Image Structure

notation: A = {F, </>}. In sloppy form: A = J dx f (x) </>( x). Samples are always finite, as opposed to point values of the underlying source field. Apart from this they may be positive, zero, or negative. A positive sample does not imply positivity of the underlying source field; a source is positive if all samples obtained by positive definite detectors tum out positive. If we want to compare different samples, we have to gauge our detectors by a conventional normalisation. For the moment we will require neither positivity nor normalisation. Since Definition 3.1 claims to define a local sample, we have to be able to tell what its base point is. In order to do so we need an explicit definition of a projection map 7r : ~ -+ M which associates each detector element </> E ~ with its corresponding base point x = 7r[</>]. This in tum assumes that we can perceive of ~ as a "bundle" of local device spaces ~x, comprising one "fibre" for each base point: ~ = UxEM~x' The inverse image 7r1(x) of a base point x is, by definition, the entire local device space ~x at that point. A local state space ~x is then established as the physical degrees of freedom probed by a local device space ~x, i.e. "what we are looking at" with a localised detector. We will return to a precise definition of 7r in Section 3.9. For the moment it suffices to think of the base point as a the "centre of gravity" for the filter </>. The base point we would like to attribute to a sample is of course the one corresponding to the detector, but note that there is no way of telling from the value of a local sample "where it's at"; the geometric notion of a base point is established as an extrinsic detector property (a label). Obviously, local samples are obtained at finite resolution. Again, being a spatiotemporal property, resolution cannot be inferred from a sample's value, only 3.1 Local Samples 41 from its underlying aperture. A precise definition of the resolution of a local detector requires us to define a notion of extent or inner scale for that detector. A definition of inner scale will also be postponed until Section 3.9; think of it for the moment as the width of a central region, containing the filter's base point, where most of the filter's weight is concentrated (it is clearly not very useful to relate inner scale to detector support, since by construction this may be all of spacetime). See also Problem 3.2. We can consider the transformation (push forward) of a detector under an arbitrary spacetime automorphism, i.e. a "warping", or a smooth transformation of spacetime with smooth inverse. Definition 3.2 (Push Forward) Let () : M -t M : x 1-+ ()(x) be a smooth spacetime automorphism. The push forward of a filter is then defined as the mapping with Jacobian determinant J'X. == I det Vx I· This induces a natural, so-called pull back (also called "reciprocal image") of the source. Definition 3.3 (Pull Back) With the automorphism () and its push forward ()* as defined in Definition 3.2, the pull back of the source is defined as the mapping ()* : Eo(x) -t Ex : F 1-+ ()* F defined by ()* F[¢] ~f F[()*¢]. In sloppy form this states that ()* f == f 0 (), which physicists tend to refer to as "scalar field transformation" (Problem 3.3). Note that if ¢ lives at base point x, then its push forward ()*¢ is associated with the mapped point ()(x), which explains its name. Naturally, pull back works the other way around. Push forward and pull back are instances of a so-called" carry along" principle. If we have two communicating objects-i.c. sources plus detectors producing a response-then a change of either will in general be reflected in the output. Reversely, a given change in output can be explained as being caused by a change in either object. For example, shifting a patient underneath a scanner will have the same effect as moving the scanner in opposite sense over a stationary patient. This principle generalises to arbitrary deformations beyond rigid transformations (at least conceptually: one of the options is not necessarily in the interest of the patient). The idea is that at least one of these dual views is practicable and legitimate (e.g. processing scanner output). It would be formally more correct to attach base points to sources and detectors matching the labels of E and tl in Definitions 3.2 and 3.3, but that would yield rather cumbersome notations. There ought to be no confusion if we simply 42 Local Samples and Images keep in mind the following commutative diagram: