 A metric representation of shape is preserved by a Fourier analysis of the cumulative angular
bend of a shape’s contour. Three experiments examined the relationship between variation in
Fourier descriptors and judgments of perceptual shape similarity. Multidimensional scaling of
similarity judgments resulted in highly ordered solutions for matrices of shapes generated by
a Fourier synthesis of a few frequencies. Multiple regression analyses indicated that particular
Fourier components best accounted for the recovered dimensions. In addition, variations in
the amplitude and the phase of a given frequency, as well as the amplitudes of 2 different
frequencies, produced independent effects on perceptual similarity. These results suggest that
a Fourier representation is consistent with the perceptual similarity of shapes, at least for the
relatively lowdimensional Fourier shapes considered.”  Although many things are useful for object recognition (color, texture, etc) earlier work shows outline (contour) shape being the most important
 Mention approach for shape representation as having an alphabet of shapepiece prototypes that are then assembled – can be represented hierarchically or spatially in some other manner.
 Pinker was a proponent of this
 But there hasn’t been any real traction on this from a practical sense, as “The difficulty lies in representing he infinite variety of shapes with a small set of primitives. Typically, the parts are distinguished only by qualitative differences in shape. ”
 Idea of geons, codons, but they dont deal with metric variations which seems important
 Marr also had idea of some form of decomposition
 An alternative is to use a system that doesn’t involve parsing an object into parts
 Fourier descriptors is one system from computer vision
 ” In this method, given an arbitrary starting point on a closed contour, the function relating cumulative arc length to local contour orientation is expanded in a Fourier series ”
 Has some nice properties, including that global shape characteristics can be determined just by the first few lowfrequency terms, also its basically invariant to starting point
 Fourier descriptors were used in early computer vision and have been considered in biological vision as well
 One study found “…hat approximately half of the visually responsive neurons in the inferior temporal cortex were selectively tuned to the frequency of FD stimuli ”
 “… all frequencies were about equally represented, except for a reduced incidence of the frequency 64 cycles per perimeter. ” Fits werent quite linear but were still good
 “In the present experiments, we tested this prediction [that FDs are related to categorization] by obtaining ratings of perceived shape similarity and subjecting them to multidimensional scaling”
 “…if, on the one hand, perceived shape similarity is related to variation in the amplitude and phase parameters of the contour, then vectors representing these Fourier components should account for the dimensions of the recovered similarity space. If, on the other hand, qualitative
stimulus attributes are used to represent shape (e.g., smoothness, number of parts, or orientation), then vectors representing these qualities should account for the majority of variability in similarity judgments. For this reason, we also obtained ratings on a number of unidimensional scales representing qualitative aspects of the stimuli.”  “In Experiment 1, we varied the amplitude and the phase of a single FD frequency. A Fourier representation of shape would predict that the perceptual similarity space should reflect variation of these two parameters. Also, because of the independence of amplitude and phase in a Fourier representation, we made an additional prediction: The amplitude and the phase of a given FD frequency should show independent effects on perceived similarity.”
 Participants were shown 45 pairs, and were told to rate them for similarity on a numeric scale, and then after that they rated each shape on 7 independent numeric scales (width, straightness, smoothness, # of parts, complexity, symmetry, orientation) – these criteria were intended to be alternatives for doing classification
 MDS using euclidian distance on the similarity ratings – there was a sharp elbow with 2 dimensions
 This reproduces almost exactly the earlier figure (just rotated and flipped), “….which suggests that perceived dissimilarity is monotonically related to distance in a 2D Euclidean space with, in this case, amplitude and phase as the two dimensions. Indeed, the relationship between distances in this space and perceived dissimilarities may be linear: A linear multidimensionalscaling analysis produced a 2D solution with virtually the same pattern as that for the monotonic analysis”
 ” that the phase and the amplitude of Frequency 6 accounted for more variability in the judgments of similarity than did any of the unidimensional scales, with the exception of smoothness.”

Experiment 2
 ” Fourier theory also predicts another pattern of effects on similarity judgments: the independence of amplitude values at different frequencies. The purpose of Experiment 2 was to test this prediction…”
 Based on MDS “it appears that the perceived dissimilarities of these shapes are monotonically related to distance in a Fourier space, with amplitude of frequency 2 and amplitude of frequency 4 as the two dimensions “
 “Fitted vectors for the amplitudes of the two frequency components were found to be orthogonal
(angular difference = 88.8°, suggesting that there were independent perceptual effects of variations in amplitude on two different frequencies. This observation, along with the observed independence of amplitude and phase in Experiment 1, is consistent with a representation of shape based on FDs.”  Variation in amplitude of freq 2 was highly correlated with judgements of “width” and freq 4 was with “smoothness”

Experiment 3
 “Experiment 3 tested the effects of variation in the phases of two different frequencies on
judgments of similarity. As in Experiments 1 and 2, this was an investigation of the perceptual effects of variation in two parameters of the Fourier expansion. However, unlike the previous experiments, the parameters manipulated in this experiment did not exhibit independent effects on the shape of the contour, because the relative phases, and not the absolute phases, determined the shape”  Here stimuli were constructed from freqs of 4,6,8 cycles/perimiter, with amplitudes held constant.
 Phases of freqs 6, 8 were varied indep (need an extra freq of 4 around for the comparison to work)
 Here stress plot from MDS didn’t have a clear elbow, but plotting with 2 dimensions made items in a ‘U’ shape (a linear manifold), implying a onedimensional solution – they are dependent
 “This relationship, taken together with the results of Experiments 1 and 2, which found a significant relationship between number of parts and amplitude of frequencies 4 and 6, suggests that object parsing may be related to the amplitude and the relative phase of frequencies in this range (4 to 8 cycles per perimeter). “
 “Of particular importance was the evidence found for independent perceptual effects for variations of amplitude and phase on a single frequency and for variations of amplitudes on two different frequencies. Both of these results predicted by a Fourier theory.”
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