Figure 1.
The sketch summarizes the main steps that were taken in [20] for computing the slope values. Four types of spectra were considered (yielding four respective sets of slope values): (i) spectra of the original face images ( = raw), (ii) raw spectra corrected for truncation artifacts with inward diffusion ( = corrected raw), (iii) spectra of minimum 4-term Blackman-Harris (B.H.) windowed face images to suppress external face features [90], (iv) B.H. spectra corrected for the spectral “fingerprint” left by the application of the Blackman-Harris window. Now, in order to compute oriented slopes, a spectrum was subdivided into 12 pie slices (denoted by different shades of gray in the last image in the top row). Spectral amplitudes with equal spatial frequencies lie on arcs in the spectrum (schematically indicated by ). Amplitudes on arcs were averaged, either by “normal” statistical measures (i.e., location = mean & spread = standard deviation), or by outlier-insensitive “robust” measures (median & median absolute deviation MAD). Averaging yields a one dimensional (1-D) isotropic spectrum at each orientation (bottom right). A line with slope
was then fitted to the double logarithmic representation of the 1-D spectra.
Figure 2.
Illustration of the various steps involved in processing the face images (i) (size 256×256 pixels) with Gabor wavelets of orientation and spatial frequency
(ii), where five response maps (iii) (size 256×256 pixels) are obtained at each
. Response maps are subsequently centered at the four feature positions and cropped as illustrated with Figure 3. The aligned and cropped maps are averaged, giving rise to corresponding feature maps (iv) (size 127×127 pixels). Feature maps are parameterized by feature (4 possible values), gender (2), spatial frequency (39), orientation (12), and response type (5), what amounts to a total of 18720 feature maps.
Figure 3.
This figure illustrates the alignment procedure with a face image (note that this procedure is actually used for aligning response maps, cf. previous figure). Four subregions are extracted from each face image as shown, such that the corresponding feature of interest (left eye, right eye, nose, or mouth) is in the image center. Feature coordinates are indicated by crosshairs in the big ( = original) image, and were obtained through manual marking.
Figure 4.
The regions over which representative feature map values were computed by spatial averaging (“ROI-compacting”) are highlighted in averaged face images. Note that the face images are shown only to illustrate ROI locations, since representative values were computed from feature maps. (Note furthermore that “full compacting” involves averaging across the entire feature map). For each feature type, a ROI thus defines a suitable set of spatial indices , which contained
points (left and right eye), 2511 (mouth), and 2687 (nose), respectively, of a total of 127×127 feature map positions. The ROIs were selected manually. Identical ROIs were used for both gender.
Figure 5.
The display shows feature maps , with response types positve odd (
) and negative odd (
) being displayed simultaneously according to
. Brighter grey levels correspond to
, and darker grey levels correspond to
. Thus, the mid grey level of each feature map indicates the zero response level. Each image represents the average of 868 response maps centered at the position of the nose. Along rows, orientation
varies from top 0° to bottom 330° in steps of
. Along columns, the spatial frequency
increases from left 8 to right 80 cycles per image in steps of
cycles per image, thus showing only 13 of a total of 39 spatial frequencies that were used in the analysis. For displaying, each feature map was normalized individually in order to improve the overall view.
Figure 6.
Symbols denote oriented spectral slopes from the four amplitude spectra (circles = raw, squares = corrected raw, diamonds = Blackman-Harris (B.H.), and triangles = corrected B.H. – see methods section and Figure 1). The solid curve centered in the light-colored area denotes maximum entropy slopes
of feature map amplitudes (label “mean (FM)”). The light-colored area indicates ±1 standard deviation. Open symbols indicate where spectral slopes and maximum entropy slopes are significantly different from each other (one-way ANOVA at each orientation,
). Filled symbols denote the opposite case (
). A further ANOVA test served to compare whether orientation-averaged slope values were drawn from the same underlying distribution. The respective probabilities are
(raw spectrum versus maximum entropy slopes),
(corrected raw),
(B.H.), and
(corrected B.H.). Notice that slope values in the angular domain from 180° to 360° are equivalent to those being shown.
Figure 7.
Response distributions with and without ROI.
Both plots show pooled feature map amplitudes vs. spatial frequency (“response distribution”) for the left eye. Corrected Blackman-Harris slopes of individual face images were averaged by computing a mean slope value (as opposed to computing the median), which was used for whitening in this figure (see Figure S1 for whitening with uncorrected B.H. slopes). Line fitting was carried out by “normal” averaging of spectral amplitudes with equal frequencies (as explained in Figure 1). Response distributions were pooled across gender (male, female) and response type (positive even, negative even, positive odd, negative odd, local energy). Symbol sizes were scaled in proportion to standard deviations ( = overall standard deviation from averaging response maps, compacting and subsequent pooling). The standard deviations (s.d.) are usually very high with maxima exceeding often 100% (see text for an explanation). (A) Compacting the full feature map (inset, “ROI = off”). Relative s.d. lie between 105.4% (smallest symbol size) and 156.9% (biggest symbol). (B) Compacting over the circular region highlighted in the inset (“ROI = on”). Relative s.d. were 93.2% (minimum) and 142.1% (maximum). Crosshairs “⊕” indicate valid maxima (summarized in Figure 8). The average spatial frequency (±1 s.d.) of the valid maxima is shown at the top of each figure (robust: median ±1 MAD). Notice that mathematically curves at orientations 0°,30°,60°,90°,120°,150° are equivalent to the respective curves in the angular domain from 180° to 330°. However, numerical errors (especially due to sampling artifacts associated with the convolution kernels) can cause small deviations. The relative absolute deviations (mean ±1 s.d. in %) are 0.26±0.15 (normal partition; maximum 0.52%) and 0.15±0.15 (robust partition; maximum 0.46%).
Figure 8.
Summarising the maxima of response distributions.
Data points indicate spatial frequencies associated with a peak of a response distribution curve (“valid maximum”). Whitening was performed with slopes from the corrected Blackman-Harris amplitude spectra. Oriented spectral slopes were either computed according to the (i) normal partition (amplitude-averaging according to mean) or the (ii) robust partition (median). The spectral slopes in turn were either averaged by (iii) computing their mean value or by (iv) computing their median
. Symbol colors (and sizes) indicate corresponding combinations: yellow = normal partition & mean of slopes (i.e.,i together with iii), violet = robust & mean, green = normal & median, and red = robust & median. Symbol shapes, on the other hand, denote the different features: ○ = left eye, □ = right eye, ◊ = mouth, and
. The mean value (median value) of the data points at each orientation is indicated by the solid red line (dash dotted line), with the shaded area indicating ±1 standard deviation. The error bars denote a robust estimate of standard deviation (by means of median absolute deviation MAD) with respect to the median value ( = dash dotted line). (A) “ROI = off”, pooling together response type and gender. (B) Same as with a, but for “ROI = on”.
Figure 9.
Estimating bandwidths of response distributions.
Estimated full bandwidths at half height (in octaves) of response distributions, where gender and response type were pooled. Each response distribution curve was considered individually. A bandwidth estimate (“sample”) was proposed by the computer, which had to be accepted or rejected by user interaction. Less than half of the curves had a shape which allowed for a reasonable estimation (47.5% of
curves), with most samples at 90°. Bandwidths from 210° to 360° are equivalently to the corresponding shown bandwidths (0 to 180°). This fact was exploited to remove inconsistently accepted or rejected bandwidth estimates, as user interaction proceeded across the full angular domain. The red curve shows the mean of all samples at an orientation. The lightly colored area indicates ±1 standard deviation. The dash-dotted line is the mean of the samples with “ROI = on” (filled symbols), the dotted line for the “ROI = off” samples (open symbols). Symbol shape denotes feature type as with the previous figure, and symbol color denotes spectra: yellow = raw, violet = corr.raw, green = B.H. & red = corr.B.H. The partition/slope-averaging combinations listed in the previous figure (items i to iv) are not further distinguished here, meaning that the same symbols were used for all of these combinations.