Excerpt from
L.Kleine-Horst: Empiristic theory of visual gestalt perception. Hierarchy and interactions of visual functions. (ETVG), Part 11, III, 1

Similarity, discriminability, and their hierarchical dependences

Theory of similarity and discriminability

The experience of sameness, difference, and similarity plays an important role in perception and cognition. In particular, any comprehensive theory of visual perception has to account for, among others, the facts of similarity, detectability, and discriminability, for grouping and segregation. This group of facts was theoretically so important to the early Gestalt psychologists that they elevated "similarity" to one of their famous "grouping factors", without, however, being able to explain what "similarity" is. Mach (1903, p.58) considered "similarity" to be "partial sameness", a definition which was, however, rejected by the gestaltists. But neither Mach nor the gestaltists could answer the essential question with the necessary distinctness: "similar in respect to what?" or "sameness of what?" "Similar" is an experience, and an experience is always a holistic experience, and such an experience is just as hard to define as the experience "red". The only possibility is to investigate the "aspects" in respect to which the experience "similar" can occur, and under which conditions this experience becomes stronger, or weaker.

 If one accepts Mach's quasi-definition as a first attempt to account for similarity, one should at least complete it by saying "partial sameness and partial difference", as Tversky (1977) did. The complete set of "aspects" needed is provided by the ETVG: the "aspects" are the 25 dimensions of visual qualities, in respect to which two stimuli can appear "same" or "different". There can be no objections to the "definition" of "similarity" by Mach/Tversky, as "similarity" is experienced only when a part of the 25 qualities are the same, and the others are different; if two stimuli do not seem "different" in any respect (i.e. quality), they do not appear "similar" but "absolutely the same". Note, we are not dealing with the sameness or difference of the stimuli themselves, but with the sameness and difference in the experience, i.e. appearance, of the stimuli.

I propose five hypotheses to account for similarity:

1. Two stimuli appear more (less) similar,
        the less (more) their values differ on the scale of the same quality dimension.

As an example: a rectangle with side measurements of 1 cm and 1.2 cm appears more similar to a rectangle with side measurements of 1 cm and 1.4 cm than to one with side measurements of 1 cm and 3 cm.

2 Two stimuli appear more (less) similar,
       the higher (lower) on the scale of the same quality dimension the quality intensities lie in which the stimuli differ.

As an example: two rectangles with side measurements of 1 cm and 1.5 cm respectively appear to be more different than two rectangles with side measurements of 4 cm and 4.5 cm, although the objective length difference of the latter equals that of the first.

3. Two stimuli appear more (less) similar,
    the more (less) qualities they share.

As an example: two rectangles with side measurements of 3 cm and 4 cm, and 3 cm and 5 cm, respectively, are more similar to each other than to a disk with a diameter of 3 cm, as they share more qualities with each other (Pm to R) than with the disk (Pm to Fl).

4. Two stimuli appear more (less) similar,
        the less (more) qualities they do not share.

As an example: a square and an upright rectangle with a length/width ratio of 1.3:1 appear more similar than a disk and an upright ellipse with the same  length/width ratio of 1.3:1. The filled square and rectangle have the same values for all qualities from P to R except for the dimension "elongatedness" (E), which is phenomenally "more intense" ( i.e. functionologically more highly developed) in the rectangle. The filled disk and ellipse share only the factors from P to Fl, whereas the ellipse additionally contains the qualities of verticality (V) and extendedness/ elongatedness (E).

5. Two stimuli appear more (less) similar,
     the lower (higher) the hierarchy level at wich their qualities of the same quality dimension are equal, and the higher (lower) the hierary level at which their qualities of the same dimension are different.

As an example: a red square and a red disk are more similar than a red square and a green square, as colors belong to the lowest, and therefore strongest, factor: Pm.

This is a theory of similarity. Since similarity and discriminability are related to each other, a theory of discriminability can be derived from these hypotheses by simply changing their formulations.

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