AbstractIntroductionGoalsAchievmentsDeliverablesIndividual ContributionsLessons LearnedAppendix- 1 - Three-Dimensional Data Recovery Using Image-Based Modeling Jeremy W. Cannon Jonathan C. Derryberry Vitaly Y. Kulikov [email protected] [email protected] [email protected] 6.837: Introduction to Computer Graphics MASSACHUSETTS INSTITUTE OF TECHNOLOGY Final Project Report—Team 13 December 6, 2002 Abstract Extraction of a three-dimensional model from a set of two-dimensional projections is a well-known problem of contemporary computer science. Termed image-based modeling, solutions to this problem have a number of practical applications ranging from virtual tours and image recognition to generation of physical models from image data. However, this problem remains the subject of active research as it has not yet been solved in the general case. Although this general case has proven very challenging, there are certain special cases where a satisfactory solution can be achieved with minimal human intervention. The following report describes our approach to a general solution to the problem of inferring geometric information from a photographic image. A detailed description of our algorithm and the method of implementation are provided along with sample results demonstrating the capabilities of this approach. I. Introduction Since the initial work of Horn in 1970 [1], the use of photograph images for constructing physical models has evolved into a range of new disciplines in the fields of both computer graphics and computer vision. This classical work has been termed “shape from shading” as it uses the reflectance equation (1) to relate image brightness, I to the surface normal, N: (,) ( )NLIRpqρ==⋅ (1) where R(p,q) is the reflectance function in terms of the surface gradient, ρ is the composite albedo, and L is the light source direction. To derive the surface normals, the radiosity at a point P on the surface of the object is given by (2): () ()()NLBP P Pρ=⋅ (2) where ρ(P) is the surface albedo, N is the surface normal, and L is the light source vector. Assuming the camera response is linear with respect to the surface radiosity, the intensity value of each pixel can be written as (, ) (, )(, ) (, )(, )NLgVIxy kBxykxy xyxyρ==⋅=⋅ (3) where k is the constant relating camera response to surface radiance thereby making V a vector containing elements related to both the scene lighting and the camera. Although the surface normal is not uniquely- 2 - determined in this expression, it can be obtained by assuming a convex surface. Because N is a unit normal, ρ(x,y) is simply the 2-norm of the surface vector g(x,y). Thus, N can be found as 21(, ) (, )(, )Nggxyxyxy= (4) A surface model can then be determined from this reference normal by recognizing that the normal can also be written as a homogeneous vector: 2211(, ) 1TNffxyffxyxy∂∂++∂∂∂∂=−−∂∂ (5) where f(x,y) is the equation for the parameterized surface which can then be integrated over x and y to yield the final model. Subsequent work by Chen and Williams, McMillan, Debevec, and others has spawned the fields of Image-based Modeling and Rendering (IBMR) which seeks to enhance the realism of computer graphics scenes by extracting environmental information about a scene from photographs [2,3]. This environmental information typically goes far beyond derivation of realistic geometry to include new approaches to visibility, modeling view-dependent variations in the appearance of materials, and the extraction of more accurate lighting models for complex scenes [3]. Indeed, many of these new approaches to model generation view photographic images as measurements which can inform the realism of any given scene. Although the general concepts of shape from shading have been studied for decades, this field remains quite active as a research discipline due to the wide range of complex issues that have been uncovered as research in this field has progressed. Examples of these complex issues include variable albedo within an object which confounds the relationship expressed in Equation 1 [4], interreflections which lead to dramatically different appearance from that predicted by local lighting models [5], and ambiguous geometries which cannot be resolved based on shading alone [6]. In summary, a general solution to the complete extraction of three-dimensional geometric and environmental data has not been described in part because of the complexity of this problem and in part due to the diversity of subject matter and modeling objectives held by those employing the techniques of IBMR. II. Goals 1.1 Image-based Modeling A variety of techniques for solving image-based modeling problems have been developed since the original methods described by Horn [1,7]. More recent techniques include using an array of silhouettes to reconstruct object geometry, using surface curves from object profiles to create the model geometry, and using stereoscopic imaging to extract a so-called “depth map” of the object [8]. In this project, we aimed to reconstruct a graphical model of physical objects based on photographic images of the object. Our goals for this phase included: • Implementing an algorithm that identifies the boundaries of the two-dimensional projection of the model and that ensures that the RGB values of pixels within the boundaries of the projection are “smooth” functions of their position. • Implementing an algorithm that, given a reference normal or normals, scans the area within the boundaries of the preprocessed projection and restores the depth and the normal direction at each vertex of the generated 3D model..- 3 - • Implementing an algorithm that generates a complete model from a set of two or more partial models where each partial model corresponds to one two-dimensional projection (only provided enough time remaining in the term). 1.2 Using the Generated Model Once the model is extracted, it needs to be rendered and, if the results are desirable, exported for use in other applications. Therefore, we needed to develop a flexible interface for rendering the model as well as include the option to convert the extracted model to a universal format. Our goals for this phase included: • Providing the user with simple tools to control different parameters of the image-based modeling process such as the granularity of the model (i.e.
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