Nonlinear Magnification

Abstract:

In this dissertation we will explore nonlinear magnification as a generalization of the familiar concepts of "fisheye" or "distortion-oriented" views, which produce in-place magnification while preserving visual context. We will formalize this generalization in order to provide a rigorous basis on which to understand and develop spatial nonlinear magnification systems. At the core of our theoretical basis for nonlinear magnification lies the distinction between transformation functions which directly transform the spatial coordinates, and magnification functions which reflect the degree of magnification that is implicit within the transformation. This relationship was previously established for 1D functions, and we will now extend this to functions of higher dimensions. Doing so allows us to quantify the effects of transformations from a wide variety of nonlinear magnification systems, rather than merely observing them as visual or implementational phenomena.

We will examine two new systems for producing nonlinear magnification. The first method uses a RISC-type approach to construct a transformation pipeline composed of sequences of simple and modular 2D transformations. This transformation pipeline is able to produce complex transformations, and is also very efficient computationally. The expressiveness and efficiency of this system is greatly facilitated by the use of piecewise linear functions which reduce complex transformations to simple table look-up operations.

The second method introduces the nonlinear magnification field as a low-level representation based on our formalism of the relation between transformation and magnification functions. We will see how the implicit magnification field of a transformation can be computed, and will also provide an iterative method for constructing a transformation from a specified magnification field. The scalar magnification field representation is particularly amenable to user and program manipulation. Because there are no restrictions of explicit foci or multi-dimensional dependencies, direct specification of the desired magnification values is now possible.

We will also develop a general framework for describing the levels on which nonlinear magnification can be applied, and examine the issue of how to effectively synchronize detail rendering functions to take advantage of the extra space produced by nonlinear magnification transformations.

Available as:

• Postscript (compressed 10.5 MB)
• PDF (1.4 MB)
• This work is Copyright © 1997 T. Alan Keahey. All rights reserved.

Other Publications

T. Alan Keahey