Where does dilation collide?

Time dilation is the slowing of a moving or gravitationally lower clock as read by another observer: in special relativity a clock moving at speed v ticks slow by the Lorentz factor, and in general relativity a clock deeper in a gravitational potential runs slow. Einstein's 1905 Zur Elektrodynamik bewegter Körper derived the effect from the constancy of the speed of light, and Hafele and Keating's 1972 flights of atomic clocks around the world confirmed it directly. The referent is a relation between elapsed times on two clocks, a dimensionless ratio set by relative motion or potential, not a widening aperture, a scaled figure, or an expanded image region.[1][2]

Unit

A clock's slowing relative to an observer, as a dimensionless ratio (the Lorentz factor)

Recovery means

Not applicable: between inertial observers the effect is symmetric rather than a return to a baseline

Measured by

Compared against a reference clock, for example atomic clocks flown around the world or the lengthened lifetime of fast muons

Cervical dilation is the progressive opening of the cervix during labor, from closed to about ten centimetres, that lets the fetus pass: it is one of the cardinal measures of labor progress and is charted against time. Friedman's 1954 The graphic analysis of labor plotted dilation over time and gave the sigmoid labor curve, with its latent and active phases, that still frames obstetric practice. The referent is the physical widening of an anatomical opening, measured in centimetres, and it shares nothing with a slowed clock, a scaled triangle, or a set operator beyond the word.[3]

Unit

The cervix's opening during labor, in centimetres from 0 to about 10

Recovery means

After delivery the cervix closes and reconstitutes over the postpartum weeks

Measured by

Assessed by digital vaginal examination and plotted against elapsed time as a Friedman labor curve

In geometry a dilation is a similarity transformation that scales a figure by a fixed factor k about a fixed center, so each point moves along the ray from the center to a distance multiplied by k: also called a homothety, it enlarges or shrinks a figure while preserving its shape, angles, and parallelism. Martin's 1982 Transformation Geometry develops dilatations as the maps that carry every line to a parallel line and that, with the isometries, generate the group of similarities. The referent is a shape-preserving map of the plane fixed by a center and a ratio, not a physical opening, a slowed clock, or an operator on image pixels.[4]

Unit

A similarity transformation, given by a center and a scale factor k

Recovery means

The inverse dilation, with reciprocal factor 1/k about the same center, restores the original figure

Measured by

Specified by its center and ratio, and checked by the constant factor relating corresponding distances from the center

In mathematical morphology, dilation is the operator that grows the foreground of an image by combining it with a small probe called a structuring element: it is the Minkowski sum of the image set with that element, so each foreground point is replaced by a translated copy of the probe and regions swell, gaps close, and thin features thicken. Haralick, Sternberg, and Zhuang's 1987 Image Analysis Using Mathematical Morphology set out dilation and its dual erosion as the primitives from which opening, closing, and shape filters are built. The referent is a set-theoretic operator on image pixels defined by a structuring element, not a scaling of a figure, a widened aperture, or a slowed clock.[5]

Unit

A set operator, the Minkowski sum of an image with a structuring element

Recovery means

Erosion is the dual operator that shrinks regions, and a dilation followed by an erosion forms a morphological closing

Measured by

Computed by translating the structuring element across the image and taking the union of its copies over the foreground