2D Spatial Frequency Research Articles

Fresnel diffractograms from pure-phase wave fields under perfect spatio-temporal coherence: Non-linear/non-local aspects and far-field behavior

Recently, the diffractogram, that is, the Fourier transform of the intensity contrast induced by Fresnel free-space propagation of a given (exit) wave field, was investigated non-perturbatively in the phase-scaling factor S (controlling the strength of phase variation) for the special case of a Gaussian phase of width sqrt{{bf{w}}}. Surprisingly, an additional low-frequency zero σ* = σ*(S, F) >0 emerges critically at small Fresnel number F (σ proportional to square of 2D spatial frequency). Here, we study the S-scaling behavior of the entire diffractogram. We identify a valley of maximum S-scaling linearity in the F − σ plane corresponding to a nearly universal physical frequency ξml = (0:143 ± 0.001)w−1/2. Large values of F (near field) are shown to imply S-scaling linearity for low σ but nowhere else (overdamped non-oscillatory). In contrast, small F values (far field) entail distinct, sizable s-bands of good S-scaling linearity (damped oscillatory). These bands also occur in simulated diffractograms induced by a complex phase map (Lena). The transition from damped oscillatory to overdamped non-oscillatory diffractograms is shown to be a critical phenomenon for the Gaussian case. We also give evidence for the occurrence of this transition in an X-ray imaging experiment. Finally, we show that the extreme far-field limit generates a σ-universal diffractogram under certain requirements on the phase map: information on phase shape then is solely encoded in S-scaling behavior.

Single-exposure two-dimensional superresolution in digital holography using a vertical cavity surface-emitting laser source array

We present a new implementation capable of producing two-dimensional (2D) superresolution (SR) imaging in a single exposure by aperture synthesis in digital lensless Fourier holography when using angular multiplexing provided by a vertical cavity surface-emitting laser source array. The system performs the recording in a single CCD snapshot of a multiplexed hologram coming from the incoherent addition of multiple subholograms, where each contains information about a different 2D spatial frequency band of the object's spectrum. Thus, a set of nonoverlapping bandpass images of the input object can be recovered by Fourier transformation (FT) of the multiplexed hologram. The SR is obtained by coherent addition of the information contained in each bandpass image while generating an enlarged synthetic aperture. Experimental results demonstrate improvement in resolution and image quality.

A new image distortion measure based on a data-driven multisensor organization

This paper describes a visual model that gives a perceptual distortion measure between an input image and that of reference based on a human-image representational model. We study an approach in which once a few active recognizers tuned to significant orientation and spatial-frequency components of the reference spectrum are obtained, any input image to be compared with the reference one is passed through an operator designated to compare its excitation levels given by the active recognizers, to the corresponding excitation levels for the reference image. Hence, the distortion between a pair of complex images is measured as the weighted sum of the distortion in each filter of a bank of strongly responding recognizers, each tuned to a certain 2D spatial-frequency data in the reference picture, with the weighting of each filter modulating its amplitude response.

The two-dimensional spectral structure of simple receptive fields in cat striate cortex.

1. A quantitative, general purpose method was developed for measuring the responses of visual neurons to stimuli distributed with high resolution over the two-dimensional (2D) spatial frequency domain. The stimuli consisted of drifting sinusoidal gratings of nonsaturating contrasts whose spatial frequency and orientation were drawn in random order from a 16 X 16 array of coordinates covering each neuron's responsive area. This method was applied to a population of 36 simple cells in area 17 of cat. 2. The response of each simple cell to drifting sinusoidal gratings appeared as a rectified sinusoidal modulation of the spike frequency. The degree of rectification varied from cell to cell, but for each cell, the form of the response was constant irrespective of stimulus spatial frequency, orientation, or contrast. The amplitude of the average response at the stimulus temporal frequency was used as the response metric at all spectral coordinates. Variations in this amplitude over two spectral dimensions forms a surface that we call the 2D spectral response profile. 3. For each cell, the 2D spectral response profile was localized to a limited region of the complete 2D spatial frequency domain. In bidirectionally responsive cells, there were two lobes in the surface disposed with mirror symmetry about the origin. In all cells, each lobe exhibited a single maximum and the response decayed smoothly in every direction away from the maximum. Isoresponse amplitude contours were elliptical and often, but not always, elongated about an axis of symmetry passing through the origin. 4. We tested the hypothesis that orientation and spatial frequency tuning are independent by forming scaled radial and angular sections through 2D spectral response profiles. In virtually every case polar separability did not obtain, that is, orientation selectivity depended on spatial frequency and vice versa. 5. In contrast, more than half the cells had 2D spectral response profiles that were Cartesian separable. The 2D spectral response profiles of most of the remaining cells were neither polar nor Cartesian separable, because the response profiles were elongated about an axis of symmetry that did not pass through the origin. 6. These results are discussed in terms of the constraints they place on models of the contributions simple cells make toward the neural representation of images.

Spatial visual channels in the fourier plane

Properties of human spatial visual channels were studied in two-dimensional form by a signal detection masking paradigm. Tuning surfaces of contrast threshold elevation induced by a sinusoidal mask were generated for four Subjects, interpolated from an 11 × 11 Cartesian grid over the Fourier plane, and numerically Fourier transformed in two dimensions to infer putative filter profiles in the 2D space domain. Among the main findings in the 2D frequency domain were: (1) Threshold elevation surfaces are highly polar nonseparable—they cannot be described as the product of a spatial frequency tuning curve times an orientation tuning curve. (2) Iso-half-amplitude contours of the spectral tuning surfaces have a length/width elongation ratio of about 2:1. (3) Necessarily, resolution for spatial frequency and for orientation are in fundamental competition with 2D spatial resolution. By calculating the occupied area of the inferred filters both in the 2D space domain and in the 2D frequency domain, it was estimated that these mechanisms approach within a factor of 2.5 of the theoretical limit of joint resolution in the two 2D domains that can be derived by 2D generalization of Gabor's famous Theory of Communication (1946). Other classes of 2D filters, such as an ideal 2D bandpass filter, have joint 2D entropies which are suboptimal by a factor of 13 or more. Subject to the inherent constraints on inference from these 2D masking experiments, the evidence suggests that 2D spatial frequency channels can be described as elongated 2D spatial wave-packets which crudely resemble optimal forms for joint information resolution in the 2D spatial and 2D frequency domains.