Image analysis – Image transformation or preprocessing – Mapping 2-d image onto a 3-d surface
Reexamination Certificate
2000-07-28
2004-02-24
Mehta, Bhavesh M. (Department: 2625)
Image analysis
Image transformation or preprocessing
Mapping 2-d image onto a 3-d surface
C708S270000, C716S030000
Reexamination Certificate
active
06697538
ABSTRACT:
BACKGROUND OF THE INVENTION
In general, the present invention relates to computer vision and image processing and more specifically, the mapping of an image stored in a computer-teadable format onto a computer generated surface. Basic techniques have been proposed for the limited purpose of obtaining a flattened computerized image or representation of a convoluted surface such as the cortical surface of a mammalian brain to aid in the visualization of this very complex structure. The computer graphics process known as ‘texture mapping’ has been limited to the process of transferring a two-dimensional image (a digitized photograph or artificially produced picture or graphic) onto a three-dimensional computer generated surface. However, here more particularly, the invention relates to a novel technique of conformally mapping any image that can be stored in a computer-readable format as two-, three-, or four- dimensional dynamic coordinate data, onto two-, three-, or a dynamically-varying family of surfaces, respectively. Of particular interest is the application of a very unique flattening function (derived from numerically approximating a selected partial differential equation, PDE) to surface data of an original digitized image. The surface data can comprise triangulated or other-shaped elements, cells, patches, segments, or portions thereof, that has been extracted as necessary from original image data to remove handles, holes, self-intersections, and so on, to produce a generally smooth manifold upon which the flattening function can be performed to produce a flattening map. The novel computerized mapping technique of the invention generally preserves angles of the original image as mapped and the mapping performed is bijective (onto and one to one).
Prior techniques have been proposed to obtain a flattened/planar representation of a ‘real’ object such as the complex cortical surface of a human brain (as described in the medical imaging literature): The collaborators hereof have addressed limitations to these proposed/known methods. For example, in one published paper the scientific authors fit a parameterized deformable digitized surface of a brain whose topology is mappable onto a sphere and then ‘flatten’ the sphere to create a planar map by using spherical coordinates. In other algorithms proposed for the purpose of ‘flattening’ a brain image using quasi-isometrics and quasi-conformal flattenings approaches, the scientists started with a triangulated representation of a given image of the cortical surface and employ a relaxation method to discretely minimize an energy functional. Unfortunately, bijectivity cannot be preserved in when using this prior approach, and in particular, there is a chance that the tiny triangle shaped patched of the triangulated surface will flip during the quasi-isometrics or quasi-conformal process (and if any number of the tiny triangles do flip, the resulting flattened representation is not true and, in fact, can become quite distorted). By way of review for reference, bijection occurs when: A mapping f from a set A onto a set B which is both an injection and a surjection; that is, for every element b of B there is a unique element a of A for which f(a)=b.
The technique of the invention is very distinguishable from prior mapping approaches. As it will be better appreciated, the novel technique provides for an explicit construction of the bijective conformal equivalence on a continuous model of the surface being mapped, and only then is there a move toward the discrete implementation. Key and surprising differences include: the flattening function performed according to the invention is preferably obtained as the solution of a selected second-order elliptic partial differential equation (PDE) on the surface to be flattened; this selected PDE can then readily be approximated using a finite element approximation on a triangulated (or other discrete-element shaped) representation of the original surface; and further, the novel flattening function performed on a constructed set of surface data (extracted and/or smoothed to reduce effects of aliasing, as desired), can be derived from identifying solutions to two sparse systems of linear equations treated using the conjugate gradient method. In the event the surface on which the flattening function of the invention is performed is constructed as a triangulated surface, one can use a known fast segmentation method to adequately represent the iso-surface as a triangulated surface of a digitized synthetic or ‘real’ image of an object (e.g., a cortical surface).
The ingenious technique described herein produces a mapped image, or a series of dynamically-mapped sequential images that represent changes in time to the original image (for example, a beating human heart or when digestive byproduct moves through a colon causing the colon surfaces to change shape), that locally preserves certain important characteristics (shape and angles) of each original image. And, unlike known prior graphics flattening algorithms or computerized graphics texture mapping processes, original image characteristics are preserved even when the surface on which the image is being mapped is quite different from the surface of the original image (for example, an image of brain white matter conformally maps quite clearly onto a sphere and a spherical image of the earth is readily mapped onto an odd synthetic ‘blob’ shape). The instant invention permits the construction, among other things, of a bijective conformal equivalence from a given image's surface (generally having holes or handles removed) onto a sphere, onto a planar domain (such as a rectangle), or onto other preselected two-, three-, or a dynamically-varying family of surfaces. The conformal flattening process maps the surface in a manner that preserves both the angles of and, locally, the shape of the original image.
In order to more-fully understand the invention, details of the rigorous mathematical analysis which has been done of this new technique follow along with illustrations of several conformal mapping examples. As one will better appreciate, one can employ graphics coloring techniques after performing the novel method of producing a flattening map of complex, time-varying digitized surfaces to create beautiful motional graphical images for a wide variety of applications—from medial diagnostic, product and process design, to pure entertainment. The flattening function and conformal mapping can be performed, with an original digitized image stored as two-, three-, and four-dimensional dynamic coordinate data onto a two-, three-, plus the dynamically-varying family of surfaces—thus, the invention is not limited to 2-D (planar) to 3-D texture mapping of still images.
SUMMARY OF THE INVENTION
It is a primary object of this invention to provide a computerized apparatus and associated method and program code on a storage medium, for producing a flattening map of a digitized image, whether this image is initially synthetically produced as discrete data (for example, a computer generated graphic) or originates as quasi-discrete image data of a real object (e.g., produced as a result of a digital photo, an x-ray, diagnostic scan, document scan, and so on)—and whether the original image data is stored as two-, three-, or four-dimensional dynamic coordinate data. The flattening map can be conformally mapped onto a computer generated surface (whether 2-D, 3-D, or any dynamically-varying family of surfaces) that can be displayed on a computer-assisted display apparatus in communication with a processor. The apparatus and associated method and program code include constructing a first set of data comprising a plurality of discrete surface-elements to represent at least a portion of a surface of the digitized image, and performing a flattening function on the first set of data to produce the flattening map.
Certain advantages of providing the novel computerized apparatus and associated new method and program code stored on a computer readable storage medium, as d
Angenent Sigurd B.
Haker Steven
Kikinis Ron
Tannenbaum Allen R.
Macheledt Bales LLP
Mehta Bhavesh M.
Patel Kanji
Wisconsin Alumni Research Foundation
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