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100-413: The Event Horizon Telescope ( EHT ) is a telescope array consisting of a global network of radio telescopes . The EHT project combines data from several very-long-baseline interferometry (VLBI) stations around Earth, which form a combined array with an angular resolution sufficient to observe objects the size of a supermassive black hole 's event horizon . The project's observational targets include

200-847: A discrete-time process or a real number for a continuous-time process). Then X t {\displaystyle X_{t}} is the value (or realization ) produced by a given run of the process at time t {\displaystyle t} . Suppose that the process has means μ X ( t ) {\displaystyle \mu _{X}(t)} and μ Y ( t ) {\displaystyle \mu _{Y}(t)} and variances σ X 2 ( t ) {\displaystyle \sigma _{X}^{2}(t)} and σ Y 2 ( t ) {\displaystyle \sigma _{Y}^{2}(t)} at time t {\displaystyle t} , for each t {\displaystyle t} . Then

300-545: A phased array , a virtual telescope which can be pointed electronically, with an effective aperture which is the diameter of the entire planet, substantially improving its angular resolution. The effort includes development and deployment of submillimeter dual polarization receivers, highly stable frequency standards to enable very-long-baseline interferometry at 230–450 GHz, higher-bandwidth VLBI backends and recorders, as well as commissioning of new submillimeter VLBI sites. Each year since its first data capture in 2006,

400-632: A 1.3 mm wavelength. On May 12, 2022, the EHT Collaboration revealed an image of Sagittarius A* , the supermassive black hole at the center of the Milky Way galaxy . The black hole is 27,000 light-years away from Earth; it is thousands of times smaller than M87*. Sera Markoff , Co-Chair of the EHT Science Council, said: "We have two completely different types of galaxies and two very different black hole masses, but close to

500-445: A black hole brings observations even closer to the event horizon. Relativity predicts a dark shadow-like region, caused by gravitational bending and capture of light, which matches the observed image. The published paper states: "Overall, the observed image is consistent with expectations for the shadow of a spinning Kerr black hole as predicted by general relativity." Paul T.P. Ho, EHT Board member, said: "Once we were sure we had imaged

600-499: A black hole, at the center of galaxy Messier 87, was published by the EHT Collaboration on April 10, 2019, in a series of six scientific publications. The array made this observation at a wavelength of 1.3 mm and with a theoretical diffraction-limited resolution of 25 microarcseconds . In March 2021, the Collaboration presented, for the first time, a polarized-based image of the black hole which may help better reveal

700-483: A dramatic radio outburst in 1997, during which its flux density at 14.5 GHz exceeded 10 Jy, while the average value is ~2 Jy. Since 2002, NRAO 530 has been monitored by the Submillimeter Array (SMA; Maunakea, Hawaii) at 1.3 mm and 870 μm. NRAO 530 has a redshift of z = 0.902 (Junkkarinen 1984), for which 100 μas corresponds to a linear distance of 0.803 pc. The source contains a supermassive black hole,

800-633: A feature in f {\displaystyle f} at t {\displaystyle t} also occurs later in g {\displaystyle g} at t + τ {\displaystyle t+\tau } , hence g {\displaystyle g} could be described to lag f {\displaystyle f} by τ {\displaystyle \tau } . If f {\displaystyle f} and g {\displaystyle g} are both continuous periodic functions of period T {\displaystyle T} ,

900-426: A few years. Progressing quantum computing might eventually allow more extensive use of interferometry, as newer proposals suggest. Cross-correlation In signal processing , cross-correlation is a measure of similarity of two series as a function of the displacement of one relative to the other. This is also known as a sliding dot product or sliding inner-product . It is commonly used for searching

1000-411: A fractional milliarcsecond have been achieved at visible and infrared wavelengths. One simple layout of an astronomical interferometer is a parabolic arrangement of mirror pieces, giving a partially complete reflecting telescope but with a "sparse" or "dilute" aperture. In fact, the parabolic arrangement of the mirrors is not important, as long as the optical path lengths from the astronomical object to

1100-616: A further improvement in resolution, and allowing even higher resolution imaging of stellar surfaces . Software packages such as BSMEM or MIRA are used to convert the measured visibility amplitudes and closure phases into astronomical images. The same techniques have now been applied at a number of other astronomical telescope arrays, including the Navy Precision Optical Interferometer , the Infrared Spatial Interferometer and

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1200-424: A long signal for a shorter, known feature. It has applications in pattern recognition , single particle analysis , electron tomography , averaging , cryptanalysis , and neurophysiology . The cross-correlation is similar in nature to the convolution of two functions. In an autocorrelation , which is the cross-correlation of a signal with itself, there will always be a peak at a lag of zero, and its size will be

1300-782: A positive scalar. Normalized correlation is one of the methods used for template matching , a process used for finding instances of a pattern or object within an image. It is also the 2-dimensional version of Pearson product-moment correlation coefficient . NCC is similar to ZNCC with the only difference of not subtracting the local mean value of intensities: 1 n ∑ x , y 1 σ f σ t f ( x , y ) t ( x , y ) {\displaystyle {\frac {1}{n}}\sum _{x,y}{\frac {1}{\sigma _{f}\sigma _{t}}}f(x,y)t(x,y)} Caution must be applied when using cross correlation for nonlinear systems. In certain circumstances, which depend on

1400-537: A resolution up to ten times greater than the Hubble Space Telescope, and complementing images made with the VLT interferometer. Optical interferometers are mostly seen by astronomers as very specialized instruments, capable of a very limited range of observations. It is often said that an interferometer achieves the effect of a telescope the size of the distance between the apertures; this is only true in

1500-507: A resolution which would be given by a hypothetical single dish with an aperture thousands of kilometers in diameter. At the shorter wavelengths used in infrared astronomy and optical astronomy it is more difficult to combine the light from separate telescopes, because the light must be kept coherent within a fraction of a wavelength over long optical paths, requiring very precise optics. Practical infrared and optical astronomical interferometers have only recently been developed, and are at

1600-487: A series of six papers published in The Astrophysical Journal Letters . A clockwise rotating black hole was observed in the 6σ region. The image provided a test for Albert Einstein 's general theory of relativity under extreme conditions. Studies have previously tested general relativity by looking at the motions of stars and gas clouds near the edge of a black hole. However, an image of

1700-475: A single telescope – an interferometer. An additional compact array of four 12-metre and twelve 7-meter antennas will complement this. The antennas can be spread across the desert plateau over distances from 150 metres to 16 kilometres, which will give ALMA a powerful variable "zoom". It will be able to probe the Universe at millimetre and submillimetre wavelengths with unprecedented sensitivity and resolution, with

1800-494: A star. This is equivalent to resolving the head of a screw at a distance of 300 km (190 mi). Notable 1990s results included the Mark III measurement of diameters of 100 stars and many accurate stellar positions, COAST and NPOI producing many very high resolution images, and Infrared Stellar Interferometer measurements of stars in the mid-infrared for the first time. Additional results include direct measurements of

1900-503: Is 1 n ∑ x , y 1 σ f σ t ( f ( x , y ) − μ f ) ( t ( x , y ) − μ t ) {\displaystyle {\frac {1}{n}}\sum _{x,y}{\frac {1}{\sigma _{f}\sigma _{t}}}\left(f(x,y)-\mu _{f}\right)\left(t(x,y)-\mu _{t}\right)} where n {\displaystyle n}

2000-527: Is standard deviation of f {\displaystyle f} . In functional analysis terms, this can be thought of as the dot product of two normalized vectors . That is, if F ( x , y ) = f ( x , y ) − μ f {\displaystyle F(x,y)=f(x,y)-\mu _{f}} and T ( x , y ) = t ( x , y ) − μ t {\displaystyle T(x,y)=t(x,y)-\mu _{t}} then

2100-733: Is a 3 × 2 {\displaystyle 3\times 2} matrix whose ( i , j ) {\displaystyle (i,j)} -th entry is E ⁡ [ X i Y j ] {\displaystyle \operatorname {E} [X_{i}Y_{j}]} . If Z = ( Z 1 , … , Z m ) {\displaystyle \mathbf {Z} =(Z_{1},\ldots ,Z_{m})} and W = ( W 1 , … , W n ) {\displaystyle \mathbf {W} =(W_{1},\ldots ,W_{n})} are complex random vectors , each containing random variables whose expected value and variance exist,

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2200-490: Is a vector of kernel functions k ( ⋅ , ⋅ ) : C M × C M → R {\displaystyle k(\cdot ,\cdot )\colon \mathbb {C} ^{M}\times \mathbb {C} ^{M}\to \mathbb {R} } and T i ( ⋅ ) : C M → C M {\displaystyle T_{i}(\cdot )\colon \mathbb {C} ^{M}\to \mathbb {C} ^{M}}

2300-662: Is an affine transform . Specifically, T i ( ⋅ ) {\displaystyle T_{i}(\cdot )} can be circular translation transform, rotation transform, or scale transform, etc. The kernel cross-correlation extends cross-correlation from linear space to kernel space. Cross-correlation is equivariant to translation; kernel cross-correlation is equivariant to any affine transforms, including translation, rotation, and scale, etc. As an example, consider two real valued functions f {\displaystyle f} and g {\displaystyle g} differing only by an unknown shift along

2400-427: Is equivalent to ( f ⋆ g ) ( τ )   ≜ ∫ t 0 t 0 + T f ( t − τ ) ¯ g ( t ) d t {\displaystyle (f\star g)(\tau )\ \triangleq \int _{t_{0}}^{t_{0}+T}{\overline {f(t-\tau )}}g(t)\,dt} Similarly, for discrete functions,

2500-505: Is equivalent to ( f ⋆ g ) ( τ )   ≜ ∫ − ∞ ∞ f ( t − τ ) ¯ g ( t ) d t {\displaystyle (f\star g)(\tau )\ \triangleq \int _{-\infty }^{\infty }{\overline {f(t-\tau )}}g(t)\,dt} where f ( t ) ¯ {\displaystyle {\overline {f(t)}}} denotes

2600-636: Is equivalent to: ( f ⋆ g ) [ n ]   ≜ ∑ m = 0 N − 1 f [ ( m − n ) mod   N ] ¯ g [ m ] {\displaystyle (f\star g)[n]\ \triangleq \sum _{m=0}^{N-1}{\overline {f[(m-n)_{{\text{mod}}~N}]}}g[m]} For finite discrete functions f ∈ C N {\displaystyle f\in \mathbb {C} ^{N}} , g ∈ C M {\displaystyle g\in \mathbb {C} ^{M}} ,

2700-1091: Is important both because the interpretation of the autocorrelation as a correlation provides a scale-free measure of the strength of statistical dependence , and because the normalization has an effect on the statistical properties of the estimated autocorrelations. For jointly wide-sense stationary stochastic processes, the cross-correlation function has the following symmetry property: R X Y ⁡ ( t 1 , t 2 ) = R Y X ⁡ ( t 2 , t 1 ) ¯ {\displaystyle \operatorname {R} _{XY}(t_{1},t_{2})={\overline {\operatorname {R} _{YX}(t_{2},t_{1})}}} Respectively for jointly WSS processes: R X Y ⁡ ( τ ) = R Y X ⁡ ( − τ ) ¯ {\displaystyle \operatorname {R} _{XY}(\tau )={\overline {\operatorname {R} _{YX}(-\tau )}}} Cross-correlations are useful for determining

2800-429: Is linearly polarized, with a fractional polarization of ~5%–8%, and it has a substructure consisting of two components. Their observed brightness temperature suggests that the energy density of the jet is dominated by the magnetic field. The jet extends over 60 μas along a position angle ~ −28°. It includes two features with orthogonal directions of polarization (electric vector position angle), parallel and perpendicular to

2900-405: Is most widely used in radio astronomy , in which signals from separate radio telescopes are combined. A mathematical signal processing technique called aperture synthesis is used to combine the separate signals to create high-resolution images. In Very Long Baseline Interferometry (VLBI) radio telescopes separated by thousands of kilometers are combined to form a radio interferometer with

3000-2206: Is not well-defined for all time series or processes, because the mean or variance may not exist. Let ( X t , Y t ) {\displaystyle (X_{t},Y_{t})} represent a pair of stochastic processes that are jointly wide-sense stationary . Then the cross-covariance function and the cross-correlation function are given as follows. R X Y ⁡ ( τ ) ≜   E ⁡ [ X t Y t + τ ¯ ] {\displaystyle \operatorname {R} _{XY}(\tau )\triangleq \ \operatorname {E} \left[X_{t}{\overline {Y_{t+\tau }}}\right]} or equivalently R X Y ⁡ ( τ ) = E ⁡ [ X t − τ Y t ¯ ] {\displaystyle \operatorname {R} _{XY}(\tau )=\operatorname {E} \left[X_{t-\tau }{\overline {Y_{t}}}\right]} K X Y ⁡ ( τ ) ≜   E ⁡ [ ( X t − μ X ) ( Y t + τ − μ Y ) ¯ ] {\displaystyle \operatorname {K} _{XY}(\tau )\triangleq \ \operatorname {E} \left[\left(X_{t}-\mu _{X}\right){\overline {\left(Y_{t+\tau }-\mu _{Y}\right)}}\right]} or equivalently K X Y ⁡ ( τ ) = E ⁡ [ ( X t − τ − μ X ) ( Y t − μ Y ) ¯ ] {\displaystyle \operatorname {K} _{XY}(\tau )=\operatorname {E} \left[\left(X_{t-\tau }-\mu _{X}\right){\overline {\left(Y_{t}-\mu _{Y}\right)}}\right]} where μ X {\displaystyle \mu _{X}} and σ X {\displaystyle \sigma _{X}} are

3100-658: Is the L ² norm . Cauchy–Schwarz then implies that ZNCC has a range of [ − 1 , 1 ] {\displaystyle [-1,1]} . Thus, if f {\displaystyle f} and t {\displaystyle t} are real matrices, their normalized cross-correlation equals the cosine of the angle between the unit vectors F {\displaystyle F} and T {\displaystyle T} , being thus 1 {\displaystyle 1} if and only if F {\displaystyle F} equals T {\displaystyle T} multiplied by

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3200-917: Is the expected value operator. Note that this expression may be not defined. Subtracting the mean before multiplication yields the cross-covariance between times t 1 {\displaystyle t_{1}} and t 2 {\displaystyle t_{2}} : K X Y ⁡ ( t 1 , t 2 ) ≜   E ⁡ [ ( X t 1 − μ X ( t 1 ) ) ( Y t 2 − μ Y ( t 2 ) ) ¯ ] {\displaystyle \operatorname {K} _{XY}(t_{1},t_{2})\triangleq \ \operatorname {E} \left[\left(X_{t_{1}}-\mu _{X}(t_{1})\right){\overline {(Y_{t_{2}}-\mu _{Y}(t_{2}))}}\right]} Note that this expression

3300-416: Is the first time astronomers have been able to measure polarisation so close to the edge of a black hole. The lines on the photo mark the orientation of polarisation, which is related to the magnetic field around the shadow of the black hole. In August 2022, a team led by University of Waterloo researcher Avery Broderick released a "remaster[ed]" version of original image generated from the data collected by

3400-393: Is the number of pixels in t ( x , y ) {\displaystyle t(x,y)} and f ( x , y ) {\displaystyle f(x,y)} , μ f {\displaystyle \mu _{f}} is the average of f {\displaystyle f} and σ f {\displaystyle \sigma _{f}}

3500-463: Is up to 25 times better than the resolution of a single VLT unit telescope. The VLTI gives astronomers the ability to study celestial objects in unprecedented detail. It is possible to see details on the surfaces of stars and even to study the environment close to a black hole. With a spatial resolution of 4 milliarcseconds, the VLTI has allowed astronomers to obtain one of the sharpest images ever of

3600-489: The Blandford–Znajek process . Producing an image from data from an array of radio telescopes requires much mathematical work. Four independent teams created images to assess the reliability of the results. These methods included both an established algorithm in radio astronomy for image reconstruction known as CLEAN , invented by Jan Högbom , as well as self-calibrating image processing methods for astronomy such as

3700-556: The CHIRP algorithm created by Katherine Bouman and others. The algorithms that were ultimately used were a regularized maximum likelihood (RML) algorithm and the CLEAN algorithm. In March 2020, astronomers proposed an improved way of seeing more of the rings in the first black hole image. In March 2021, a new photo was revealed, showing how the M87 black hole looks in polarised light. This

3800-520: The COVID-19 pandemic , weather patterns, and celestial mechanics, the 2020 observational campaign was postponed to March 2021. The Event Horizon Telescope Collaboration announced its first results in six simultaneous press conferences worldwide on April 10, 2019. The announcement featured the first direct image of a black hole, which showed the supermassive black hole at the center of Messier 87 , designated M87*. The scientific results were presented in

3900-660: The IOTA array. A number of other interferometers have made closure phase measurements and are expected to produce their first images soon, including the VLT I, the CHARA array and Le Coroller and Dejonghe 's Hypertelescope prototype. If completed, the MRO Interferometer with up to ten movable telescopes will produce among the first higher fidelity images from a long baseline interferometer. The Navy Optical Interferometer took

4000-650: The Keck Interferometer and the Palomar Testbed Interferometer . In the 1980s the aperture synthesis interferometric imaging technique was extended to visible light and infrared astronomy by the Cavendish Astrophysics Group , providing the first very high resolution images of nearby stars. In 1995 this technique was demonstrated on an array of separate optical telescopes for the first time, allowing

4100-633: The South Pole Telescope arrived. Data collected on hard drives are transported by commercial freight airplanes (a so-called sneakernet ) from the various telescopes to the MIT Haystack Observatory and the Max Planck Institute for Radio Astronomy , where the data are cross-correlated and analyzed on a grid computer made from about 800 CPUs all connected through a 40 Gbit/s network. Because of

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4200-570: The Very Long Baseline Array imaged the distant blazar J1924-2914. They operated at 230 GHz, 86 GHz and 2.3+8.7 GHz, respectively, the highest angular resolution images of polarized emission from a quasar ever obtained. Observations reveal a helically bent jet and the polarization of its emission suggest a toroidal magnetic field structure. The object is used as calibrator for Sagittarius A* sharing strong optical variability and polarization with it. In February 2023,

4300-420: The complex conjugate of f ( t ) {\displaystyle f(t)} , and τ {\displaystyle \tau } is called displacement or lag. For highly-correlated f {\displaystyle f} and g {\displaystyle g} which have a maximum cross-correlation at a particular τ {\displaystyle \tau } ,

4400-702: The conjugate of f {\displaystyle f} ensures that aligned peaks (or aligned troughs) with imaginary components will contribute positively to the integral. In econometrics , lagged cross-correlation is sometimes referred to as cross-autocorrelation. For random vectors X = ( X 1 , … , X m ) {\displaystyle \mathbf {X} =(X_{1},\ldots ,X_{m})} and Y = ( Y 1 , … , Y n ) {\displaystyle \mathbf {Y} =(Y_{1},\ldots ,Y_{n})} , each containing random elements whose expected value and variance exist,

4500-2037: The cross-correlation matrix of X {\displaystyle \mathbf {X} } and Y {\displaystyle \mathbf {Y} } is defined by R X Y ≜   E ⁡ [ X Y ] {\displaystyle \operatorname {R} _{\mathbf {X} \mathbf {Y} }\triangleq \ \operatorname {E} \left[\mathbf {X} \mathbf {Y} \right]} and has dimensions m × n {\displaystyle m\times n} . Written component-wise: R X Y = [ E ⁡ [ X 1 Y 1 ] E ⁡ [ X 1 Y 2 ] ⋯ E ⁡ [ X 1 Y n ] E ⁡ [ X 2 Y 1 ] E ⁡ [ X 2 Y 2 ] ⋯ E ⁡ [ X 2 Y n ] ⋮ ⋮ ⋱ ⋮ E ⁡ [ X m Y 1 ] E ⁡ [ X m Y 2 ] ⋯ E ⁡ [ X m Y n ] ] {\displaystyle \operatorname {R} _{\mathbf {X} \mathbf {Y} }={\begin{bmatrix}\operatorname {E} [X_{1}Y_{1}]&\operatorname {E} [X_{1}Y_{2}]&\cdots &\operatorname {E} [X_{1}Y_{n}]\\\\\operatorname {E} [X_{2}Y_{1}]&\operatorname {E} [X_{2}Y_{2}]&\cdots &\operatorname {E} [X_{2}Y_{n}]\\\\\vdots &\vdots &\ddots &\vdots \\\\\operatorname {E} [X_{m}Y_{1}]&\operatorname {E} [X_{m}Y_{2}]&\cdots &\operatorname {E} [X_{m}Y_{n}]\end{bmatrix}}} The random vectors X {\displaystyle \mathbf {X} } and Y {\displaystyle \mathbf {Y} } need not have

4600-431: The (circular) cross-correlation is defined as: ( f ⋆ g ) [ n ]   ≜ ∑ m = 0 N − 1 f [ m ] ¯ g [ ( m + n ) mod   N ] {\displaystyle (f\star g)[n]\ \triangleq \sum _{m=0}^{N-1}{\overline {f[m]}}g[(m+n)_{{\text{mod}}~N}]} which

4700-640: The EHT array has moved to add more observatories to its global network of radio telescopes. The first image of the Milky Way's supermassive black hole, Sagittarius A*, was expected to be produced from data taken in April 2017, but because there are no flights in or out of the South Pole during austral winter (April to October), the full data set could not be processed until December 2017, when the shipment of data from

4800-479: The EHT reported on the observations of the quasar NRAO 530. NRAO 530 (1730−130, J1733−1304) is a flat-spectrum radio quasar (FSRQ) that belongs to the class of bright γ-ray blazars and shows significant variability across the entire electromagnetic spectrum. The source was monitored by the University of Michigan Radio Observatory at 4.8, 8.4, and 14.5 GHz for several decades until 2012. The quasar underwent

4900-405: The EHT so far. The team reconstructed the first images of the source at 230 GHz, at an angular resolution of ~20 μas, both in total intensity and in linear polarization (LP). Source variability was not detected, that allowed to represent the whole data set with static images. The images reveal a bright feature located on the southern end of the jet, which was associated with the core. The feature

5000-466: The EHT. This image "resolve[d] a fundamental signature of gravity around a black hole," with it showing a displaying photon ring around M87* .The claim has been subsequently disputed. In 2023, EHT released new, sharper images of the M87 black hole, reconstructed from the same 2017 data but created using the PRIMO algorithm. In April 2020, the EHT released the first 20 microarcsecond resolution images of

5100-500: The above sum is equal to ⟨ F ‖ F ‖ , T ‖ T ‖ ⟩ {\displaystyle \left\langle {\frac {F}{\|F\|}},{\frac {T}{\|T\|}}\right\rangle } where ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } is the inner product and ‖ ⋅ ‖ {\displaystyle \|\cdot \|}

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5200-402: The archetypal blazar 3C 279 it observed in April 2017. These images, generated from observations over 4 nights in April 2017, reveal bright components of a jet whose projection on the observer plane exhibit apparent superluminal motions with speeds up to 20 c. Such apparent superluminal motion from relativistic emitters such as an approaching jet is explained by emission originating closer to

5300-559: The beam combiner (focus) are the same as would be given by the complete mirror case. Instead, most existing arrays use a planar geometry, and Labeyrie 's hypertelescope will use a spherical geometry. One of the first uses of optical interferometry was applied by the Michelson stellar interferometer on the Mount Wilson Observatory 's reflector telescope to measure the diameters of stars. The red giant star Betelgeuse

5400-406: The best angular resolution at astronomical facilities on Earth. The EHT is composed of many radio observatories or radio-telescope facilities around the world, working together to produce a high-sensitivity, high-angular-resolution telescope. Through the technique of very-long-baseline interferometry (VLBI), many independent radio antennas separated by hundreds or thousands of kilometres can act as

5500-436: The black hole is not large enough for its photon sphere to be observed, as in EHT images of Messier M87*, but its jet extends even beyond its host galaxy while staying as a highly collimated beam which is a point of study. Edge-brightening of the jet was also observed which would exclude models of particle acceleration that are unable to reproduce this effect. The image was 16 times sharper than previous observations and utilized

5600-435: The brightness of the image and template can vary due to lighting and exposure conditions, the images can be first normalized. This is typically done at every step by subtracting the mean and dividing by the standard deviation . That is, the cross-correlation of a template t ( x , y ) {\displaystyle t(x,y)} with a subimage f ( x , y ) {\displaystyle f(x,y)}

5700-444: The correlations of the various temporal instances of X {\displaystyle \mathbf {X} } are known as autocorrelations of X {\displaystyle \mathbf {X} } , and the cross-correlations of X {\displaystyle \mathbf {X} } with Y {\displaystyle \mathbf {Y} } across time are temporal cross-correlations. In probability and statistics,

5800-1337: The cross-correlation function to get a time-dependent Pearson correlation coefficient . However, in other disciplines (e.g. engineering) the normalization is usually dropped and the terms "cross-correlation" and "cross-covariance" are used interchangeably. The definition of the normalized cross-correlation of a stochastic process is ρ X X ( t 1 , t 2 ) = K X X ⁡ ( t 1 , t 2 ) σ X ( t 1 ) σ X ( t 2 ) = E ⁡ [ ( X t 1 − μ t 1 ) ( X t 2 − μ t 2 ) ¯ ] σ X ( t 1 ) σ X ( t 2 ) {\displaystyle \rho _{XX}(t_{1},t_{2})={\frac {\operatorname {K} _{XX}(t_{1},t_{2})}{\sigma _{X}(t_{1})\sigma _{X}(t_{2})}}={\frac {\operatorname {E} \left[\left(X_{t_{1}}-\mu _{t_{1}}\right){\overline {\left(X_{t_{2}}-\mu _{t_{2}}\right)}}\right]}{\sigma _{X}(t_{1})\sigma _{X}(t_{2})}}} If

5900-409: The cross-correlation is defined as: ( f ⋆ g ) ( τ )   ≜ ∫ − ∞ ∞ f ( t ) ¯ g ( t + τ ) d t {\displaystyle (f\star g)(\tau )\ \triangleq \int _{-\infty }^{\infty }{\overline {f(t)}}g(t+\tau )\,dt} which

6000-886: The cross-correlation is defined as: ( f ⋆ g ) [ n ]   ≜ ∑ m = − ∞ ∞ f [ m ] ¯ g [ m + n ] {\displaystyle (f\star g)[n]\ \triangleq \sum _{m=-\infty }^{\infty }{\overline {f[m]}}g[m+n]} which is equivalent to: ( f ⋆ g ) [ n ]   ≜ ∑ m = − ∞ ∞ f [ m − n ] ¯ g [ m ] {\displaystyle (f\star g)[n]\ \triangleq \sum _{m=-\infty }^{\infty }{\overline {f[m-n]}}g[m]} For finite discrete functions f , g ∈ C N {\displaystyle f,g\in \mathbb {C} ^{N}} ,

6100-598: The cross-correlation matrix of Z {\displaystyle \mathbf {Z} } and W {\displaystyle \mathbf {W} } is defined by R Z W ≜   E ⁡ [ Z W H ] {\displaystyle \operatorname {R} _{\mathbf {Z} \mathbf {W} }\triangleq \ \operatorname {E} [\mathbf {Z} \mathbf {W} ^{\rm {H}}]} where H {\displaystyle {}^{\rm {H}}} denotes Hermitian transposition . In time series analysis and statistics ,

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6200-404: The cross-correlation of f ( t ) {\displaystyle f(t)} and g ( − t ) {\displaystyle g(-t)} ) gives the probability density function of the sum X + Y {\displaystyle X+Y} . For continuous functions f {\displaystyle f} and g {\displaystyle g} ,

6300-435: The cross-correlation of a pair of random process is the correlation between values of the processes at different times, as a function of the two times. Let ( X t , Y t ) {\displaystyle (X_{t},Y_{t})} be a pair of random processes, and t {\displaystyle t} be any point in time ( t {\displaystyle t} may be an integer for

6400-493: The cross-covariance and cross-correlation are independent of t {\displaystyle t} is precisely the additional information (beyond being individually wide-sense stationary) conveyed by the requirement that ( X t , Y t ) {\displaystyle (X_{t},Y_{t})} are jointly wide-sense stationary. The cross-correlation of a pair of jointly wide sense stationary stochastic processes can be estimated by averaging

6500-459: The cutting edge of astronomical research. At optical wavelengths, aperture synthesis allows the atmospheric seeing resolution limit to be overcome, allowing the angular resolution to reach the diffraction limit of the optics. Astronomical interferometers can produce higher resolution astronomical images than any other type of telescope. At radio wavelengths, image resolutions of a few micro- arcseconds have been obtained, and image resolutions of

6600-426: The definition of correlation always includes a standardising factor in such a way that correlations have values between −1 and +1. If X {\displaystyle X} and Y {\displaystyle Y} are two independent random variables with probability density functions f {\displaystyle f} and g {\displaystyle g} , respectively, then

6700-628: The definition of the cross-correlation between times t 1 {\displaystyle t_{1}} and t 2 {\displaystyle t_{2}} is R X Y ⁡ ( t 1 , t 2 ) ≜   E ⁡ [ X t 1 Y t 2 ¯ ] {\displaystyle \operatorname {R} _{XY}(t_{1},t_{2})\triangleq \ \operatorname {E} \left[X_{t_{1}}{\overline {Y_{t_{2}}}}\right]} where E {\displaystyle \operatorname {E} }

6800-425: The diameter of its event horizon to be approximately 40 billion kilometres (270 AU; 0.0013 pc; 0.0042 ly), roughly 2.5 times smaller than the shadow that it casts, seen at the center of the image. Previous observations of M87 showed that the large-scale jet is inclined at an angle of 17° relative to the observer's line of sight and oriented on the plane of the sky at a position angle of −72°. From

6900-564: The different telescopes to the astronomical instruments where it is combined and processed. This is technically demanding as the light paths must be kept equal to within 1/1000 mm (the same order as the wavelength of light) over distances of a few hundred metres. For the Unit Telescopes, this gives an equivalent mirror diameter of up to 130 metres (430 ft), and when combining the auxiliary telescopes, equivalent mirror diameters of up to 200 metres (660 ft) can be achieved. This

7000-416: The edge of these black holes they look amazingly similar. This tells us that General Relativity governs these objects up close, and any differences we see further away must be due to differences in the material that surrounds the black holes." On March 22, 2024, the EHT Collaboration released an image of Sagittarius A* in polarized light. In August 2022, the EHT together with Global Millimeter VLBI Array and

7100-416: The enhanced brightness of the southern part of the ring due to relativistic beaming of approaching funnel wall jet emission, EHT concluded the black hole, which anchors the jet, spins clockwise, as seen from Earth. EHT simulations allow for both prograde and retrograde inner disk rotation with respect to the black hole, while excluding zero black hole spin using a conservative minimum jet power of 10 erg/s via

7200-406: The entries of X {\displaystyle \mathbf {X} } itself, those forming the correlation matrix of X {\displaystyle \mathbf {X} } . If each of X {\displaystyle \mathbf {X} } and Y {\displaystyle \mathbf {Y} } is a scalar random variable which is realized repeatedly in a time series , then

7300-641: The first step in this direction in 1996, achieving 3-way synthesis of an image of Mizar ; then a first-ever six-way synthesis of Eta Virginis in 2002; and most recently " closure phase " as a step to the first synthesized images produced by geostationary satellites . Astronomical interferometry is principally conducted using Michelson (and sometimes other type) interferometers. The principal operational interferometric observatories which use this type of instrumentation include VLTI , NPOI , and CHARA . Current projects will use interferometers to search for extrasolar planets , either by astrometric measurements of

7400-409: The forces giving rise to quasars . Future plans involve improving the array's resolution by adding new telescopes and by taking shorter-wavelength observations. On 12 May 2022, astronomers unveiled the first image of the supermassive black hole at the center of the Milky Way , Sagittarius A* . Recently EHT Project has reported to have reached the resolution of 870 μm at 345 GHz, that is pair to 19 μas,

7500-497: The four 8.2-metre (320 in) unit telescopes, four mobile 1.8-metre auxiliary telescopes (ATs) were included in the overall VLT concept to form the Very Large Telescope Interferometer (VLTI). The ATs can move between 30 different stations, and at present, the telescopes can form groups of two or three for interferometry. When using interferometry, a complex system of mirrors brings the light from

7600-1149: The function ρ X X {\displaystyle \rho _{XX}} is well-defined, its value must lie in the range [ − 1 , 1 ] {\displaystyle [-1,1]} , with 1 indicating perfect correlation and −1 indicating perfect anti-correlation . For jointly wide-sense stationary stochastic processes, the definition is ρ X Y ( τ ) = K X Y ⁡ ( τ ) σ X σ Y = E ⁡ [ ( X t − μ X ) ( Y t + τ − μ Y ) ¯ ] σ X σ Y {\displaystyle \rho _{XY}(\tau )={\frac {\operatorname {K} _{XY}(\tau )}{\sigma _{X}\sigma _{Y}}}={\frac {\operatorname {E} \left[\left(X_{t}-\mu _{X}\right){\overline {\left(Y_{t+\tau }-\mu _{Y}\right)}}\right]}{\sigma _{X}\sigma _{Y}}}} The normalization

7700-749: The integration from − ∞ {\displaystyle -\infty } to ∞ {\displaystyle \infty } is replaced by integration over any interval [ t 0 , t 0 + T ] {\displaystyle [t_{0},t_{0}+T]} of length T {\displaystyle T} : ( f ⋆ g ) ( τ )   ≜ ∫ t 0 t 0 + T f ( t ) ¯ g ( t + τ ) d t {\displaystyle (f\star g)(\tau )\ \triangleq \int _{t_{0}}^{t_{0}+T}{\overline {f(t)}}g(t+\tau )\,dt} which

7800-450: The jet axis, consistent with a helical structure of the magnetic field in the jet. The outermost feature has a particularly high degree of LP, suggestive of a nearly uniform magnetic field. The EHT Collaboration consists of 13 stakeholder institutes: The EHT Collaboration receives funding from numerous sources including: Additionally, Western Digital and Xilinx are industry donors. Astronomical interferometer Interferometry

7900-787: The kernel cross-correlation is defined as: ( f ⋆ g ) [ n ]   ≜ ∑ m = 0 N − 1 f [ m ] ¯ K g [ ( m + n ) mod   N ] {\displaystyle (f\star g)[n]\ \triangleq \sum _{m=0}^{N-1}{\overline {f[m]}}K_{g}[(m+n)_{{\text{mod}}~N}]} where K g = [ k ( g , T 0 ( g ) ) , k ( g , T 1 ( g ) ) , … , k ( g , T N − 1 ( g ) ) ] {\displaystyle K_{g}=[k(g,T_{0}(g)),k(g,T_{1}(g)),\dots ,k(g,T_{N-1}(g))]}

8000-595: The limited sense of angular resolution . The amount of light gathered—and hence the dimmest object that can be seen—depends on the real aperture size, so an interferometer would offer little improvement as the image is dim (the thinned-array curse ). The combined effects of limited aperture area and atmospheric turbulence generally limits interferometers to observations of comparatively bright stars and active galactic nuclei . However, they have proven useful for making very high precision measurements of simple stellar parameters such as size and position ( astrometry ), for imaging

8100-498: The mass of which is currently uncertain, with estimates ranging from 3 × 10 M☉ to 2 × 10 M☉. It was observed with the Event Horizon Telescope on 2017 April 5−7, when NRAO 530 was used as a calibrator for the EHT observations of Sagittarius A*. The observations were performed with the full EHT 2017 array of eight telescopes located at six geographical sites. At z = 0.902, this is the most distant object imaged by

8200-417: The mean and standard deviation of the process ( X t ) {\displaystyle (X_{t})} , which are constant over time due to stationarity; and similarly for ( Y t ) {\displaystyle (Y_{t})} , respectively. E ⁡ [   ] {\displaystyle \operatorname {E} [\ ]} indicates the expected value . That

8300-523: The nearest giant stars and probing the cores of nearby active galaxies . For details of individual instruments, see the list of astronomical interferometers at visible and infrared wavelengths . At radio wavelengths, interferometers such as the Very Large Array and MERLIN have been in operation for many years. The distances between telescopes are typically 10–100 km (6.2–62.1 mi), although arrays with much longer baselines utilize

8400-409: The observer (downstream along the jet) catching up with emission originating further from the observer (at the jet base) as the jet propagates close to the speed of light at small angles to the line of sight. In July 2021, high resolution images of the jet produced by the supermassive black hole sitting at the center of Centaurus A were released. With a mass around 5.5 × 10   M ☉ ,

8500-574: The photon orbit and first simulations of what a black hole would look like progressed to predictions of VLBI imaging for the Galactic Center black hole, Sgr A*. Technical advances in radio observing moved from the first detection of Sgr A*, through VLBI at progressively shorter wavelengths, ultimately leading to detection of horizon scale structure in both Sgr A* and M87. The collaboration now comprises over 300 members, and 60 institutions, working in over 20 countries and regions. The first image of

8600-436: The probability density of the difference Y − X {\displaystyle Y-X} is formally given by the cross-correlation (in the signal-processing sense) f ⋆ g {\displaystyle f\star g} ; however, this terminology is not used in probability and statistics. In contrast, the convolution f ∗ g {\displaystyle f*g} (equivalent to

8700-477: The product of samples measured from one process and samples measured from the other (and its time shifts). The samples included in the average can be an arbitrary subset of all the samples in the signal (e.g., samples within a finite time window or a sub-sampling of one of the signals). For a large number of samples, the average converges to the true cross-correlation. It is common practice in some disciplines (e.g. statistics and time series analysis ) to normalize

8800-630: The reciprocal motion of the star (as used by the Palomar Testbed Interferometer and the VLT I), through the use of nulling (as will be used by the Keck Interferometer and Darwin ) or through direct imaging (as proposed for Labeyrie 's Hypertelescope). Engineers at the European Southern Observatory ESO designed the Very Large Telescope VLT so that it can also be used as an interferometer. Along with

8900-636: The same dimension, and either might be a scalar value. Where E {\displaystyle \operatorname {E} } is the expectation value . For example, if X = ( X 1 , X 2 , X 3 ) {\displaystyle \mathbf {X} =\left(X_{1},X_{2},X_{3}\right)} and Y = ( Y 1 , Y 2 ) {\displaystyle \mathbf {Y} =\left(Y_{1},Y_{2}\right)} are random vectors, then R X Y {\displaystyle \operatorname {R} _{\mathbf {X} \mathbf {Y} }}

9000-435: The shadow, we could compare our observations to extensive computer models that include the physics of warped space, superheated matter, and strong magnetic fields. Many of the features of the observed image match our theoretical understanding surprisingly well." The image also provided new measurements for the mass and diameter of M87*. EHT measured the black hole's mass to be 6.5 ± 0.7 billion solar masses and measured

9100-419: The signal energy. In probability and statistics , the term cross-correlations refers to the correlations between the entries of two random vectors X {\displaystyle \mathbf {X} } and Y {\displaystyle \mathbf {Y} } , while the correlations of a random vector X {\displaystyle \mathbf {X} } are the correlations between

9200-766: The sizes of and distances to Cepheid variable stars, and young stellar objects . High on the Chajnantor plateau in the Chilean Andes, the European Southern Observatory (ESO), together with its international partners, is building ALMA, which will gather radiation from some of the coldest objects in the Universe. ALMA will be a single telescope of a new design, composed initially of 66 high-precision antennas and operating at wavelengths of 0.3 to 9.6 mm. Its main 12-meter array will have fifty antennas, 12 metres in diameter, acting together as

9300-724: The techniques of Very Long Baseline Interferometry . In the (sub)-millimetre, existing arrays include the Submillimeter Array and the IRAM Plateau de Bure facility. The Atacama Large Millimeter Array has been fully operational since March 2013. Max Tegmark and Matias Zaldarriaga have proposed the Fast Fourier Transform Telescope which would rely on extensive computer power rather than standard lenses and mirrors. If Moore's law continues, such designs may become practical and cheap in

9400-400: The time delay between two signals, e.g., for determining time delays for the propagation of acoustic signals across a microphone array. After calculating the cross-correlation between the two signals, the maximum (or minimum if the signals are negatively correlated) of the cross-correlation function indicates the point in time where the signals are best aligned; i.e., the time delay between

9500-411: The two black holes with the largest angular diameter as observed from Earth: the black hole at the center of the supergiant elliptical galaxy Messier 87 , and Sagittarius A* , at the center of the Milky Way . The Event Horizon Telescope project is an international collaboration that was launched in 2009 after a long period of theoretical and technical developments. On the theory side, work on

9600-511: The two signals is determined by the argument of the maximum, or arg max of the cross-correlation , as in τ d e l a y = a r g m a x t ∈ R ( ( f ⋆ g ) ( t ) ) {\displaystyle \tau _{\mathrm {delay} }={\underset {t\in \mathbb {R} }{\operatorname {arg\,max} }}((f\star g)(t))} Terminology in image processing For image-processing applications in which

9700-512: The value of ( f ⋆ g ) {\displaystyle (f\star g)} is maximized. This is because when peaks (positive areas) are aligned, they make a large contribution to the integral. Similarly, when troughs (negative areas) align, they also make a positive contribution to the integral because the product of two negative numbers is positive. With complex-valued functions f {\displaystyle f} and g {\displaystyle g} , taking

9800-407: The x-axis. One can use the cross-correlation to find how much g {\displaystyle g} must be shifted along the x-axis to make it identical to f {\displaystyle f} . The formula essentially slides the g {\displaystyle g} function along the x-axis, calculating the integral of their product at each position. When the functions match,

9900-542: Was extended to measurements using separated telescopes by Johnson, Betz and Townes (1974) in the infrared and by Labeyrie (1975) in the visible. In the late 1970s improvements in computer processing allowed for the first "fringe-tracking" interferometer, which operates fast enough to follow the blurring effects of astronomical seeing , leading to the Mk I, II and III series of interferometers. Similar techniques have now been applied at other astronomical telescope arrays, including

10000-546: Was the first to have its diameter determined in this way on December 13, 1920. In the 1940s radio interferometry was used to perform the first high resolution radio astronomy observations. For the next three decades astronomical interferometry research was dominated by research at radio wavelengths, leading to the development of large instruments such as the Very Large Array and the Atacama Large Millimeter Array . Optical/infrared interferometry

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