Introduction
Functional imaging is a critical tool in neuroscience. For example, calcium imaging methods are used routinely in hundreds of labs, generating largescale video datasets whose characteristics (cell shapes, signaltonoise levels, background activity, signal timescales, etc.) can vary widely depending on the imaging modality and the details of the brain region and cell types being imaged. To handle this data, scientists must solve two basic tasks: we need to extract signals from the raw video data with minimal noise, and we need to store (and share) the data. A number of papers have focused on the first task [Mukamel et al., 2009, Maruyama et al., 2014, Pnevmatikakis et al., 2016, Pachitariu et al., 2016, Friedrich et al., 2017, Inan et al., 2017, Reynolds et al., 2017, Petersen et al., 2017, Zhou et al., 2018, Mishne et al., 2018]; however, somewhat surprisingly, very little work has focused on the second task.
For both of these tasks, it is critical to denoise and compress the data as much as possible. Boosting the signaltonoise ratio (SNR) is obviously important for detecting weak signals, performing singletrial analyses (where noise cannot be averaged over multiple trials), and for realtime experiments (where we may need to make decisions based on limited data  i.e., averaging over time is not an option). The benefits of compression are perhaps less obvious but are just as numerous: compression would facilitate much more widespread, routine open data sharing, enhancing reproducible neuroscience research. Compression will also be critical for in vivo imaging experiments in untethered animals, where data needs to be transmitted wirelessly, making data bandwidth a critical constraint. Finally, many signal extraction methods based on matrix factorization can be sped up significantly if run on suitably compressed data.
Previous methods for denoising and compressing functional data have several drawbacks. Generic video compression approaches do not take advantage of the special structure of functional imaging data and produce visible artifacts at high compression rates; more importantly, these approaches do not denoise the data, since they focus on compressing the full data, including noise, whereas our goal here is to discard the noise. Conversely, generic image denoising approaches do not offer any compression (and also fail to take advantage of strong structured correlations in the video data). Constrained nonnegative matrix factorization (CNMF) [Pnevmatikakis et al., 2016]
approaches provide state of the art denoising and demixing of calcium imaging data, but these methods can leave significant visible signal behind in the residual (discarding potentially valuable signal) and are highly dependent on the initialization of the matrix factorization; thus it would be dangerous to keep only the matrix factorization output and discard the raw data. Principal components analysis (PCA) is often employed as a compression and denoising method
[Mukamel et al., 2009, Pachitariu et al., 2016], but PCA is based on a rather unstructured signal model and therefore provides a suboptimal encoder of functional data (we will discuss this point in further depth below). In addition, the computation time of PCA scales quadratically with the number of pixels (assuming a long video dataset) and therefore naive applications of PCA are rather slow [Friedrich et al., 2017]. Finally, importantly, it is difficult to automatically choose the number of principal components that should be retained in a given video (and the “correct” number of components can vary widely across different datasets).Here we introduce a new simple approach to denoising and compressing functional video data. We apply a variant of penalized matrix decomposition [Witten et al., 2009] that operates locally in space, and encourages smoothness in both the spatial and temporal dimensions. This method offers multiple advantages over previous approaches. It is based on a signal model that is wellmatched to the structure of the data: cells are local in space, there aren’t too many of them compared to the number of pixels (leading to a lowrank signal model), and cellular activity is smoother than the dominant noise sources, which are spatially and temporally uncorrelated. The approach is scalable (scaling linearly in the number of frames and pixels), and has modest memory requirements (because all processing is only performed in local spatial patches). All parameters (including the local matrix rank and the degree of smoothness of the output) are chosen automatically. Empirically we find that the method is highly effective, leaving behind minimal visible structure in the residual, while achieving 20300x compression rates and 24x improvements in SNR. We demonstrate the method’s effectiveness on a wide variety of functional imaging datasets (both calcium and voltage imaging; one, two and threephoton imaging; and data including somas and dendrites) and show that the method is also effective on widefield imaging data, where singlecell resolution is not available. Finally, we develop a new constrained NMF approach based on the denoised and compressed representation of the data, and apply this new demixing method to two challenging applications: dendritic calcium imaging data and voltage imaging data in the context of optogenetic stimulation. In both cases, we show that our new approach leads to faster and much more robust extraction of activity from the video data.
Methods
We begin by defining notation. Our starting point is an imaging dataset that has been motioncorrected (i.e., we assume that there is no motion of visible cellular components from frame to frame of the movie) and then “unfolded” into a matrix , where is the number of frames in the movie and is the number of pixels per frame (or voxels per frame if we are performing imaging in three dimensions). Now the typical approach is to model the data as , where the columns of model the locations of each source (with sources total), the rows of model the timevarying fluorescence of each source, is a “background” term to handle signals that can not easily be split into singleneuronal components, and denotes temporally and spatially uncorrelated noise.
It is useful to break the processing pipeline into three subproblems:

Denoising: separation of neural signal from noise ;

Compression of signal ;

Demixing: factorization of into its constituent components , and .
Most prior work has attempted to solve these subproblems simultaneously, e.g., to recover and directly from the raw data . As emphasized above, this direct approach involves a challenging nonconvex optimization problem; the solution to this problem typically misses some structure in , is highly sensitive to initialization and hyperparameter settings, and can be particularly unstable in lowSNR regimes. We have found empirically that a sequential approach is more robust and effective. First we compute the compressed and denoised estimate ; here and are chosen so that captures all of the signal in while retaining minimal noise (i.e., ) and also and are highlystructured, compressible matrices, but we do not enforce any constraints between and . The computation of and essentially solves subproblems 1 and 2 simultaneously. Second, we exploit , , and the resulting denoised to facilitate the solution of problem 3. We discuss each of these steps in turn below.
Denoising & Compression
To achieve good compression and denoising we need to take advantage of three key properties of functional imaging data:

Signal sources are (mostly) spatially local;

Signal is structured both temporally and spatially, whereas noise is temporally and spatially uncorrelated;

Signal is (mostly) lowrank.
Given these structural assumptions, it is natural to construct and via a local penalized matrix decomposition approach^{1}^{1}1One important note: many matrix factorizations are possible here to obtain a compressed representation . This nonuniqueness does not pose an issue for either compression or denoising. This makes these problems inherently easier than the demixing problem, where the identifiability of , , and (perhaps up to permutations of the rows and columns of and ) is critical.: we break the original data matrix into a collection of overlapping spatial patches, then decompose each of these matrix patches (in parallel) using a factorization method that enforces smoothness in the estimated spatial and temporal factors, then combine the resulting collection of spatial and temporal factors over all the patches into a final estimate of and . (See CaImAn for a similar patchwise approach to the demixing problem.)
We have experimented with several approaches to penalized matrix decomposition (PMD), and found that an iterative rankone deflation approach similar to the method described in [Witten et al., 2009]
works well. We begin by standardizing the data within a patch: for each pixel, we subtract the mean and normalize by an estimate of the noise variance within each pixel; the noise variance is estimated using the frequencydomain method described in
[Pnevmatikakis et al., 2016], which exploits the fact that the signal and noise power are spectrally separated in movies with sufficiently high frame rates. After this normalization we can model the noise as roughly spatially and temporally homogeneous. Denote this standardized data matrix within a patch as , and Frobenius norm as . Then at the iteration PMD extracts the best rankone approximation to the current residual , as determined by the objective(1) 
followed by a temporal debiasing update . The objective (1) can be ascended via alternating minimization on and .
Note that if we drop the and constraints above then we can solve for and
directly by computing the rank1 singular value decomposition (SVD) of
; in other words, by performing PCA within the patch. Since we have normalized the noise scale within each pixel, PCA should identify the signal subspace within the patch, given enough data (because the normalized projected data variance in any direction will be equal to one plus the signal variance in this direction; since PCA searches for signal directions that maximize variance, PCA will choose exactly the signal subspace in the limit of infinite data). Indeed, as discussed in the results section, simple patchwise PCA (with an appropriate adaptive method for choosing the rank) often performs well, but incorporating spatial and temporal penalties in the optimization can push and closer to the signal subspace, resulting in improved compression and SNR.How should we define the penalties and , along with the corresponding constraints and ? The simplest option would be to use quadratic smoothing penalties; this would lead to a simple closedform linear smoothing update for each and . However, the signals of interest here have inhomogeneous smoothness levels — an apical dendrite might be spatially smooth in the apical direction but highly nonsmooth in the orthogonal direction, and similarly a calcium signal might be very smooth except at the times at which a spike occurs. Therefore simple linear smoothing is typically highly suboptimal, often resulting in both undersmoothing and oversmoothing in different signal regions. We have found total variation (TV) [Rudin et al., 1992] and trend filtering (TF) [Kim et al., 2009] penalties to be much more empirically effective. We let
and
Here denotes the onedimensional discrete second order difference operator and the incidence matrix of the nearestneighbor pixeladjacency graph (pixels are in the edge set if the pixels are nearest neighbors).
Similarly to [Pnevmatikakis et al., 2016], we define the smoothing constraints and implicitly within the alternating updates by the simple reformulation
(2) 
and
(3) 
where (resp. ) estimates the noise level of the unregularized update (resp. ), and we are using the fact that if the residual contains just noise then its squared norm should be close to
, by the law of large numbers (and similarly for equation
3). See Algorithm 1 for a summary.To solve the constrained problems on the righthand side we use the line search approach described in [Langer, 2017]. We solve the primal form of the TV optimization problem (2) using the proxTV package [Barbero and Sra, 2014], and of the TF optimization problem (3) using the PrimalDual Active Set method in [Han and Curtis, 2016]. Both of these methods can exploit warm starts, leading to major speedups after a good initial estimate is found. Empirically the TF optimization scales linearly with the movie length ; since the scale of the TV problem is bounded (because we work in local spatial patches) we have not explored the scaling of the TV problem in depth.
Figure 1 illustrates the effect of trend filtering on a couple components. One important difference compared to previous denoising approaches [Haeffele et al., 2014, Pnevmatikakis et al., 2016]
is that the TF model is more flexible than the sparse autoregressive model that is typically used to denoise calcium imaging data: the TF model does not require the estimation of any sparsity penalties or autoregressive coefficients, and can handle a mixture of positive and negative fluctuations, while the sparse nonnegative autoregressive model can not (by construction). This is important in this context since each component in
can include multiple cellular components (potentially with different timescales), mixed with both negative and positive weights.To complete the description of the algorithm on a single patch we need an initialization and a stopping criterion to adaptively choose the rank of and . For the latter, the basic idea is that we want to stop adding components
as soon as the residual looks like uncorrelated noise. To make this precise, we define a pair of spatial and temporal “roughness” test statistics
and compute these statistics on each extracted and
. We accept or reject each component according to a onesided hypothesis test under the null hypothesis that
consists of uncorrelated Gaussian noise of variance one. (We compute the critical region for this test numerically.) In the compression stage we are aiming to be rather conservative (we are willing to accept a bit of extra noise or a slightly higherrank and in order to ensure that we are capturing the large majority of the signal), so we terminate the outer loop (i.e., stop adding more components ) after we reject a couple components in a row. See Algorithm 2 for a summary.To initialize, we have found that setting works well. To speed up early iterations, it is natural to iterate the projections while skipping the denoising steps; this corresponds to intializing with an approximate rank1 SVD as computed by power iterations. Initializing in this manner can reduce the total number of iterations needed for
to converge. Matrixvector multiplications are a rate limiting step here; thus, these initial iterations can be sped up using spatial and temporal decimation on
. Empirically, decimation has the added benefit of boosting signal (by averaging out noise in neighboring timepoints and pixels) and can be useful for extracting weak components in low SNR regimes; see [Friedrich et al., 2017] for a related discussion.The method described so far handles a single spatial patch of data. We can process patches in parallel; a multicore implementation of this method (assigning different patches to different cores) achieves nearly linear speedups. We have found that for some datasets edge artifacts can appear near patch boundaries if the patches do not overlap spatially. These boundary artifacts can be eliminated by performing a overcomplete blockwise decomposition of using halfoffset grids for the partitions (so that each pixel
lies within the interior of at least one patch). Then we combine the overlapping patches together via linear interpolation (see
[Pnevmatikakis and Giovannucci, 2017] for a similar approach): setwhere indexes the patches (so denotes the denoiser output in the th patch) and is a “pyramid” function composed of piecewise linear functions that start at at the patch boundaries and increase linearly to at the center of the patch.
The above is equivalent to starting with a collection of overlapping sparse local factorizations , forming elementwise products between the individual spatial components and the pyramid functions , and then forming the union of the result to obtain a new factorization . Typically this will result in some redundancy due to the overlapping spatial components; we remove this redundancy in a final backwards model selection step that tests whether each temporal component can be explained as a weighted sum of its neighbors. More precisely, we sort the components in ascending order according to the norms of . For each in this order we then regress onto the collection of temporal components whose corresponding spatial components overlap with , i.e., approximate . We then test the signal strength of the residual (using the temporal test statistic defined previously); the component is rejected if the residual is indistinguishable from noise according to this test statistic. If component is rejected then we distribute its energy to the remaining spatial components according to the regression weights: .
We conclude with a few final implementation notes. First, the results do not depend strongly on the precise patch size, as long as the patch size is comparable to the spatial correlation scale of the data: if the patches are chosen to be much smaller than this then the components in neighboring patches are highly correlated, leading to excessive redundancy and suboptimal compression. (Conversely, if the patch size is too big then the sparsity of is reduced, and we lose the benefits of patchwise processing.)
Second, in some datasets (e.g., widefield imaging, or microendoscopic imaging data), large background signals are present across large portions of the field of view. These background signals can be highly correlated across multiple spatial patches, leading to a suboptimal compression of the data if we use the simple independentpatch approach detailed above. Thus in some cases it is preferable to run a couple iterations of PMD(TV, TF) on the full and then subtract the resulting components away before moving on to the independent block processing scheme. We have found that this effectively subtracts away dominant background signals; these can then be encoded as a small number of dense columns in the matrix , to be followed by a larger number of sparse columns (corresponding to the small patches), resulting in an overall improvement in the compression rate. See the microendoscopic imaging background video for an example.
The patchwise PMD(TV,TF) approach results in an algorithm that scales linearly in three critical parameters: (due to the sparse nature of the seconddifference operator in the TF step), (due to the patchwise approach), and the rank of and . We obtain further speedups by exploiting warm starts and parallel processing over patches. Additional speedups can be obtained for very long datasets by computing on a subset of the data and then updating on the remainder of the movie; the latter step does not require any PMD iterations (since the spatial signal subspace has already been identified) and is therefore very fast, just requiring a single temporal update call per element of .
Demixing
The methods described above provide a compressed and denoised representation of the original data : the output matrices and are lowrank compared to , and is additionally highly sparse (since is formed by appending spatial components from multiple local spatial patches, and each is zero outside of its corresponding patch). How can we exploit this representation to improve the demixing step?
It is useful to first take a step back to consider the strengths and weaknesses of current state of the art demixing methods, most of which are based on NMF. The NMF model is very natural in calcium imaging applications, since each neuron has a shape that is fixed over the timescale of a typical imaging experiment (and these shapes can be represented as nonnegative images, i.e., an element of the matrix), and a corresponding timevarying calcium concentration that can be represented as a nonnegative vector (an element of ): to form a movie we simply take a product of each of these terms and add them together with noise and background, i.e., form .
However, current NMFbased approaches leave room for improvement in several key directions. First, since NMF is a nonconvex problem, good initializations are critical to obtain good results via the standard alternating optimization approaches (similar points are made in [Petersen et al., 2017]). Good initialization approaches have been developed for somatic or nuclear calcium imaging, where simple Gaussian shape models are useful crude approximations to the elements of [Pnevmatikakis et al., 2016], but these approaches do not apply to dendritic or axonal imaging. Second (related), it can be hard to separate weak components from noise using current NMFbased approaches. Finally, voltage imaging data does not neatly fit in the NMF framework, since voltage traces typically display both positive and negative fluctuations around the baseline resting potential.
To improve the robustness of NMF approaches for demixing functional data, we make use of the growing literature on “guaranteed NMF” approaches — methods for computing a nonnegative matrix factorization that are guaranteed to output the “correct” answer under suitable conditions and assumptions [Donoho and Stodden, 2004, Recht et al., 2012, Arora et al., 2012, Li et al., 2016]
. In practice, these methods work well on clean data of sufficiently small dimensionality, but are not robust to noise and scale poorly to highdimensional data. We can solve both of these issues by “superpixelizing” the denoised version of
; the resulting NMF initialization method improves significantly on state of the art methods for processing dendritic and axonal data. We also take advantage of the sparse, lowrank structure of and to speed up the NMF iterations.Initialization via pure superpixels
The first step of the initialization procedure is to identify groups of highly correlated spatially connected pixels – “superpixels.” The idea is that a pixel within a neuron should be highly correlated with its neighbors, while a pixel containing mostly noise should have a much lower neighbor correlation. These neighbor correlations, in turn, can be estimated much more accurately from the denoised compared to the raw data. The superpixelization procedure results in a set of nonoverlapping groups of pixels which are likely to be contained in good neural components. Then we want to extract “pure” superpixels, i.e., the subset of superpixels dominated by signal from just one neural component. We will use the temporal signals extracted from these pure superpixels to seed in the NMF decomposition.
To identify superpixels, we begin with the denoised data . Since the compression process discussed in the previous section is rather conservative (aiming to preserve the full signal, at the expense of retaining a modest amount of noise), there is room to apply a more aggressive lossy denoiser in the initialization stage to further reduce any remaining noise in . We softthreshold signals in each pixel that are not sufficiently large — less than the median plus the median absolute deviation (MAD) within each pixel, with or . (This thresholding serves to extract mostly spiking activity from functional imaging data.) We identify two neighboring pixels to be from the same superpixel if their resulting denoised, softthresholded temporal signals have a correlation larger than a threshold , with . Superpixels that contain fewer than pixels are discarded to further reduce noise and the total number of superpixels. We then apply rank 1 NMF on the signals from each superpixel to extract their (thresholded) temporal activities.
To extract pure superpixels, we apply the Successive Projection Algorithm (SPA) [Gillis and Vavasis, 2014] to the temporal activities of superpixels. This algorithm removes “mixed” superpixels whose temporal activity can be modeled as a nonnegative linear combination of activity in other superpixels (up to some Rsquared level larger than , where we use ) and outputs the remaining “pure” superpixels. See Algorithm 3 for pseudocode.
Note that running SPA on superpixels rather than raw pixels improves performance significantly here, since averaging signals within superpixels boosts SNR (making it easier to separate signal from noise and isolate pure from mixed pixels) and also greatly reduces the dimensionality of the nonnegative regression problem SPA has to solve at each iteration. (To keep the problem size small we also run SPA just on small local spatial patches, as in the previous section.) Finally, while we have obtained good results with SPA, other approaches are available [Gillis and Luce, 2018] and could be worth further exploration in the future. See Figure 2 for a visual summary of the full procedure.
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Local NMF
Next we run NMF, using the temporal signals extracted from the “pure” superpixels to initialize . Given the initial , the typical next step is to regress onto the data to initialize . (Note that pure superpixels typically capture just a subset of pixels within the corresponding neuron, so it is not efficient to initialize with the pure superpixels.) However, given the large number of pixels in a typical functional imaging video, direct regression of onto is slow and overfits, providing poor estimates of .
This issue is wellunderstood [Pnevmatikakis et al., 2016], and several potential solutions have been proposed. For somatic imaging it makes sense to restrict the support of to remain close to their initial values (we could use a dilation of the superpixel support for this). But for data with large dendritic or axonal components this approach would cut off large fractions of these components. Sparse regression updates are an option here, but these do not enforce spatial structure in the resulting directly; this often results in “speckle” noise in the estimated spatial components (c.f. Figure 15 below).
We have found the following approach to be more effective. We initialize the support set as the support of the th “pure” superpixel. Given , we compute the correlation image for each component as the correlation between the denoised data and the th temporal component, . We truncate this correlation image below a certain threshold to zero, then update as the connected component of the truncated correlation image which overlaps spatially with the previous . We use the modified fastHALS algorithm in [Friedrich et al., 2017] to update , , and to locally optimize the objective
(4) 
Here we have modeled the background as a simple temporallyconstant vector; we discuss generalizations to timevarying backgrounds below. Also note that we are approximating directly here, not the thresholded version we used to extract the superpixels above.
Finally, we incorporate a merge step: we truncate the correlation image below certain threshold to zero, and automatically merge neurons if their truncated correlation images are highly overlapped. The full algorithm is shown in Algorithm 4.
Further implementation details
Multi pass strategy: As in [Zhou et al., 2018], we find it effective to take a couple passes over the data; particularly in datasets with high neuron density, the first NMF pass might miss some dim neurons. We decrease the MAD threshold and rerun Algorithm 3 on the residual to find additional components, and then run a final merge and NMF update to complete the pipeline.
Improvements from denoising and compression: Compressed data leads to faster NMF updates, since we can replace as ; in fastHALS, we can regress each on or on first instead of directly onto . Similarly, when calculating the correlation image, we can compute the correlation between the low rank and first. As emphasized above, denoising also improves the estimation of the correlation images, which in turn improves the estimation of the support sets .
Timevarying background: It is straightforward to generalize the objective 4 to include a timevarying background, using either a lowrank model (as in [Pnevmatikakis et al., 2016]) or a ringstructured model (as in [Zhou et al., 2018]). For the lowrank background model, we have found that performing an SVD on the data excluding the support of the superpixels provides an efficient initialization for the background temporal components.
Incorporating temporal penalties: Note that we are only imposing nonnegativity in here; after denoising to obtain , we have found that this simple nonnegative constraint is sufficient for the datasets examined here. However, it is certainly possible to incorporate temporal penalties or constraints on (e.g., a TF penalty or a nonnegative autoregressive penalty as in [Pnevmatikakis et al., 2016]), either within each iteration or as a final denoising step.
Postprocessing: We find that sorting the extracted components by their “brightness,” computed as
, serves to separate dim background components from bright singleneuronal components. We also found it useful to drop components whose temporal trace has skewness less than 0.5; traces with high skewness correspond to components with significant spiking activity, but lowskewness traces corresponded to noise.
Results
Denoising
Dataset  Dimensions  Method  Compression  Total  SNR  

Frames  FOV  Patch  ratio  runtime (s)  metric  
Endoscopic  6000  256x256  16x16  Patchwise PMD  23  220.4  2.3 
16x16  Patchwise PCA*  X  X  X  
NA  Standard PCA  2  595.5  1.3  
Dendritic  1000  192x192  16x16  Patchwise PMD  52  3.2  3.7 
16x16  Patchwise PCA  32  1.2  2.5  
NA  Standard PCA  2  18.3  1.1  
Threephoton  3650  160x240  20x20  Patchwise PMD  94  12.4  1.8 
20x20  Patchwise PCA  44  3.5  1.4  
NA  Standard PCA  2  187.2  1.0  
Widefield  1872  512x512  32x32  Patchwise PMD  298  12.5  3.5 
32x32  Patchwise PCA  265  10.1  3.4  
NA  Standard PCA  10  80.1  1.6  
Voltage  6834  80x800  40x40  Patchwise PMD  180  30.5  2.8 
40x40  Patchwise PCA  213  8.7  2.7  
NA  Standard PCA  8  185.1  1.0 
We have applied the denoising and compression approach described above to a wide variety of functional imaging datasets (See Appendix for full details):

Endoscopic: onephoton microendoscopic calcium imaging in dorsal striatum of behaving mouse

Dendritic: twophoton Besselbeam calcium imaging of dendrites in somatosensory cortex of mouse in vivo

Threephoton: threephoton calcium imaging of visual cortex of mouse in vivo

Widefield: onephoton widefield wholecortex calcium imaging in behaving mouse

Voltage: onephoton in vitro voltage imaging under optogenetic stimulation.
The proposed methods perform well in all cases with no parameter tuning. We obtain compression ratios (defined as , where counts the number of nonzero elements of the matrix ) of 20x200x, and SNR improvements typically in the range of about 2x but ranging up to 10x, depending on the dataset and the region of interest (we find that SNR improvements are often largest in regions of strongest activity, so SNR improvements vary significantly from pixel to pixel). See Table 1 and Figures 312 for details.
In terms of runtime, we observed the expected scaling: the proposed method scales linearly in , , and the number of extracted components. In turn, the number of estimated components scales roughly proportionally to the number of neurons visible in each movie (in the datasets with singlecell resolution). Total runtimes ranged from a few seconds to a few minutes (for the “Endoscope” dataset, which had the largest number of extracted components); these runtimes are fast enough for the proposed method to be useful as a preprocessing step to be run prior to demixing.
We also performed comparisons against two simpler baselines: standard PCA run on the full dataset, and “patchwise PCA” run on the same patches as used by PMD. For patchwise PCA, we used the same stopping rule for choosing the rank of as described above for PMD, but did not apply the TV or TF penalty. We find that using the same rank selection criterion for PCA applied to the full dataset performs relatively poorly: in each of the five datasets examined, this approach left significant visible signal behind in the residual. Thus, to make the comparisons as favorable as possible for standard PCA, we chose the rank manually, to retain as much visible signal as possible while keeping the rank as low as possible. Nonetheless, we found that the PMD approach outperformed standard PCA significantly on all three metrics examined here (compression ratio, SNR improvement, and runtime), largely because PCA on the full image outputs dense matrices (leading to slower computation and worse noise suppression) whereas the matrices output by the patchwise approaches are highly sparse.
The patchwise PCA approach has much stronger performance than standard PCA applied to the full data. In four out of five datasets (the “Endoscope” dataset was the exception) patchwise PCA captured all the visible signal in the dataset and did not leave any visible signal behind in the residual. In these four datasets PMD performed comparably or significantly better than patchwise PCA in terms of SNR improvement and compression score, but patchwise PCA was faster. Thus there may be some room to combine these two approaches, e.g., to use PCA as a fast initial method and then PMD to provide further denoising and compression. We leave this direction for future work.
Demixing
Voltage imaging data
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Next we turn to the problem of demixing. We begin with an analysis of a challenging voltage imaging dataset. Voltage imaging (VI) data presents a few important challenges compared to calcium imaging (CI) data: currentlyavailable VI data typically has much lower SNR and displays much stronger bleaching effects than CI data. The dataset we focus on here has another challenging feature: the preparation was driven with timevarying fullfield optogenetic stimulation, resulting in highly correlated subthreshold activity in the visible cells, which are highly overlapping spatially. In preliminary analyses of this data we applied variants of CNMFE [Zhou et al., 2018] but did not obtain good results (data not shown), due to the strong bleaching and optogenetic stimulationinduced correlations present in this data.
Thus we preprocessed this data by applying a splinebased detrending to each pixel (see Appendix for full details). This served to attenuate the highlycorrelated bleaching signals and subthreshold fluctuations in the raw data, leaving behind spiking signals (which were not perfectly correlated at the millisecond resolution of the video data here) along with uncorrelated noise as the dominant visible signals in the data. Figure 2 shows that the denoiser (followed by softthresholding) serves to significantly improve the separability of neural signals from noise in this data: the superpixels obtained after denoising and softthresholding provide excellent seeds for the constrained NMF analysis. Figures 13 (and the corresponding video) and 14 demonstrate that the full demixing pipeline achieves good performance, extracting components with high spatial and temporal SNR and leaving relatively little residual signal behind despite the limited SNR and the multiple overlapping signals visible in the original (detrended) data. Note that in the final step we project the estimated spatial components back onto the original data, recovering the (highly correlated) temporal components including strong bleaching components (panel D of Figure 14). Finally, we achieved a speedup in the NMF iterations here that was roughly proportional to the ratio of the rank of compared to the rank of .
Bessel dendritic imaging data
Proposed pipeline  NMF on  NMF on 

Ground truth  Proposed pipeline  NMF on  NMF on 

Spatial components  Spatial components support  Temporal components 

The VI dataset analyzed in the preceding subsection contained a number of large visible axonal and dendritic components, but also displayed strong somatic components. For our next example we focus on a CI dataset dominated by dendritic components, where the simple Gaussian spatial filter approach introduced in [Pnevmatikakis et al., 2016] for initializing somatic components is ineffective. (Indeed, in dendritic or axonal imaging datasets, a search for “hotspots” in the images is biased towards pixels summing activity from multiple neurons — and these “nonpure” pixels are exactly those we wish to avoid in the demixing initialization strategy proposed here.)
Figure 15 illustrates several of the spatial components extracted by our pipeline (again, see the corresponding video for a more detailed illustration of the demixing performance); these components visually appear to be dendritic segments and match well with the signals visible in the data movie. Notably, no parameter tuning was necessary to obtain good demixing performance on both the VI and CI datasets, despite the many differences between these data types. Additionally, as a baseline comparison we applied a simple sparse NMF approach with random initialization (similar to the method described in [Pnevmatikakis et al., 2016]) to both the denoised and raw data ( and , respectively). As shown in the right columns of Figure 15, this baseline approach extracted components that were much more mixed and noisy than the components extracted by our proposed demixing pipeline; we also found that the baseline approach was more prone to missing weaker, dimmer components than was the proposed pipeline (data not shown).
The above analyses depended on qualitative visual examinations of the obtained components and demixing video. We also generated simulated data with characteristics closely matched to the raw data, in order to more quantitatively test the demixing performance against a known (albeit simulated) ground truth. To generate simulated data , we used the and estimated from the raw data, and further estimated the conditional distribution of the residual as a function of the denoised data in the corresponding pixel and time bin ; then we added independent noise samples from this signaldependent conditional distribution (but with the noise scale multiplied 2x, to make the simulation more challenging) to . See the simulated Bessel dendritic imaging video for comparison of real and simulated data. We ran the three demixing pipelines on this simulated data. Typical results of these simulations are shown in Figure 16: again we see that the proposed pipeline captures the ground truth components much more accurately than do the baseline methods, similar to the results shown in Figure 15. Quantitatively, components extracted by proposed pipeline have higher correlation with ground truth components than do those extracted by sparse NMF approaches, as shown in Figure 17.
Discussion
We have presented new scalable approaches for compressing, denoising, and demixing functional imaging data. The compression and denoising methods presented are generally applicable and can serve as a useful generic step in any functional video processing pipeline, following motion correction and artifact removal. The new demixing methods proposed here are particularly useful for data with many dendritic and axonal processes, where methods based on simple sparse NMF are less effective.
Related work
Other work [Haeffele et al., 2014, Pnevmatikakis et al., 2016, de Pierrefeu et al., 2018] has explored penalized matrix decomposition incorporating sparsity or total variation penalties in related contexts. An important strength of our proposed approach is the focus on highly scalable patchwise computations (similar to CaImAn); this leads to fast computations and avoids overfitting by (implicitly) imposing strong sparsity constraints on the spatial matrix . We also employ a constrained optimization approach using the trendfiltering (TF) penalty, which is more flexible e.g. than the sparse convolutional temporal penalty used in [Haeffele et al., 2014], since the constrained TF approach doesn’t require us to fit a specific convolutional model or to estimate any Lagrange multipliers for the sparsity penalty.
There are also some interesting connections between the demixing approach proposed in [Petersen et al., 2017] and our approach to initializing NMF, which is based on the sparse projection algorithm (SPA). [Fu et al., 2015, Gillis and Luce, 2018] discuss the relationships between SPA and groupsparse dictionary selection methods related to the approach used in [Petersen et al., 2017]; thus the methods we use to compute “pure” superpixels and the methods used in [Petersen et al., 2017] to select neural dictionary elements are closely related. However, our denoisethensuperpixelize approach to seeding the dictionary of neural temporal components is in a sense converse to the clustering approach developed in [Petersen et al., 2017] for seeding the dictionary of neural spatial components. There may be room to fruitfully combine these two approaches in the future.
Future work
Realtime online updates for and should be possible, which would enable the incorporation of the compression and denoising approach into [Giovannucci et al., 2017] for improved online demixing of neural activity. We are also continuing to explore alternative methods for spatial and temporal denoising of and
, e.g. artificial neural network denoisers.
In the near future we plan to incorporate our code into the CaImAn and CNMFE packages for calcium imaging analysis. We hope that the proposed compression methods will help facilitate more widespread and routine public sharing of these valuable datasets and lead to more open and reproducible neuroscience.
Code
Open source code is available at https://github.com/paninskilab/funimag.
Video captions

Microendoscopic imaging background video
(left) Raw movie ; (middle) background estimated via rank5 PMD; (right) estimated foreground . Ticks along the horizontal and vertical axis (in this video and in the videos below) indicate patch borders; note that no edge artifacts are visible at these borders. 
Microendoscopic imaging video
(left) Foreground; (middle) denoised foreground ; (right) residual . 
Threephoton imaging video
(left) Raw movie ; (middle) denoised movie ; (right) residual . 
Widefield imaging video
Same format as previous video. 
Superpixelization video
Panels from top to bottom: (1) detrended movie ; (2) denoised movie ; (3) MAD softthresholded movie; (4) rank1 NMF approximation within superpixels; (5) superpixels; (6) pure superpixels. 
Voltage imaging demixing video
Panels from top to bottom: (1) detrended movie ; (2) denoised movie ; (3) estimated signal ; (4) background ; (5) residual ; (6) estimated noise . 
Bessel dendritic imaging demixing video
Top: (left) motion corrected movie ; (middle) denoised movie ; (right) estimated signal ; Bottom: (left) background , (middle) residual , and (right) estimated noise . 
Simulated Bessel dendritic imaging video
Top: (left) Motion corrected real movie; (right) simulated movie. Bottom: (left) estimated noise from real movie; (right) simulated noise.
Acknowledgments
We thank Shay Neufeld and Bernardo Sabatini for generously sharing their microendoscopic data with us, and Andrea Giovanucci, Eftychios Pnevmatikakis, Ziqiang Wei, Darcy Peterka, Jack Bowler, and Uygar Sumbul for helpful conversations. We also thank our colleagues in the International Brain Laboratory for motivating our efforts towards compressing functional imaging data. This work was funded by Army Research Office W911NF1210594 (MURI; EH and LP), the Simons Foundation Collaboration on the Global Brain (LP), National Institutes of Health R01EB22913 (LP), R21EY027592 (LP), 1U01NS10348901 (NJ and LP), R01NS063226 (EH), R01NS076628 (EH), RF1MH114276 (EH), and U19NS10464901 (EH and LP); in addition, this work was supported by the Intelligence Advanced Research Projects Activity (IARPA) via Department of Interior/ Interior Business Center (DoI/IBC) contract number D16PC00003 (LP). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Author contributions
EKB and LP conceived the project. EKB led development of the local PCA compression and denoising approach, including the 4x overcomplete approach for avoiding block artifacts. IK led development of the PMD(TF,TV) approach. DZ led development of the superpixelization and local NMF demixing approach. RZ developed a preliminary version of the PMD approach. PZ contributed to the development of the demixing approach. FG, JF, and GD contributed the voltage imaging dataset. JR, PF, TM, and AT contributed the threephoton imaging dataset. YL, RL, and NJ contributed the Bessel dendritic dataset. YM, SK, MS, and EH contributed the widefield dataset. EKB, IK, DZ, and LP wrote the paper, with input from PZ. LP supervised the project.
Appendix: dataset details
Microendoscopic imaging data
This dataset was analyzed previously in [Zhou et al., 2018]; see the “Dorsal Striatum Data” subsection of the Methods section of that paper for full experimental details. Briefly, a 1 mm gradient index of refraction (GRIN) lens was implanted into dorsal striatum of a mouse expressing AAV1SynGCaMP6f; imaging was performed using a miniature onephoton microscope with an integrated 475 nm LED (Inscopix) while the mouse was freely moving in an openfield arena. Images were acquired at 30 Hz and then down sampled to 10 Hz.
Bessel dendritic imaging data
All surgical procedures were in accordance with protocols approved by the Howard Hughes Medical Institute Janelia Research Campus Institutional Animal Care and Use Committee. C57BL/6J mice over 8 weeks old at the time of surgery were anesthetized with isoflurane anesthesia (1–2%). A craniotomy over nearly the entire left dorsal cortex (from Bregma +3 mm to Bregma 4.0 mm) was performed with the dura left intact, with the procedure described in detail previously in [Sofroniew et al., 2016]. AAV2/9synapsinflexGCaMP6s (2.5 GC/ml) was mixed with AAV2/1synapsinCre (1.5 GC/ml, 1000dilution with PBS) at 1:1 to make the working viral solution for intracerebral injections. 30 nl viral solution was slowly injected into exposed cortex at 0.5 mm below dura. Injection sites were evenly spaced (at 0.70.9 mm separation) along two lines at 2.3 mm and 3.3 mm parallel to the midline. A custommade glass coverslip (450 m thick) was embedded in the craniotomy and sealed in place with dental acrylic. A titanium head bar was attached to the skull surrounding the coverslip. After recovery from surgery, the mice were habituated to head fixation. Four weeks after surgery, the headfixed mouse was placed on a floating ball in the dark. The spontaneous neural activity as indicated by GCaMP6s fluorescence signal was recorded in the somatosensory cortex.
Volumetric imaging of dendrites was achieved by scanning an axially extended Bessel focus in [Lu et al., 2018] and [Lu et al., 2017]. An axiconbased Bessel beam module was incorporated into a 2photon random access mesoscope (2pRAM) in [Lu et al., 2018]. Details of the 2pRAM have been described previously in [Sofroniew et al., 2016]. Briefly, the system was equipped with a 12kHz resonant scanner (24 kHz line rate) and a remote focusing unit that enabled fast axial movements of the focal plane. The system has an excitation numerical aperture (NA) of 0.6 and a collection NA of 1.0. The measured lateral full width at half maximum (FWHM) of the Gaussian focus at the center of the field of view was 0.65 m. The lateral and axial FWHMs of the Bessel focus were 0.60 m and 71 m, respectively. Scanning the Bessel focus in two dimensions, therefore, probed brain volumes within a 100 m axial range. The volumetric dendritic data presented in this paper were obtained by placing the center of the Bessel focus at 62 m below dura to probe structures at 12 m to 112 m below dura (figure 18). Dendrites within this volume were imaged at an effective volume rate of 3.7 Hz, with each image having 19242104 pixels at 0.33 m/pixel in the xy plane. The wavelength of the excitation light was 970 nm and the postobjective excitation power was 120 mW. Images were spatially decimated and cropped for the analyses shown here.
Threephoton imaging data
All procedures were carried out in accordance with the ethical guidelines of the National Institutes of Health and were approved by the Institutional Animal Care and Use Committee (IACUC) of Baylor College of Medicine. Cranial window surgeries over visual cortex were performed as described previously [Reimer et al., 2014]. Briefly, a 4 mm cranial window was opened under isoflurane anesthesia and sealed with a 4 mm glass coverslip and surgical glue. The dura was removed before applying the coverslip to increase optical access to the cortex. Imaging was performed in a tripletransgenic mouse (Slc17a7Cre x Dlx5CreER x Ai148) expressing GCaMP6f panneuronally throughout cortex. Threephoton imaging data was collected as described previously [Ouzounov et al., 2017]. Threephoton excitation of GCaMP6 was at 1320nm, which also enabled visualization of unlabeled vasculature and white matter via THG (third harmonic generation). Power was calibrated prior to each day of scanning and carefully maintained below 1.5nJ at the focal plane. For this study, scans were collected at 680 microns and 710 microns below the cortical surface with a 540 x 540 micron field of view at 0.59 pixels/micron spatial resolution and a frame rate of 5 Hz. Imaging was performed at the border of V1 and LM during presentation of oriented noise stimuli.
Widefield imaging data
See [Ma et al., 2016b, Ma et al., 2016a] for full details.
Voltage imaging data
QState’s proprietary Optopatch alloptical electrophysiology platform was used to record fluorescence recordings from induced pluripotent stem (iPS) cellderived NGN2 excitatory neurons from a cohort of human subjects [Werley et al., 2017]. Stimulation of action potentials was achieved with a blue lightactivated channelrhodopsin (CheRiff). Fluorescent readout of voltage was enabled by an Archaerhodopsin variant (QuasAr). NGN2 neurons were produced at QState using a transcriptional programming approach. Recordings were performed with an ultrawidefield instrument with a resolution of 800x80 pixels (corresponding field of view of 2 mm) at a frame rate of 987 Hz.
The obtained data displayed breaks during stimulus resets and photobleaching. To remove these effects from the raw data, we removed frames during stimulus resets, extracted slow trends with a robust Bspline regression (with knots chosen to allow for nondifferentiability at stimulus changepoints and discontinuity at stimulus resets), and then a quadratic regression against frames with no stimuli to capture and then remove photobleaching effects.
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