# What is the rate of expansion of an image as it aproaches the human eye?

My intuition about the scaling (x,y) of the 2d representation of the 3d object generated by our native visual processing system tells me it accelerates as it approaches the visual sensor (our eye).

Is this true? What is the rate of expansion of objects within our visual field?

The magnification of an object at distance $g$ from a thin lens with focal length $f$ is given as

$$\beta=\frac{f}{g-f}$$

which is generally negative for objects at distances greater than the focal length. This means the image is upside down.

The derivative of the magnification with respect to the object distance is

$$\frac{\partial \beta}{\partial g}=-\frac{f}{(g-f)^2}$$

So for constant $f$ the magnification diverges like $x^{-2}$ where $x=(g-f)$ is the difference between focal length and object distance.

So yes, the magnification grows in an accelerated way and even approaches infinity.

However, the relationship is only strictly true if the image at the image plane is sharp and the aperture is unlimited. So for your eye, at some point the image will become blurry and the magnification will grow in an asymptotic way that is much harder to describe.

Note that passed the optical system, images are transformed to a form of natural log-polar mapping* such that scaling in the Cartesian domain becomes analogous to a translation.

This mitigates greatly the effect of scaling in human perception. The fact that the radial dimension is logarithmically mapped means a 10-fold scaling corresponds to a single-unit translation on the logarithmic scale. My understanding is that this allows human vision to be very good at scale-invariant pattern recognition.

*See: G. Sandini, and V. Tagliasco, ”A anthropomorphic retina-like structure for scene analysis”, Computer, Graphics and Image Process- ing, Vol. 14, 1980, pp. 365-372. or E. Schwartz, ”Anatomical and Physi- ological Correlates of Visual Computation from Striate to Infero-Temporal Cortex” , IEEE Trans. on Systems, Man, and Cybernetics, Vol. SMC-14, No. 2, March/April 1984, pp. 257-271.