A Brief Tour of Nonlinear Magnification

Written by T. Alan Keahey

Sponsored by:
Visualytics, the Visual Analysis Company

Overview

There are many different methods for producing nonlinear magnification, pointers to in-depth discussion of these methods can be found from the nonlinear magnification home page. This non-technical introduction simply shows some of the visual effects which can be achieved by applying nonlinear magnification to a regular grid like the one shown here (the FAD toolkit was used to construct the specific examples on this page).

Nonlinear magnification has advantages over traditional (linear) magnification techniques in that it allows for non-occluding in-place magnification. This provides a region of increased resolution while still preserving a view of the global context.

1D Transformations

As mentioned before, there are many different methods for producing nonlinear magnification. One of the simplest and most common methods however, uses a simple one dimensional function (like the one shown here) as the starting point for performing more sophisticated magnification in higher dimensions.

The function shown here is the hyperbolic tangent function tanh(x), although many other similar functions could also be used. The key requirement is to have a varying slope on the function so that some ranges with have slope greater than one (areas of magnification) and other areas will have slope less than one (areas of demagnification or minification). We can add a variable parameter to this 1D function which controls the shape and slope of the function, such as tanh(b x).

Basic 2D Transformations

Given a suitable 1D function like the one above, it is a simple matter to apply this 1D function to 2D spaces. The image on the left shows the effect of applying the 1D function to the x and y axes independently, this is called the orthogonal transformation. In the right image you see the radial transformation, which is the result of applying the 1D function to the radius component of the polar coordinates of the grid points.

Constrained Transformations

You can also constrain the domain of a transformation so that only certain regions of your visual domain are affected. This gives the advantage of helping to preserve a static, unmoving global context. The effect is similar to moving a magnifying glass over a region.

Combining Linear and Nonlinear Magnification

One of the disadvantages of nonlinear magnification is that it produces distortion at the area of highest magnification. Traditional (linear) magnification does not have this problem however, and results only in a constant scaling of the object being magnified. It is possible to combine the best properties of linear and nonlinear magnification within a single magnification routine, as shown below. The image on the left shows this combination in an infinite (unconstrained) domain. The figure on the right illustrates how this effect can be used within a constrained domain transformation

Multiple Transformations

It is possible to combine multiple transformations in a number of different ways. Each transformation is independently controlled, although several different effects can result from combining them in different ways. The image on the left shows the result of taking the average of multiple transformations. The middle image shows a technique which computes voronoi regions between the transformations, and then each transformation function affects only those points which are within its corresponding voronoi region. The rightmost image shows the result of compositing several transformations in succession, which produces an effect not unlike a stack of overlapping magnifying lenses.

Other Methods

Of course there are many different methods for producing transformations similar to the ones shown here. For example, some techniques use perspective projections and graphics hardware to map 2D spaces onto manifolds which are projected back into 2D screen coordinates. Other techniques involve N-Dimensional piecewise linear functions to produce efficient transformations of near-arbitrary complexity. A fuller description of these and other techniques is available from the literature and the nonlinear magnification home page.

Application: Image Magnification

A commonly used technique with nonlinear magnification involves a straight-forward application of traditional graphics techniques. Texture mapping can be used to distort images simply by mapping image coordinates to the transformed coordinates of a regular grid. An example of this technique is shown below.