R. Kimmel, N. Sochen, R. Malladi
A general geometrical framework for image processing is presented.
We consider intensity images as surfaces in the 
space.
The image is thereby a two dimensional surface in three dimensional
space for gray level images.
The new formulation unifies many classical schemes, algorithms, and measures
via choices of parameters in a ``master" geometrical measure.
More important, it is a simple and efficient tool for the design
of natural schemes for image enhancement, segmentation, and scale space.
Here we give the basic motivation and apply the scheme to enhance
images.
We present the concept of an image as a surface in dimensions higher
than the three dimensional intuitive space.
This will help us
handle movies, color, and volumetric medical images.
Motivated by Alvarez and Morel,
we consider low level vision as an input to output process.
For example,
the most common input is a gray level image; namely a map from
a two dimensional surface to a
three dimensional space 
. We have
at each point of the xy coordinate plane an intensity I(x,y).
The 
space-feature has Cartesian coordinates
(x,y,I) where x and
y are the spatial coordinates and I is the feature coordinate.
The output of the low level process in most models consists of
1). A smoothed image from which reliable features can be
extracted by local, and therefore differential operators.
2). A segmentation, that is, either a decomposition of the image
domain into homogeneous regions with boundaries, or a set of boundary points
- an ``edge map".
The research on the low level vision process in the retina and the brain indicate the existence of layers serving as operators such that the information is processed locally in the layers and forwarded to the next layer with no interaction between distance layers. This means that the low level vision process can be described by a local differential operator. This process is called scale space where t is the scale (layer) parameter.
There are many definitions for scale spaces of images aiming to arrive at a coherent framework that unifies many assumptions. One such assumption is that ``only isophotes matter''. We argue that this assumption, though leading to many interesting results in many cases, seems to fail in many other natural cases. Let us demonstrate it with a simple example: In Fig. 1 we see two images of a bright square on a darker background.
Two images of a bright square on dark background
In fact, we notice that (see Fig. 2) in the second image the lower left corner of the `bright square' is much darker than the upper right corner of the `dark' background. Furthermore, even the upper right corner of the `bright' square is darker than the upper right corner of the `dark' background. The boundary of the inner square in the left image is closely related to one of the isophotes of the gray level image in that image, as shown in the upper row of Fig. 2. In the second case, we added a smooth function - a tilted plane - to the first intensity function. This additional smooth function might be the result of non-uniform lighting conditions. It is obvious that in the second intensity image (the right image) the isophotes play only a minor role in the perception process of the image. The importance of edges in scale space construction is obvious. Our view is consistent with the rest of the vision community in that boundaries between objects should survive as long as possible along the scale space, while homogeneous regions should be simplified and flattened in a more rapid way. On the other hand, we still want to preserve the geometry and mathematical integrity that results in some interesting non-linear `scale spaces'. Another important question, for which there are only partial answers, is how to treat multi valued images. A color image is a good example since one actually talks about 3 images (Red, Green, Blue) that are composed into one. Should one treat such images as multi valued functions?
We attempt to answer some of the above questions by viewing images as embedding maps, that flow towards minimal surfaces. We consider two dimensions higher than most of the classical schemes, and instead of dealing with isophotes as planar curves we deal with the whole image as a surface. For example, a gray level image is no longer considered as a function but as a two dimensional surface in three dimensional space. In another example, we will show how to treat color images as a 2D surfaces in 5D: e.g. (x,y,R,G,B) space.
