Synthetic Fourier Domain Optical Coherence Tomography

A novel approach for image formation in optical coherence tomography (OCT) and microscopy is presented. The depth resolution of OCT, including recently developed nanosensitive OCT (nsOCT), is limited by the spectral bandwidth of the light source used for illumination. The proposed approach, synthetic OCT (synOCT), permits label‐free, depth‐resolved quantitative visualization of the subwavelength‐sized structures with nanosensitivity. Using synOCT it is possible to estimate the contribution of axial Fourier components of an object's structure in image formation at each small volume within the image. The size of such areas can be smaller than the resolution limit of the imaging system that provides potential for super‐resolution imaging. Visualization of the subwavelength periodic structures and quantitative visualization of the subwavelength internal structures of highly scattering biological samples, within voxels smaller than resolution limit of the imaging system, are demonstrated. In contrast to nsOCT, the trade‐off between spectral and spatial resolution is removed which results in dramatic improvement of both spectral and spatial resolution in synOCT relative to nsOCT.


Introduction
Optical coherence tomography (OCT) provides noninvasive, contactless, depth-resolved imaging of an object's internal structure. [1]Nowadays, Fourier domain OCT (FDOCT), including both spectral domain OCT (SDOCT) and swept source OCT (SSOCT), is the most popular of the OCT techniques and is based on the spectral detection of light interference signals and the inverse Fourier transform of these signals to reconstruct images of the depth-wise light scattering profile within a sample.[4] However, the resolution and sensitivity of the typical intensity-based OCT system is about 5-10 μm which is not sufficient in many applications, including the early detection of biological tissue pathologies, such as cancer, where the visualization and analysis of morphological features below the typical resolution of OCT is necessary.
The common approach to improve the axial resolution of OCT is to use a light source with a broader spectral bandwidth.[7] However, this approach can be cumbersome, impractical, and adds complexity to the imaging process and system configuration, introducing problems such as chromatic dispersion.Another group of techniques for improvement of axial resolution at given spectral bandwidths, usually called super-resolution OCT, have been published.This group includes deconvolution methods, [8,9] different sampling methods in K-space, [10] obtaining super-resolved OCT images from recording multiple lower resolution images, [11] spectral estimation OCT to avoid digital Fourier transform, [12,13] modeling of OCT images by the scale mixture models, [14] and generative adversarial network-based techniques. [15]However, methods which extract more information from the limited spectral bandwidth of OCT interference signals are still not widely recognized and should be further explored.
OCT usually works in reflection configuration, where the presented spectral interference signal from an object contains information about its small, subwavelength-sized structures which backscatter incident light.18][19][20][21][22][23][24][25][26] Here, we introduce and demonstrate a novel imaging approach which permits the estimation of the contribution of each detected axial spatial Fourier component of the object's structure in image formation.This can be achieved at any small areas within the image and provides quantitative access to subwavelength-sized structures in an object with nanosensitivity to structural changes.The size of each area, where the axial spatial frequency profiles are reconstructed, can be smaller than the DOI: 10.1002/adpr.202300260A novel approach for image formation in optical coherence tomography (OCT) and microscopy is presented.The depth resolution of OCT, including recently developed nanosensitive OCT (nsOCT), is limited by the spectral bandwidth of the light source used for illumination.The proposed approach, synthetic OCT (synOCT), permits label-free, depth-resolved quantitative visualization of the subwavelength-sized structures with nanosensitivity.Using synOCT it is possible to estimate the contribution of axial Fourier components of an object's structure in image formation at each small volume within the image.The size of such areas can be smaller than the resolution limit of the imaging system that provides potential for super-resolution imaging.Visualization of the subwavelength periodic structures and quantitative visualization of the subwavelength internal structures of highly scattering biological samples, within voxels smaller than resolution limit of the imaging system, are demonstrated.In contrast to nsOCT, the trade-off between spectral and spatial resolution is removed which results in dramatic improvement of both spectral and spatial resolution in synOCT relative to nsOCT.resolution limit of the imaging system, which provides the basis for super-resolution imaging.We call this approach synthetic OCT (synOCT).

Principle of synOCT
The concept of the synthetic aperture was first applied to increase the resolution of the radio telescope and to synthetic aperture radar. [27,28]Later this concept was applied to improve the lateral resolution of optical imaging systems, [29][30][31][32][33] including optical coherence microscopy. [34,35]The synthetic aperture approach improves lateral resolution by extending the spectrum of collected lateral spatial frequencies, corresponding to a larger aperture in the spatial domain than the objective aperture of the imaging system.In contrast to synthetic aperture techniques, the proposed synOCT approach deals with the range of axial spatial frequencies, as opposed to lateral.The axial spatial frequencies are encoded as corresponding wavelengths, and so information about high spatial frequencies can be passed through the imaging system independently of the numerical aperture.The synthetic nature of synOCT refers to the synthesis of the full spectrum of collected axial spatial frequencies, limited by the bandwidth of the illuminating source, including the negative part of the spectrum, and to the synthesis of corresponding axial spatial harmonics.Therefore, synOCT provides new imaging possibilities, namely, permitting quantitative access to the subwavelength structure within an object at areas, smaller than the resolution limit of the imaging system, with nanosensitivity to structural changes.
It is known that an object's structure can be represented as a 3D Fourier transform of spatial frequencies of its scattering potential. [36]Physically, these frequencies correspond to spatial, harmonically varying refractive indices with different periodicities, amplitudes, phases, and orientations within an object.The complex amplitude of the scattered wave contains information about the entire 3D structure of the object which is described by the spatial frequency vector. [22,36,37]The complex amplitude of the scattered wave at a given wavelength in the far zone for a given direction depends entirely on only one Fourier component (one spatial frequency) of the 3D scattering potential, labeled by the vector K, Figure 1a, where s and s 0 are unit vectors of scattered and illumination waves. [36]][38] At constant incident angle, θ = 0, and for broadband illumination, the Ewald's spheres for different wavelengths are presented in Figure 1 as Sp1, where calculation was done for a typical OCT system with central wavelength 1300 nm.The axial, along z, components of the spatial frequency K-vector can be written as [39] where n is the refractive index, θ is the incident angle, and α is the scattering angle.
For OCT, all accessible spatial frequencies are along the black arrow representing the K-vector.In this case, this vector can be written as Only a narrow bandwidth of high spatial frequencies, Δv z , which depends on the spectral bandwidth, can be detected using OCT (Figure 1, Sp1).The OCT interference signal at each wavelength is generated from back scattered light diffracted by structures with the corresponding high spatial frequency/periodicity.Thus, information about subwavelength structures is present in the OCT signal at the wavelength corresponding to this spatial periodicity. [22]So, the OCT signal should be placed in the region of high spatial frequencies in the Fourier domain. [16,17,22]If the OCT signal at some spatial frequency can be detected, then the harmonic structure with the corresponding spatial periodicity is present within the object.However, information about small, subwavelength-sized structures is lost in conventional OCT during the Fourier transform. [16,22]o improve the resolution, additional regions of K-space can be detected, for example, by increasing the spectral bandwidth of the light source.Another option is changing the incident and/or collection angles. [36]Corresponding Ewald's spheres for different illumination and collection angles at a single wavelength are shown in Figure 1 in red.If the object is illuminated from the opposite side, i.e., the incident angle is θ = π, then, as it follows from (1), the region of detected spatial frequencies will be extended by an additional set of Ewald's spheres, see Figure 1b and Sp2.These spheres are mirror symmetrical relative to the initial spheres in Figure 1b and Sp1, and provide similar spatial frequency content, but with negative frequencies.For OCT, with θ = 0 and α = 0, Equation (1) will be simplified as v z ¼ 2n λ , and for illumination and collection from the opposite side, for θ = π and α = π, Equation (1) will be written as v z ¼ À 2n λ .This means that waves scattered from the opposite direction, which provides information about negative spatial frequencies of the object's structure, are similar to the initial scattering wave, which provides information about positive spatial frequencies.Thus, instead of illuminating and collecting from opposite side, the scattered from opposite direction wave, at the first Born approximation, can be formed numerically from the initial wave by changing positive spatial frequencies to negative.
Let us consider the OCT signal formed by the light backscattered from just one spatial frequency component (spatial harmonic structure) of an object, which is within the detected bandwidth of spatial frequencies, presented in Figure 2. The Fourier spectrum of such spatial harmonics consists of two components, positive and negative.In conventional OCT, only the positive spatial frequency component of the Fourier spectrum, specified by the illuminating source wavelength bandwidth, is detected and used for imaging (Figure 2a,b).Thus, it is impossible to reconstruct the corresponding harmonic spatial periodicity of the structure within the object, where the light is diffracted, and so this structure cannot be resolved, as shown in Figure 2c.To reconstruct such a spatial harmonic within an OCT image, it is necessary to have both the positive and negative components of the Fourier spectrum (Figure 2e).As it was shown above (Figure 1), instead of illuminating the object from the opposite side and detecting the complex amplitude of the reflected light, it is possible to form the negative part of the Fourier spectrum numerically.After this, the synthetic Fourier spectrum of axial spatial frequencies can be formed.
The first step in forming a synOCT image of the object with resolvable harmonic structure, after conventional OCT preprocessing, is to create a spectrum of spatial frequencies with intensity values of zero from zero spatial frequency (ν z = 0) to the minimum spatial frequency (ν zmin ), which is determined by the source bandwidths maximum wavelength.Then the spectral interference values corresponding to the light scattered from the structure over the available spatial frequency range (ν zmin to ν zmax ) are imposed by concatenating with the preceding zeros.Second, the corresponding signal at negative spatial frequencies (Àν zmax to 0) is synthesized, as shown in Figure 2e.In this case, the signal will be fixed at the proper location in the region of high spatial frequencies of the Fourier domain, with its conjugate spectrum.As a result, both components of the Fourier spectrum are formed.After computing the Fourier transform of this synthesized spectrum, the fine structure, which produces signals at a given spatial frequency, will be reconstructed (Figure 2f ).In the realistic case, where an object being scanned is composed of complex structures consisting of many spatial frequency components, this procedure will reconstruct the fine structures of the object as a superposition of all spatial frequency harmonics over the given bandwidth of illumination.The synthesized Fourier spectrum of the object can be written as where U 1 (v z ) is the collected OCT signal, converted to spatial frequencies, [16,22] v zmin and v zmax are the minimum and maximum detected spatial frequencies respectively, and I 0 is the constant which determines the DC level.Thus, the fine structure of the object will be reconstructed after computing the Fourier transform (Figure 2f ) For a rectangular spectral shape of the light source and for a single scatterer within the object, and after Fourier transform of the synthesized spectrum, the depth profile can be described as [Figure S1 and Equation S(9), Supporting information] where U s ðzÞ ¼ 2U 0 sincðπΔv z zÞ cosð2πv zc zÞ (6) Δv z is the bandwidth of the axial spatial frequencies, which depends on the spectral bandwidth of the light source and v zc is the median spatial frequency, which provides information about the location of the spatial frequencies in the Fourier domain.Thus, in contrast to conventional OCT, in the synOCT image we have information about axial spatial frequency content at each location within the image.
To realize comprehensive information about spatial harmonics of the object's structure, the synthesized Fourier spectrum of spatial frequencies can be decomposed into sub-bands, where the width of each band, δv z , can be equal to the spectral width of one pixel of the photodetector array (SDOCT), or one time instance (SSOCT) (Figure 3).
The collected signal at each ith sub-band with median spatial frequency v zci = 2 nλ i À1 provides information about the corresponding spatial frequency.Then the Fourier transform can be applied to signals at each pair of sub-bands, including the positive and negative components.After Fourier transform each pair will generate the corresponding axial spatial harmonic of the object's structure.Example of such spatial harmonics is presented in Figure 3 for single scatterer.Then the depth profile can be formed as a superposition of all such spatial harmonics, detected from the spectral bandwidth of the illuminating light where I si (z) is the ith reconstructed axial spatial frequency harmonic.
Alternative approach for reconstruction of the spatial harmonics in Figure 3 could be if we convert the spectral interference FDOCT signal into the Fourier spectrum of axial spatial frequencies and put it in proper location in Fourier domain at v zc , U(v-v zc ) using zero padding.After the Fourier spectrum of spatial frequencies is decomposed into small sub-bands δv z , the Fourier transform can be applied to each sub-band.Result of Fourier transform of each sub-band using shift theorem can be written as and after applying Euler's formula where U is (z) is a result of the Fourier transform of U i (v.)If each sub-band δv z is small, for example, equal to spectral width of one pixel, then the corresponding signal can be represented as rectangle.In this case Equation ( 10) forms envelop which limits the depth range, similar to conventional FDOCT.However, in synOCT, where we have access to all spatial harmonics which form the image, this envelop could be ignored, and spatial harmonics can be extended numerically to increase the depth range.After conventional processing of the FDOCT signal, we have just I(z), so no information about captured high spatial frequencies is present.Using synOCT the corresponding spatial harmonics in Figure 3 can be formed, for example, taking the real part of (9).In this case, Equation (9) will be equivalent to Equation (6).After that the depth profile can be reconstructed as a superposition of all spatial harmonics.Extension of the spectrum by including negative frequencies permits to avoid of the imaginary part in (9).In both cases, we can see contribution of each spatial frequency to each point within the image, which permits to characterize the structure within each small region of the image.Figure 3 and the experimental video 1 in Supporting Information show how the depth profile of a single scatterer, glass surface, is formed via superposition of spatial harmonics with corresponding amplitudes and phases.
Information about spatial harmonics permits the reconstruction of the axial spatial frequency/periodicity profiles at each point within the image.At each point z i in the depth profile, it is possible to take the intensity from each spatial harmonics and plot the spatial frequency/periodicity profile I(v z ) at point z i .These profiles have significant differences from axial spatial frequency profiles obtained using nsOCT.The synOCT technique is free from the trade-off between spectral and spatial resolution, [16] so both spatial and spectral resolution are significantly improved in comparison with nsOCT.As a result, the axial spatial frequency profiles can be reconstructed for small areas, with sizes smaller than the axial resolution limit of the imaging system.In synOCT, the number of points in the spatial frequency profiles is usually equal to a number of pixels in the detector, or time instances.For example, the TELESTO III SDOCT system, used in our experiments, has 2048 points, whereas for nsOCT, due to the spectral windowing over multiple sub-bands, it is typically about 10-20 points.
Similar to nsOCT, color synOCT images can be formed as a color map of some informative parameters of these profiles, depending on application.For example, the image can be formed as a color map of the spatial period with the maximum intensity, or as a mean spatial period, etc.These colormaps add a contrast mechanism to sample images, which can aid in discriminating between samples, which are typically indistinguishable in conventional intensity-based OCT imaging.To visualize the nanoscale structural changes, the image can be formed as a correlation map between axial spatial frequency profiles, which can be compared in space or in time, [17] or as a difference between axial spatial frequency profiles, etc.In addition to this, the developed in Raman imaging spectroscopy data processing, for example, described in Table 2 in ref. [40], can be adapted to synOCT, where the axial spatial frequency profiles can be used instead of Raman spectra.

Experimental Validation of SynOCT
For experimental validation of this approach, we imaged two samples with well-known internal periodic structures (corresponding to variations in refractive index) with different spatial periods obtained from OptiGrate Corp., USA.The axial spatial periods of the structure, H z , within the samples are 431.6 and 441.7 nm, Figure 4a, the refractive index is n = 1.483, and the variation amplitude of the refractive index spatial harmonics is Δn = 0.003.
These samples are good representations of two different axial spatial frequency components that can be present within samples of complicated internal structure.The samples were imaged using the TELESTO III SDOCT system from Thorlabs, Inc. with a central wavelength 1300 nm, depth resolution of 5.5 μm, and depth range of about 3.6 mm in air.The conventional OCT image (B-scan) of these samples is presented in Figure 4b.The spatial periods of the structures within the samples are about 8.6 times smaller than the depth resolution of the SDOCT system, so it is impossible to resolve these structures using conventional OCT imaging.Application of the synOCT approach permits clear resolution of the structure within both samples, as can be seen in the magnified portions of synOCT images in comparison with OCT images in Figure 4d,e.Measured physical spatial periods averaged within selected areas of the samples are H z1 = 440.25 nm, with a standard deviation (SD) of 0.08 nm, and H z2 = 430.09nm, with SD = 0.06 nm.The difference between periods, which is just about 10 nm, can be clearly detected.It is impossible to visualize and resolve such small structures along depth and perform quantitative characterization using known optical imaging techniques.The examples of the axial spatial periodicity and absolute spatial periodicity profiles at selected points are presented in Figure 3Sb,c in Supporting information.The spectral peaks, which correspond to periods of the internal structure within the samples, can be clearly seen in these profiles.Any region of interest, corresponding to a physical volume/set of voxels, can be selected within the image and the dominant spatial periods calculated within the selected volumes to give a quantitative indication of the structure.
For further visualization of the size of structures in an object in larger regions, where difference in periodicity cannot be seen, and ease of interpretation, a color map can be created.Each pixel in the color map is assigned its dominant spatial periodicity and the color of each pixel in the image corresponds to the particular dominant spatial periodicity assigned to that pixel, similar to nsOCT images, [16][17][18][19][20][21][22][23][24][25][26] as presented in Figure 4c.However, in contrast to nsOCT, synOCT facilitates imaging of the subwavelength structure within regions of interest in the depth (axial) direction, smaller than the resolution limit of the imaging system.The dimension of the selected areas, where the periods are calculated in Figure 4c-e, is about 2 μm, below the 5.5 μm resolution limit of the SDOCT system.The structure is visualized as different colors, and differences in the structural sizes as small as 10 nm can be clearly detected.The subwavelength structures cannot be resolved in such large synOCT images, as in Figure 4c, but the dominant sizes of the structure can be seen as corresponding colors.
The presented synOCT approach can also be realized for objects with a continuous Fourier spectrum of spatial frequencies, collected using any given OCT system.Figure 5 shows reconstructed images of a glass plate having a thickness of approximately 1 mm and a refractive index of 1.52.The magnified portions of a selected region (corresponding to the blue rectangle) were reconstructed using both conventional OCT imaging (Figure 5a,b, respectively) and synOCT imaging (Figure 5c).An improvement in resolution using the synOCT approach (Figure 5c), in comparison with conventional OCT image Figure 5b, is clearly seen.The tilt of the glass surface, which is very difficult to see in conventional OCT image Figure 5b, can be clearly seen in the synOCT image Figure 5c.
More importantly, the synOCT provides quantitative information about the subwavelength structure, again unresolvable in conventional OCT.In Figure 6, an in vivo image of a human finger, formed using OCT and synOCT, and fragments of the axial spatial frequency profiles, reconstructed at selected points, are presented.Formation of the depth profiles for central lines in Figure 6c,d by superposition of the axial spatial frequency harmonics, as shown in Figure 3, is demonstrated in the Supported videos 2 and 3.As discussed above, the axial spatial frequency/periodicity profiles, which provide information about small sub-wavelength sized structure, can be reconstructed and presented for the synOCT images, Figure 6g,h.The size of regions, where the spatial frequency/periodicity profiles are reconstructed, can be selected depending on the imaging task.For example, profiles in Figure 6g,h were reconstructed for areas with an axial size of 24 nm, which is more than 200 times smaller than the depth resolution of the TELESTO III system, which is 5.5 μm.Such profiles are very sensitive to structural alterations.For example, the difference in structure between points in and out of the sweat duct channel in Figure 6e, market by red and blue arrows, can be seen in profiles in Figure 6g, whereas the difference in structure between two areas in Figure 6f is not so clear in Figure 6h.
In Figure 7, the conventional OCT images, Figure 7a,b, and quantitative color visualization of the structure within a slice of potato using synOCT, Figure 7c, are presented.The size of areas, where the subwavelength structure is visualized in Figure 4c and 7c, is about 2 μm, which is approximately 3 times smaller than the resolution limit of the OCT imaging system.The improvement in depth resolution relative to the corresponding nsOCT image in Figure 7d is demonstrated; the thickness of layers of the structure within the synOCT image in Figure 7c is significantly reduced.

Discussion and Conclusion
The presented results confirm the ability of the synOCT approach to estimate the contribution of complex amplitude of each spatial frequency (spatial harmonic) of the object, within the detected bandwidth, including amplitude and phase, in the image formation at all image regions.The size of such regions can be smaller than the resolution limit of the imaging system.The subwavelength structure can be visualized from a single frame with nanosensitivity to structural changes, which is impossible using conventional OCT.For example, to resolve subwavelength structures within the object in Figure 4a using conventional OCT with a 1300 nm central wavelength, the spectral bandwidth would need to be larger than 1150 nm, which cannot be achieved at the present time.Moreover, even with such a spectral bandwidth it would be impossible to measure the sizes of spatial periods with the presented nanometer accuracy and detect the difference between periods of 10 nm.
The proposed quantitative synOCT approach opens exciting new possibilities for OCT for biomedical imaging and other applications and can be realized using both SDOCT and SSOCT.Quantitative visualization of the subwavelength structure for biological samples with synOCT is also demonstrated.The color synOCT technique, based on a novel contrast mechanism, facilitates depth-resolved visualization of the subwavelength internal structure of the object with nanoscale sensitivity, similar to nsOCT, but with dramatically improved spectral and depth resolution.The spatial resolution is increased by about 10 times and potentially more, as demonstrated in Figure 7c, and spectral resolution is increased by more than 100 times in comparison with typical nsOCT.Quantitative information about the subwavelength structure can be visualized within areas much smaller than the resolution limit of the imaging system.
The presented results demonstrate the potential for further development of the approach, including the super-resolution abilities using just a single frame, extension of the depth range (each spatial harmonic could be extended ever further numerically), accessing the phase with spatial resolution better than the resolution limit of the imaging system by analyzing the phase of each spatial harmonic, etc.Indeed, ability to reconstruct the spatial frequency profiles at regions smaller than resolution limit of the imaging system permits to detect structural changes between regions separated by the distance shorter than resolution limit, which creates potential for super-resolution imaging.Some known OCT techniques, including speckle variance OCT [41] or temporal correlation analysis, [42,43] enhance the spatial resolution of OCT images, for example, of vascular network.Ability of the synOCT to reconstruct the spatial frequency content within regions smaller than resolution limit could permit to visualize the smallest skin blood capillary loops and possibly tracking the displacement of individual red blood cells.Instead of detection just intensity changes in time, the synOCT permits detecting changes in spatial frequencies profiles in time at each small region within the image.So, instead of detecting just changes in single intensity value at each point, we can detect changes in thousand and more values at each point.This could advance the dynamic OCT and creates potential for reconstructing the smallest details within skin which cannot be resolved using conventional OCT.
Removing the trade-off between spectral and spatial resolution may also improve the performance of the spectroscopic OCT technique. [44]Combination of the synOCT with interferometric synthetic aperture microscopy could also be interesting. [34]An improvement in axial and lateral resolution beyond the resolution limit of the imaging system, similar to the approach presented in, [45,46] is also possible.The demonstrated quantitative estimation of the submicrometer structure within biological samples and human skin in vivo with nanosensitivity to structural changes and improvement in resolution shows the potential of the proposed approach for broad applications in medicine, biology, and industry.

Figure 1 .
Figure 1.Representation of accessible spatial frequencies: a) geometry of the illumination/collection of the object and b) red circles are accessible spatial frequencies as a set of Ewald's spheres at different illumination and collection angles.Sp1 is the Ewald's spheres for different wavelengths at incident angle θ = 0; Sp2 is the Ewald's spheres for different wavelengths at incident angle θ = π.

Figure 2 .
Figure 2. Formation of the Fourier spectrum of axial spatial frequencies and reconstruction of A-scans (depth profiles in z-direction) and 2D B-scans: using conventional a,b,c) OCT and d,e,f ) synOCT.The axis labeled I illustrates intensity (arbitrary units), z illustrates depth (arbitrary), and v z illustrates axial spatial frequency.The object in this figure a,d) being illuminated shows a harmonic spatially oscillating refractive depth profile with green regions illustrating higher refractive index and purple illustrating lower, and H is the periodicity.

Figure 3 .
Figure 3. Depth profile formation in synOCT: a) detected signal, b) synthesized spatial harmonics versus depth, c) amplitude, and d) intensity depth profiles of the single scatterer.

Figure 4 .
Figure 4. Imaging of the samples with submicrometer internal structure: a) schematic of the sample; b) image of the samples with different spatial periods with magnified portions d,e), reconstructed using conventional OCT and synthetic OCT; and c) magnified color synOCT image of the selected region within the blue rectangle in b).The color bar in c) represents the dominant spatial periods in nanometers.

Figure 5 .
Figure 5. a) Conventional OCT image of the glass plate and corresponding magnified b) OCT and c) synOCT images within the selected blue rectangle.

Figure 6 .
Figure 6.a,b) Conventional OCT images of human finger; c,d) magnified portions of the OCT images within red rectangles in a,b); e,f ) corresponding synOCT images; and g,h) fragments of the axial spatial periodicity profiles at selected blue and red points, indicated by arrows.