Yet another way to accelerate the function evaluation for the offset defined by segments is to calculate and store the matrices T from Eq. 2 for each segment to be traced.
The bounding boxes are used not only for defining the region to be ray traced after each individual operation. They, as well as other simple shapes like spheres and cylinders, are also associated with computationally expensive shapes/operations to accelerate the function evaluation when points are located with a mouse or a pen on the surface of the metal. On top of this spatial organization of the functional model, another acceleration structure--a regular grid--is used when a large number of shapes/operations is involved in the model. The bounding boxes together with the spatial organization of the functional model provide the required fast function evaluation that is usually a bottleneck for functionally based shape modeling systems. The final or interim model of the object can be saved for further high quality ray tracing with the de-facto standard POV-Ray program or for later use with other models. The extension of the POV-Ray program [29] capable of visualizing objects represented with isosurfaces has been used in the project. At this point, the desired coloring can be applied simulating both the metal used and the final chemical processing of the picture. Other virtual embossed pictures made for this article are presented in Fig. 16. Over 3000 individual offset operations have been used for making their models. Refer also to the Color Plates Section.
2.5
Achieving immersion
For better immersion, a graphics tablet has been used. The graphics tablet with a pressure sensitive pen lets the author control the depth of embossing almost in the same way as when doing real embossing. The depth of the tool penetration is a function of a pressure. The tip feel is to be customized interactively for each virtual metal. Since the author knows embossing, he simulated it following closely the process of real embossing, and fine-tuned all the respective parameters to achieve eventually the required feeling of “being there”. The Pentium II 300 MHz with the Wacom Intuos graphics tablet has been used for running the embossing program. As for the time spent for making the real embossed pictures and their virtual clones, it was about the same.
3
Figure 16: Virtual embossed pictures.
MAKING VIRTUAL ENGRAVING AND CUTTING
The model and interaction technique discussed in Section 2 can easily be expanded to virtual engraving, linoleum/wood cutting and carving. To engrave is to cut or incise a line. Engraving is always done with a cutting tool, generally by pressure from the hand. It detaches material in cutting. When pressure is applied with a hammer, the process is called carving. Each of these decorative arts can be simulated by applying settheoretic subtraction of a solid object swept by the tool. Depending on the original shape and the carving tool, the swept object can be either substituted by a simple shape like an ellipsoid or a superquadric, or defined functionally like is has been done in [22]. Like in the previous Section, the resultant shape is defined as a superposition of real functions. In Figure 17, one of the author’s virtual woodcuts is presented. In this work, the swept traces of the cutters are subtracted from the functional model of the wooden board. Cutting has been done interactively with the same software that was used for embossing in Section 2. Final rendering has been made with POV-Ray [28].
Figure 17: Virtual woodcut.
4
CONCLUSION AND FUTURE WORK
VR modeling of embossing, as well as other vanishing arts, will let us preserve them, and probably will give them another birth when works of art can be created in the virtual environment, easily duplicated, and exchanged through the Internet. Rapid prototyping would let artists make “physical” copies of their virtual works.
In this article, virtual embossing and wood cutting based on the function representation of the shape and the tools have been described. The program is implemented as an interactive shape modeler where the functional model of the initial shape is subsequently modified with the offset and the set-theoretic operations. The proposed offset functions let us simulate uniformly different embossing operations and provides pseudophysical simulation of the real process. For visualization, interactive ray tracing is used. The program runs on a personal computer. The use of the function representation for modeling lets us provide any desired precision of the simulation and thus avoid artifacts that are usual for polygonal representation. Besides, relatively “small” formulae are more attractive form of representation comparing to thousands of polygons. The use of the bounding boxes together with the spatial organization of the model provided us the required fast rendering time and allowed us to use ray tracing for the VR rendering purposes. Further work will be carried out in several directions. First, implementation of more tool shapes is being planned. Second, the platform independent version of the software will be developed and made available for embossing and cutting enthusiasts. Also, the presented techniques will be expanded to virtual modeling of other decorative arts.
ACKNOWLEDGEMENTS I would like to express my sincere gratitude to my wife Olga and daughter Anna for their help and inspiration.
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Functionally Based Virtual Computer Art Alexei Sourin
