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However, excluding brightness from the metric does not exclude it from the geometrical description. On the contrary, it defines I geometrically up to acceptable transformations of the brightness line. We choose to include the linear dodging transformation, assuming linear reduction functions, introduced earlier along with varying exposure time in the set of transformations under which image content is invariant, (5) u ← u + a 1 x + a 2 y + tu , where a1 , a2 and tv are arbitrary constants, the last one representing a translation of v.

So any curve in measurement space must have its orientations and location tangents aligned. This can be achieved by creating a sub-Riemannian geometry defined by a distribution on Mw using the following cotangent equation and then restricting curves to travel along this distribution: Ω = dx2 − tan θdx1 = 0. (3) So far the proposed geometry is invariant to scaling of the visual signal since scaling the signal also scales measurements proportionally. However, in the full measurement space, width is allowed to vary.

1. Figure (a) shows an example chemical reduction function which specifies how exposure through the negative affects the density on the photo paper. The numbers in the graph only serve to show that the grid is regularly spaced and the standard direction of increasing values. Figure (b) adds bins representing a regular discretization of the photo paper density. The vertical lines represent the borders of the bins. Notice that the induced discretization of log exposures is not necessarily regular and depends on the reduction function.

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