Coherent Averaging in k-Space

• Coherent Averaging in k-Space is a phase-preserving method that sums complex-valued Fourier measurements to enable constructive interference of signal components.
• It is applied across modalities such as OCT, MRI, and dual-comb spectroscopy to improve signal-to-noise ratio and enhance image resolution through precise k-space sampling.
• Its success relies on maintaining phase stability and proper noise management, ensuring that coherent contributions lead to improved imaging performance without amplifying artifacts.

Coherent averaging in k-space is the phase-preserving combination of complex measurements indexed by spatial frequency, wavevector, or an analogous Fourier-domain coordinate. In the cited literature, the term appears in several technically distinct but mathematically related settings: the diagonal of a k-space reflection matrix in coherent image-scanning microscopy, the complex FFT output of k-linearized OCT fringes, repeated complex sampling of MRI k-space lines, spatial-frequency beamforming across an aperture, ensemble-averaged spectra in imaging through random media, and comb-mode accumulation in dual-comb spectroscopy (Sommer et al., 2023, Pfeiffer et al., 2016, Zhou et al., 2020). In all of these cases, the defining feature is that complex amplitudes are added before magnitude or intensity formation, so that relative phase is retained and can produce constructive interference of signal terms.

1. Foundational definition and formal structure

The most explicit distinction between coherent and incoherent averaging is given in MHz FDML-OCT. There, coherent averaging is defined as averaging the complex-valued output of the FFT, whereas incoherent averaging is averaging the absolute values of the FFT outputs. If $S_n(z)$ denotes the complex A-scan of the $n$-th acquisition, the coherent average is

$\bar{S}(z)=\frac{1}{N}\sum_{n=1}^{N} S_n(z),$

while incoherent averaging is performed after the magnitude operation (Pfeiffer et al., 2016). This distinction is not merely terminological: the coherent form preserves phase, so signal terms can add constructively while random-phase noise averages down.

In coherent ISM, the same principle appears in a two-coordinate reflection matrix $R(x_{in},x_{out};z)$, whose entries are complex fields rather than intensities. The ISM image is formed by summing reflection-matrix elements along anti-diagonals in real space,

$$I_{ISM}(x)=\sum_{x_{in}} R\bigl(x_{in},x_{out}=2x-x_{in};z\bigr),$$

and, by the Fourier-slice theorem, this is equivalent to taking the diagonal of the k-space reflection matrix,

$\tilde I_{ISM}(k)=\tilde R(k/2,k/2;z).$

The paper emphasizes that the complex phases in $\tilde R$ are preserved and coherently summed along this diagonal (Sommer et al., 2023).

Parallel MRI provides a more general functional-analytic formulation. In the RKHS treatment of multi-coil k-space, the reconstructed signal at a target location is written as a complex linear combination of acquired samples,

$$\hat f_n(\boldsymbol{x})=\sum_{k=1}^{|S|}\sum_{i=1}^{N} f_i(\boldsymbol{x}_k)\,u_n^{k,i}(\boldsymbol{x}),$$

where the weights are determined by a matrix-valued reproducing kernel derived from the coil sensitivities (Athalye et al., 2013). This formalizes coherent averaging in k-space as interpolation by phase-preserving superposition rather than by magnitude-only combination.

A broader implication of these formulations is that “k-space” is modality-dependent. Across the cited work it denotes, respectively, transverse spatial frequency, the diagonal of a reflection matrix in Fourier coordinates, the complex depth-domain signal after an FFT of a uniform-k interferogram, the spatial-frequency coordinate of an aperture, or the mode-index space of a dual-comb spectrum (Sommer et al., 2023, Pfeiffer et al., 2016, Vouras, 2023, Long et al., 14 Apr 2025).

2. Fourier imaging, microscopy, and inverse scattering

In coherent ISM, the k-space formulation shows that only the diagonal $\tilde R(k_{in}=k/2,k_{out}=k/2)$ is required to reconstruct the image. This leads directly to the paper’s spotlight-SAR interpretation: illuminating with a plane wave of transverse wavevector $k_{in}$ and detecting only the reflected plane wave with $n$0 traces the main diagonal of $n$1. The same image can therefore be acquired either by scanning focus positions with a detector array in real space or by scanning illumination angle with a single detector in the sample’s k-space (Sommer et al., 2023). In that setting, coherent averaging in k-space is not an auxiliary denoising step; it is the image-formation rule itself.

The unified OCT theory places the same idea in a fully 3D Fourier-diffraction framework. Under the first Born approximation, measured complex fields sample the 3D Fourier transform $n$2 of the scattering potential along Ewald-sphere manifolds. For point-scanning OCT, the 3D transfer function is obtained by coherently summing the full-field OCT transfer function over illumination angles,

$n$3

so the transfer function itself is a coherent angular average in k-space (Zhou et al., 2020). In ISAM and related inverse-scattering reconstructions, measurements are resampled onto a common 3D k-space grid and then coherently combined before a 3D inverse Fourier transform.

Imaging through random media gives a different but closely related example. There, real-space shift-and-add averaging corresponds in Fourier space to

$n$4

with the phase factors undoing random image shifts (Hwang et al., 2021). Under the stated assumptions of statistical homogeneity, isotropy, and Gaussian phase statistics, the ensemble-averaged OTF is real and even in k-space. The paper’s central claim is then that the Fourier phase of the object can be read from the averaged, shift-corrected image up to the diffraction limit when this averaging is combined with Labeyrie’s amplitude recovery (Hwang et al., 2021).

These examples establish a common pattern. Coherent averaging in k-space is often equivalent to selecting, synthesizing, or regridding a specific subset of Fourier-domain data rather than merely repeating the same measurement and averaging it.

3. MRI: repeated sampling, variable-density averaging, and parallel reconstruction

In MRI, coherent averaging traditionally means repeated acquisition of the same k-space line followed by averaging of the complex-valued measurements, assuming phase stability across repetitions. “Compressed Sensing MRI With Variable Density Averaging” implements this idea with a deliberately nonuniform profile in which the number of averages depends on k-space radius. In the center-dense scheme,

$n$5

with $n$6 used in the experiments, so low-frequency k-space receives many averages and the periphery receives fewer (Schoormans et al., 2019). Because the variance of a line with $n$7 averages is modeled as $n$8, the reconstruction uses a weighted least-squares data term,

$n$9

with diagonal weights proportional to $\bar{S}(z)=\frac{1}{N}\sum_{n=1}^{N} S_n(z),$0 (Schoormans et al., 2019).

Within a fixed scan-time budget, that paper reports that center-dense averaging can outperform full sampling at low SNR. Phantom and in vivo brain and knee results showed that more coherent averages in the center of k-space yielded better image quality than full sampling in the same scan time, with preserved soft-tissue contrast and increased anatomical detail (Schoormans et al., 2019). The theoretical rationale given is that low-frequency k-space is more important for coarse wavelet coefficients and is less favorable for compressed-sensing recovery, whereas high-frequency regions are more amenable to CS reconstruction.

The RKHS formulation of parallel MRI generalizes coherent k-space combination beyond repeated sampling. Here, interpolation from arbitrary k-space locations is treated as approximation in a vector-valued RKHS with a matrix-valued kernel defined by the receive-coil sensitivities. The kernel

$\bar{S}(z)=\frac{1}{N}\sum_{n=1}^{N} S_n(z),$1

encodes how samples across coils and k-space locations can substitute for one another (Athalye et al., 2013). In this view, reconstruction is a coherent linear combination of complex samples whose stability depends on the sampling geometry and the conditioning of the kernel system.

The same point appears operationally in 3D GRAPPA. Missing k-space samples are synthesized as complex linear combinations of neighboring acquired samples across coils,

$\bar{S}(z)=\frac{1}{N}\sum_{n=1}^{N} S_n(z),$2

so GRAPPA is a structured coherent averaging operator in k-space (Rabanillo et al., 2018). The exact g-factor analysis in that work derives noise propagation directly in k-space under stationarity and uncorrelated-noise assumptions, showing how sampling pattern and kernel structure determine noise amplification. In vivo comparisons in that paper showed higher g-factor for uniformly random patterns than for structured Cartesian patterns such as CAIPIRINHA at the same nominal acceleration (Rabanillo et al., 2018).

4. OCT and Fourier-domain ranging

In FDML-OCT, coherent averaging is implemented after standard Fourier-domain processing. The acquisition sequence is: record the time-domain fringe during the wavelength sweep, map time to a uniformly sampled wavenumber grid, compute the FFT to obtain a complex A-scan, and average the resulting complex A-scans at the same lateral position (Pfeiffer et al., 2016). The paper explicitly treats this as “averaging of complex FFT outputs,” not as averaging of raw intensities.

That work introduces “computational downscaling” of the A-scan rate. Starting from a 3.2 MHz FDML system, coherent averaging of approximately $\bar{S}(z)=\frac{1}{N}\sum_{n=1}^{N} S_n(z),$3 consecutive A-scans yields an effective rate of about $\bar{S}(z)=\frac{1}{N}\sum_{n=1}^{N} S_n(z),$4 kHz: $\bar{S}(z)=\frac{1}{N}\sum_{n=1}^{N} S_n(z),$5 The reported conclusion is that downscaling by a factor of $\bar{S}(z)=\frac{1}{N}\sum_{n=1}^{N} S_n(z),$6 leads to a sensitivity only $\bar{S}(z)=\frac{1}{N}\sum_{n=1}^{N} S_n(z),$7 dB less than that of a comparable ideal $\bar{S}(z)=\frac{1}{N}\sum_{n=1}^{N} S_n(z),$8 kHz system (Pfeiffer et al., 2016). The paper attributes the feasibility of this procedure to very good phase stability of the FDML laser and the high speed, which suppresses sample-motion effects over the averaging window.

The imaging results in that study distinguish clearly between high- and low-sensitivity regimes. In vivo finger-knuckle imaging at about $\bar{S}(z)=\frac{1}{N}\sum_{n=1}^{N} S_n(z),$9 dB system sensitivity showed that both coherent and incoherent averaging improved image quality, but coherent averaging mainly reduced uneven background and improved local contrast. In an orange sample artificially reduced to $R(x_{in},x_{out};z)$0 dB sensitivity, the difference was described as much more dramatic: the coherently averaged image revealed much more detail and substantially reduced uneven background (Pfeiffer et al., 2016). The limitation identified is that excess laser noise and ADC-related noise do not have random phase distributions and are therefore more resistant to coherent averaging.

The unified OCT k-space theory broadens this 1D depth-domain picture. The measured interferometric signal is proportional to the complex cross-term $R(x_{in},x_{out};z)$1, and conventional coherent averaging of spectra or A-lines is the 1D axial special case of a more general 3D coherent combination in $R(x_{in},x_{out};z)$2 after Ewald-sphere resampling (Zhou et al., 2020). This suggests that many OCT operations commonly described in separate terms—spectral averaging, synthetic aperture, ISAM, and aberration correction—are different realizations of the same underlying phase-preserving k-space combination.

5. Array processing, random media, spectroscopy, and wave engineering

In k-space beamforming for an array of quantum sensors, coherent averaging is indexed directly by spatial frequency. A frequency-comb transmit waveform is arranged so that each array element is tuned to a different tone; as the comb is received, the phase slope across the array evolves in time and sweeps through effective spatial frequencies $R(x_{in},x_{out};z)$3. The time axis from $R(x_{in},x_{out};z)$4 to $R(x_{in},x_{out};z)$5 is explicitly mapped to $R(x_{in},x_{out};z)$6, and the coherent sum across array elements produces a strong peak when the phases align (Vouras, 2023). The paper states that coherent summation of all array outputs yields an integration gain that increases SNR by a factor of $R(x_{in},x_{out};z)$7 for a planar array, or $R(x_{in},x_{out};z)$8 for the linear-array case discussed there.

The random-media phase-retrieval method shows that coherent averaging in k-space can also be produced by appropriate preprocessing in real space. Shift-corrected image averaging aligns random speckle PSFs so that their Fourier transforms add with corrected linear phase. The paper’s claim is that, combined with Labeyrie’s amplitude recovery, this directly yields the Fourier phase of the hidden object up to the diffraction limit without iterative phase retrieval (Hwang et al., 2021). In this context, coherent averaging is used to expose deterministic phase structure beneath statistically fluctuating distortions.

In dual-comb spectroscopy, the relevant k-space is the mode-index or frequency-domain representation of the comb. Conventional DCS relies on coherent averaging of many interferograms because per-shot SNR per RF comb line is low. “Spectral Mode Enhancement in Coherent-harmonic Dual-comb Spectroscopy” instead engineers the combs so that several sub-pulse trains add coherently at each spectral mode within a single acquisition. With a densification factor $R(x_{in},x_{out};z)$9, the experiment reported approximately $$I_{ISM}(x)=\sum_{x_{in}} R\bigl(x_{in},x_{out}=2x-x_{in};z\bigr),$$0 SNR gain per single acquisition and more than $$I_{ISM}(x)=\sum_{x_{in}} R\bigl(x_{in},x_{out}=2x-x_{in};z\bigr),$$1-fold reduction in averaging time for comparable SNR relative to conventional DCS (Long et al., 14 Apr 2025). The paper interprets this as effectively embedding coherent averaging into the comb-generation physics rather than performing it only in post-processing.

A structurally different use appears in the engineering of lattices of spin-orbit beams. There, coherent averaging in k-space means constructing a discrete set of plane-wave components with prescribed transverse wavevectors, amplitudes, and spin states, then Fourier-summing them to synthesize a real-space lattice with specified orbital and radial structure. The method uses a branching operator in k-space and design inequalities such as $$I_{ISM}(x)=\sum_{x_{in}} R\bigl(x_{in},x_{out}=2x-x_{in};z\bigr),$$2 and $$I_{ISM}(x)=\sum_{x_{in}} R\bigl(x_{in},x_{out}=2x-x_{in};z\bigr),$$3 for hexagonal lattices, and $$I_{ISM}(x)=\sum_{x_{in}} R\bigl(x_{in},x_{out}=2x-x_{in};z\bigr),$$4 and $$I_{ISM}(x)=\sum_{x_{in}} R\bigl(x_{in},x_{out}=2x-x_{in};z\bigr),$$5 for square lattices (Chahal et al., 13 Mar 2026). In this setting, coherent averaging is a wave-synthesis tool rather than a denoising strategy.

6. Assumptions, benefits, and recurring misconceptions

Across modalities, the benefits of coherent averaging in k-space fall into two categories. First, when repeated measurements correspond to the same Fourier-domain coordinate, coherent averaging improves SNR or sensitivity while preserving resolution. This is explicit in FDML-OCT, where noise reduction initially follows the expected behavior for coherent averaging, and in DCS, where equivalent averaging-time reductions are reported through per-mode enhancement (Pfeiffer et al., 2016, Long et al., 14 Apr 2025). Second, when different measurements populate complementary parts of k-space, coherent averaging produces synthetic-aperture effects, broadens effective k-space support, and can improve lateral resolution or transfer-function isotropy, as in coherent ISM, OCT synthetic-aperture formalisms, and quantum-sensor beamforming (Sommer et al., 2023, Zhou et al., 2020, Vouras, 2023).

The assumptions are correspondingly strict. The coherent-ISM paper lists coherent illumination, linearity and shift-invariance, a stable sample, and phase stability as requirements for coherent k-space synthesis (Sommer et al., 2023). The FDML-OCT work adds accurate k-linearization, consistent dispersion compensation, timing stability, and low sample motion as practical necessities (Pfeiffer et al., 2016). In MRI, repeated measurements of a k-space line are assumed to represent the same underlying complex value, corrupted by independent noise; motion or phase drift violates that model (Schoormans et al., 2019).

Several misconceptions recur in the literature. One is that any averaging performed after a Fourier transform is automatically coherent averaging. The OCT terminology is narrower: averaging complex FFT outputs is coherent averaging, whereas averaging the absolute values of FFT outputs is incoherent averaging (Pfeiffer et al., 2016). A second misconception is that coherent averaging necessarily reduces speckle. The unified OCT theory instead treats speckle as deterministic coherent interference among scatterers within the resolution cell; coherent averaging of the same field preserves those phase relations, whereas speckle reduction requires averaging across altered coherent states, such as varying illumination angle or wavefront, and then discarding phase information (Zhou et al., 2020). A third misconception is that k-space always denotes the Cartesian sampling grid of MRI; across the cited work, it is a broader Fourier-domain index whose exact physical meaning depends on the measurement architecture (Sommer et al., 2023, Vouras, 2023, Long et al., 14 Apr 2025).

The principal limitations follow directly from these requirements. Correlated or non-Gaussian noise can resist coherent averaging, as reported for excess laser noise and ADC-related noise in MHz OCT (Pfeiffer et al., 2016). Aggressive undersampling can trade SNR gain for smoothing in CS-VDA MRI (Schoormans et al., 2019). In parallel MRI and GRAPPA, unstable coherent weights amplify noise and are reflected in larger power functions, Frobenius norms, or g-factor maps (Athalye et al., 2013, Rabanillo et al., 2018). In single-detector k-space ISM, the same k-space diagonal is preserved, but only a fraction of the reflected field is captured per acquisition (Sommer et al., 2023). These limitations do not negate the utility of coherent averaging in k-space; they specify the regime in which phase-preserving combination remains physically and statistically justified.