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Spatial frequency
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Green Sea Shell image
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Spatial frequency representation of the Green Sea Shell image
Image and its spatial frequencies: Magnitude of frequency domain is logarithmically scaled, and zero frequency is in the center. Notable is the clustering of the content on the lower frequencies, a typical property of natural images.
In mathematics, physics, and engineering, spatial frequency is a characteristic of any structure that is periodic across position in space. The spatial frequency is a measure of how often sinusoidal components (as determined by the Fourier transform) of the structure repeat per unit of distance.
The SI unit of spatial frequency is the reciprocal metre (m−1),[1] although cycles per meter (c/m) is also common. In image-processing applications, spatial frequency is often expressed in units of cycles per millimeter (c/mm) or also line pairs per millimeter (LP/mm).
In wave propagation, the spatial frequency is also known as wavenumber. Ordinary wavenumber is defined as the reciprocal of wavelength and is commonly denoted by [2] or sometimes :[3] Angular wavenumber , expressed in radian per metre (rad/m), is related to ordinary wavenumber and wavelength by
Visual perception
[edit]In the study of visual perception, sinusoidal gratings are frequently used to probe the capabilities of the visual system, such as contrast sensitivity. In these stimuli, spatial frequency is expressed as the number of cycles per degree of visual angle. Sine-wave gratings also differ from one another in amplitude (the magnitude of difference in intensity between light and dark stripes), orientation, and phase.
Spatial-frequency theory
[edit]The spatial-frequency theory refers to the theory that the visual cortex operates on a code of spatial frequency, not on the code of straight edges and lines hypothesised by Hubel and Wiesel on the basis of early experiments on V1 neurons in the cat.[4][5] In support of this theory is the experimental observation that the visual cortex neurons respond even more robustly to sine-wave gratings that are placed at specific angles in their receptive fields than they do to edges or bars. Most neurons in the primary visual cortex respond best when a sine-wave grating of a particular frequency is presented at a particular angle in a particular location in the visual field.[6] (However, as noted by Teller (1984),[7] it is probably not wise to treat the highest firing rate of a particular neuron as having a special significance with respect to its role in the perception of a particular stimulus, given that the neural code is known to be linked to relative firing rates. For example, in color coding by the three cones in the human retina, there is no special significance to the cone that is firing most strongly – what matters is the relative rate of firing of all three simultaneously. Teller (1984) similarly noted that a strong firing rate in response to a particular stimulus should not be interpreted as indicating that the neuron is somehow specialized for that stimulus, since there is an unlimited equivalence class of stimuli capable of producing similar firing rates.)
The spatial-frequency theory of vision is based on two physical principles:
- Any visual stimulus can be represented by plotting the intensity of the light along lines running through it.
- Any curve can be broken down into constituent sine waves by Fourier analysis.
The theory (for which empirical support has yet to be developed) states that in each functional module of the visual cortex, Fourier analysis (or its piecewise form [8]) is performed on the receptive field and the neurons in each module are thought to respond selectively to various orientations and frequencies of sine wave gratings.[9] When all of the visual cortex neurons that are influenced by a specific scene respond together, the perception of the scene is created by the summation of the various sine-wave gratings. (This procedure, however, does not address the problem of the organization of the products of the summation into figures, grounds, and so on. It effectively recovers the original (pre-Fourier analysis) distribution of photon intensity and wavelengths across the retinal projection, but does not add information to this original distribution. So the functional value of such a hypothesized procedure is unclear. Some other objections to the "Fourier theory" are discussed by Westheimer (2001)[10]). One is generally not aware of the individual spatial frequency components since all of the elements are essentially blended together into one smooth representation. However, computer-based filtering procedures can be used to deconstruct an image into its individual spatial frequency components.[11] Research on spatial frequency detection by visual neurons complements and extends previous research using straight edges rather than refuting it.[12]
Further research shows that different spatial frequencies convey different information about the appearance of a stimulus. High spatial frequencies represent abrupt spatial changes in the image, such as edges, and generally correspond to featural information and fine detail. M. Bar (2004) has proposed that low spatial frequencies represent global information about the shape, such as general orientation and proportions.[13] Rapid and specialised perception of faces is known to rely more on low spatial frequency information.[14] In the general population of adults, the threshold for spatial frequency discrimination is about 7%. It is often poorer in dyslexic individuals.[15]
Spatial frequency in MRI
[edit]When spatial frequency is used as a variable in a mathematical function, the function is said to be in k-space. Two dimensional k-space has been introduced into MRI as a raw data storage space. The value of each data point in k-space is measured in the unit of 1/meter, i.e. the unit of spatial frequency.
It is very common that the raw data in k-space shows features of periodic functions. The periodicity is not spatial frequency, but is temporal frequency. An MRI raw data matrix is composed of a series of phase-variable spin-echo signals. Each of the spin-echo signal is a sinc function of time, which can be described by Where Here is the gyromagnetic ratio constant, and is the basic resonance frequency of the spin. Due to the presence of the gradient G, the spatial information r is encoded onto the frequency . The periodicity seen in the MRI raw data is just this frequency , which is basically the temporal frequency in nature.
In a rotating frame, , and is simplified to . Just by letting , the spin-echo signal is expressed in an alternative form
Now, the spin-echo signal is in the k-space. It becomes a periodic function of k with r as the k-space frequency but not as the "spatial frequency", since "spatial frequency" is reserved for the name of the periodicity seen in the real space r.
The k-space domain and the space domain form a Fourier pair. Two pieces of information are found in each domain, the spatial information and the spatial frequency information. The spatial information, which is of great interest to all medical doctors, is seen as periodic functions in the k-space domain and is seen as the image in the space domain. The spatial frequency information, which might be of interest to some MRI engineers, is not easily seen in the space domain but is readily seen as the data points in the k-space domain.
See also
[edit]References
[edit]- ^ "ISO 80000-3:2019 Quantities and units — Part 3: Space and time" (2 ed.). International Organization for Standardization. 2019. Retrieved 2019-10-23. [1] (11 pages)
- ^ SPIE Optipedia article: "Spatial Frequency"
- ^ The symbol is also used to represent temporal frequency, as in, e.g., Planck's formula.
- ^ Martinez LM, Alonso JM (2003). "Complex receptive fields in primary visual cortex". Neuroscientist. 9 (5): 317–31. doi:10.1177/1073858403252732. PMC 2556291. PMID 14580117.
- ^ De Valois, R. L.; De Valois, K. K. (1988). Spatial vision. New York: Oxford University Press.
- ^ Issa NP, Trepel C, Stryker MP (2000). "Spatial frequency maps in cat visual cortex". The Journal of Neuroscience. 20 (22): 8504–8514. doi:10.1523/JNEUROSCI.20-22-08504.2000. PMC 2412904. PMID 11069958.
- ^ Teller, DY (1984). "Linking propositions". Vision Research. 24 (10): 1233–1246. doi:10.1016/0042-6989(84)90178-0. PMID 6395480. S2CID 6146565.
- ^ Glezer, V. D. (1995). Vision and mind: Modeling mental functions. Lawrence Erlbaum Associates, Inc. https://doi.org/10.4324/9780203773932
- ^ Barghout, Lauren (2014). Vision: How Global Perceptual Context Changes Local Contrast Processing (Ph.D. Dissertation 2003). Updated for Computer Vision Techniques. Scholars' Press. ISBN 978-3-639-70962-9.
- ^ Westheimer, G. "The Fourier Theory of Vision"
- ^ Blake, R. and Sekuler, R., Perception, 3rd ed. Chapter 3. ISBN 978-0-072-88760-0
- ^ Pinel, J. P. J., Biopsychology, 6th ed. 293–294. ISBN 0-205-42651-4
- ^ Bar M (Aug 2004). "Visual objects in context". Nat. Rev. Neurosci. 5 (8): 617–29. doi:10.1038/nrn1476. PMID 15263892. S2CID 205499985.
- ^ Awasthi B, Friedman J, Williams MA (2011). "Faster, stronger, lateralized: Low spatial frequency information supports face processing". Neuropsychologia. 49 (13): 3583–3590. doi:10.1016/j.neuropsychologia.2011.08.027. PMID 21939676. S2CID 10037045.
- ^ Ben-Yehudah G, Ahissar M (May 2004). "Sequential spatial frequency discrimination is consistently impaired among adult dyslexics". Vision Res. 44 (10): 1047–63. doi:10.1016/j.visres.2003.12.001. PMID 15031099. S2CID 12605281.
External links
[edit]- "Tutorial: Spatial Frequency of an Image". Hakan Haberdar, University of Houston. Retrieved 22 March 2012.
- Kalloniatis, Michael; Luu, Charles (2007). "Webvision: Part IX Psychophysics of Vision. 2 Visual Acuity, Contrast Sensitivity". University of Utah. Retrieved 19 July 2009.
Spatial frequency
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Spatial frequency is a fundamental concept in signal processing and visual science that quantifies the rate at which a pattern or signal intensity varies across space, typically measured in cycles per unit distance, such as cycles per millimeter or cycles per degree of visual angle.[1] It represents the number of complete cycles (e.g., repetitions of a sine wave or grating pair) occurring within a given spatial interval, analogous to temporal frequency in time-domain signals but applied to two-dimensional images or scenes.[2] Low spatial frequencies capture broad, coarse structures like overall shapes, while high spatial frequencies encode fine details such as edges and textures.[3]
In human visual perception, spatial frequency plays a central role in how the brain analyzes images, with the visual system organized into selective channels that respond preferentially to specific frequency bands, as demonstrated in foundational psychophysical experiments using sinusoidal gratings.[4] These channels enable parallel processing of visual information, where the contrast sensitivity function—a measure of the minimum contrast detectable at each frequency—typically peaks at intermediate frequencies around 2–4 cycles per degree and declines sharply at higher frequencies, limiting visual acuity to about 50–60 cycles per degree under optimal conditions.[5] This frequency-based decomposition underlies phenomena like the visibility of patterns and the integration of low-frequency global context with high-frequency local details in scene recognition.[3]
Beyond vision, spatial frequency is essential in image processing and computer vision, where Fourier analysis decomposes images into frequency components to facilitate tasks like filtering, compression, and feature extraction.[1] In biomedical applications, techniques such as spatial frequency domain imaging (SFDI) exploit these principles to non-invasively map tissue optical properties by modulating light patterns and analyzing their frequency-domain responses.[6] Overall, the concept bridges optics, neuroscience, and engineering, influencing advancements in display technology, medical diagnostics, and artificial intelligence for visual tasks.
Fundamentals
Definition and units
Spatial frequency is defined as the number of cycles, or complete repetitions, of a spatially periodic pattern occurring per unit distance, serving as the spatial analog to temporal frequency in time-varying signals.[1] This measure quantifies the periodicity or repetition rate of variations in a signal or image across space, such as the alternating bright and dark bands in a sinusoidal grating.[7] Physically, low spatial frequencies represent coarse, gradual changes or broad patterns, such as the overall shape of an object, while high spatial frequencies capture fine details, sharp transitions, or rapid variations, like edges and textures.[8] For instance, in an image, the low-frequency components convey the general structure, whereas high-frequency components encode intricate features that contribute to perceived sharpness.[9] The standard units for spatial frequency are cycles per unit length, such as cycles per millimeter (cycles/mm) or cycles per meter (cycles/m), depending on the scale of the application.[7] In the context of human vision, it is often expressed in angular terms as cycles per degree (cpd) of visual angle, accounting for the observer's distance from the pattern.[2] To convert linear spatial frequency to angular spatial frequency in cycles per degree, multiply the linear frequency by the viewing distance (in units matching the linear frequency's inverse) and by approximately 0.0175 (the radians per degree, π/180). A practical example is a black-and-white grating pattern with five complete cycles (alternating dark and light bars) spanning 1 centimeter, yielding a spatial frequency of 5 cycles/cm.[8] If viewed from 57 cm away—where 1 degree of visual angle corresponds to about 1 cm on the pattern—this equates to approximately 5 cpd.[2] The concept of spatial frequency emerged in the mid-20th century within Fourier optics and gained prominence in the 1960s through applications in visual perception.[9]Mathematical representation
Spatial frequency in one dimension is fundamentally defined for a periodic signal , where the spatial frequency represents the number of cycles per unit distance and is given by , with denoting the spatial period or wavelength.[10] This measure arises in the context of sinusoidal variations along a single spatial axis . A pure sinusoidal signal can thus be expressed as where is the amplitude modulating the signal's intensity, and is the phase shift determining the offset of the waveform.[11] For analytical purposes, particularly in Fourier analysis, the equivalent complex exponential form serves as the basis function, enabling the decomposition of arbitrary signals into sums of these harmonics.[12] The concept of spatial frequency emerges directly from the Fourier transform, which projects the signal onto these exponential basis functions. The continuous Fourier transform of is defined by the integral where captures the amplitude and phase contributions at each frequency , revealing how spatial frequency components contribute to the overall signal structure.[12] This formulation underscores that spatial frequency quantifies the rate of oscillation in the spatial domain, analogous to temporal frequency in time-domain signals. In two dimensions, applicable to images or planar fields , spatial frequency is characterized by orthogonal components and , representing cycles per unit length along the - and -axes, respectively.[13] These components combine to yield the radial (or magnitude) frequency , which indicates the overall oscillation rate, and the orientation angle , specifying the direction of the frequency vector.[13] A two-dimensional sinusoid then takes the form , extending the one-dimensional representation to account for directional variations. For discrete signals, such as those in digital imaging, spatial frequency is normalized in cycles per pixel, reflecting the sampling grid's influence. The highest representable frequency, known as the Nyquist limit, is 0.5 cycles per pixel, beyond which aliasing occurs due to undersampling.[14] This limit ensures that each frequency component can be uniquely reconstructed, mirroring the continuous case but constrained by the pixel resolution.Signal Processing Applications
Fourier transform in spatial domains
The spatial Fourier transform provides a mathematical framework for decomposing a two-dimensional spatial signal into its constituent frequency components in the frequency domain , where and represent spatial frequencies along the respective axes.[11] The forward transform is given by the integral which converts the spatial domain representation into a complex-valued spectrum.[11] The inverse transform reconstructs the original signal via ensuring perfect reversibility under ideal conditions.[11] This decomposition reveals the spatial frequency content, where low frequencies correspond to gradual variations and high frequencies to rapid changes in the signal.[15] In the frequency domain, the magnitude quantifies the amplitude or energy at each spatial frequency pair , providing insight into the strength of periodic components, while the phase encodes information about spatial shifts and alignments necessary for accurate reconstruction.[11] Several properties of the Fourier transform are particularly relevant to spatial frequency analysis: linearity, which states that the transform of a linear combination of signals is the linear combination of their transforms, ; the shift theorem, indicating that a spatial translation by introduces a phase shift, ; and the convolution theorem, where spatial convolution corresponds to multiplication in the frequency domain, .[11] These properties facilitate efficient analysis of spatial patterns without direct computation in the spatial domain.[11] For digital images, which are discrete spatial signals, the continuous transform is approximated by the two-dimensional discrete Fourier transform (DFT), defined as where and are discrete frequency indices, and the inverse DFT reconstructs the image with appropriate normalization.[11] The fast Fourier transform (FFT) algorithm computes the DFT efficiently in time for an image, making it practical for large-scale spatial frequency analysis.[15] To prevent aliasing artifacts, where high spatial frequencies masquerade as lower ones, the sampling must satisfy the spatial Nyquist-Shannon theorem: the sampling rate in the spatial domain must exceed twice the maximum spatial frequency present in the signal.[11] A practical example of this decomposition is applied to a grayscale photograph, where the low-frequency components in the Fourier spectrum capture smooth tonal gradients and overall structure, such as broad sky areas, while high-frequency components represent fine details like edges and textures in foliage or fabric patterns.[15] Isolating these components via the transform allows visualization of how spatial frequencies contribute to perceived image content.[15]Spatial filtering techniques
Spatial filtering techniques in the frequency domain involve modifying the Fourier transform of an image or signal to selectively alter specific spatial frequencies, followed by an inverse transform to return to the spatial domain. The process begins by computing the 2D Fourier transform of the input signal . This spectrum is then multiplied pointwise by a filter function , yielding the filtered spectrum . Finally, the inverse 2D Fourier transform of produces the filtered output . This approach leverages the frequency representation to achieve effects that are computationally efficient for certain operations compared to direct spatial manipulation.[16] A classic example is the ideal low-pass filter, defined as for and otherwise, where is the cutoff frequency. This filter attenuates high spatial frequencies, effectively removing fine details and noise while preserving the overall structure dominated by low frequencies. In practice, such abrupt cutoffs are rarely used due to their undesirable side effects, but they serve as a foundational model for understanding frequency attenuation.[16] Filters are categorized by their frequency response. Low-pass filters, like the ideal or Butterworth variants, smooth images by suppressing high frequencies, which blurs edges and reduces noise but can obscure important details. High-pass filters, conversely, emphasize high frequencies to sharpen images and enhance edges, often amplifying noise in the process. Band-pass filters allow a specific range of frequencies to pass, useful for isolating textures or patterns, while notch filters target and remove narrow bands of unwanted periodic noise, such as interference patterns from electrical sources.[16][17] These frequency domain operations have direct equivalents in the spatial domain via the convolution theorem, which states that multiplication in the frequency domain corresponds to convolution in the spatial domain. Thus, applying a filter is equivalent to convolving the input with the inverse Fourier transform of , often implemented as a kernel. For instance, a Gaussian low-pass filter in the frequency domain corresponds to convolution with a Gaussian kernel in the spatial domain, providing a smooth transition that avoids sharp cutoffs. This duality allows practitioners to choose between domains based on computational needs, with frequency domain preferred for large kernels.[16] Key design considerations include the transition band, which defines how gradually the filter attenuates frequencies around the cutoff to minimize artifacts. Sharp transitions in ideal filters lead to ringing artifacts, known as the Gibbs phenomenon, where overshoots and oscillations appear near discontinuities in the signal, reaching up to about 9% of the jump magnitude. To mitigate this, filters like Butterworth or Gaussian are designed with smoother roll-offs. Practical implementations are available in software libraries; for example, MATLAB's Image Processing Toolbox supports frequency domain filtering via functions likefft2 and ifft2 combined with custom filter matrices, while OpenCV provides cv::dft for similar operations in C++ or Python.[16][18][19][20]
As an illustrative example, applying a low-pass filter to a noisy image involves computing its Fourier transform, multiplying by a circular low-pass mask centered at the origin with radius , and inverse transforming the result. This preserves low-frequency components representing the image's broad shapes and textures while attenuating high-frequency noise, resulting in a cleaner version suitable for further analysis without excessive blurring of structural elements.[19]