FourierNets enable the design of highly non-local optical encoders for computational imaging
Anonymous Author(s)
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Abstract
Differentiable simulations of optical systems can be combined with deep learningbased reconstruction networks to enable high performance computational imagingvia end-to-end (E2E) optimization of both the optical encoder and the deep learningdecoder.
This has enabled imaging applications such as 3D localization microscopy,depth estimation, and lensless photography via the optimization of local opticalencoders.
More challenging computational imaging applications, such as 3D snapshot microscopy which compresses 3D volumes into single 2D images, require ahighly non-local optical encoder.
We show that existing deep network decodershave a locality bias which prevents the optimization of such highly non-local optical encoders.
We address this with a decoder based on a shallow neural networkarchitecture using global kernel Fourier convolutional neural networks (FourierNets).
We show that FourierNets surpass existing deep network based decoders atreconstructing photographs captured by the highly non-local DiffuserCam opticalencoder.
Further, we show that FourierNets enable E2E optimization of highlynon-local optical encoders for 3D snapshot microscopy.
By combining FourierNets with a large-scale multi-GPU differentiable optical simulation, we are ableto optimize non-local optical encoders 170 × to 7372 × larger than prior state ofthe art, and demonstrate the potential for ROI-type specific optical encoding with aprogrammable microscope.
Introduction
Modern computational optics relies on the end-to-end (E2E) optimization of the optical system(optical encoder) and the computational image reconstruction algorithm (computational decoder).
This has been enabled by the development of differentiable physics-based simulations of opticalencoders, paired with deep network based decoders.
This E2E approach has been successfully appliedto the estimation of depth from defocus [1, 2, 3, 4], lensless photography [5], and particle localizationmicroscopy [6, 7].
Despite this wide success, only the space of local optical encoders has beenexplored in this fashion due to the computational expense of physics-based simulation of highlynon-local optical encoders [3].
This is particularly the case for 3D snapshot microscopy which iswell-known to require highly non-local optical encoders, and for which no E2E solutions have yetbeen demonstrated.
We show that there are actually two issues blocking the E2E optimization of highly non-local opticalencoders.
First, the computational expense of physics-based simulation of non-local optical encodersis significantly greater, since they are orders of magnitude larger than local encoders.
And second, thelocality bias inherent to deep network architectures currently used for computational imaging preventsdecoding from non-local solutions.
In this paper, we primarily focus on developing a computationalframework for engineering highly non-local optical encoders by solving both issues.
We solve the first issue by developing a large-scale multi-GPU differentiable optical simulation, and the second bydeveloping a new FourierNet architecture based deep network decoder.
In 3D snapshot microscopy, all existing methods [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 5, 19]employ a highly non-local optical encoding to optically transform a 3D volume into a single 2Dcamera image, which is computationally decoded to reconstruct the 3D volume.
Fast volumetricimaging is invaluable across biology, including for imaging neural activity across the whole brainof an animal [20, 21, 22].
Here, we focus on developing a 3D snapshot microscope for imagingneuronal activity across the whole brain of the larval zebrafish ( Danio rerio ) at camera rates exceeding100Hz.
This is two orders of magnitude faster than the fastest conventional microscopes — lightsheet imaging of whole brain neural activity can only achieve 0.5Hz - 2Hz volume rates [20].
Thisimprovement of temporal resolution is essential for imaging with fast calcium indicators [23] andvoltage indicators [24].
Newly developed programmable microscopes with up to 10 6 free parameters, e.g. pixels on a spatiallight modulator (SLM), enable the implementation of a rich space of optical encodings, and presentthe possibility of direct optimization of snapshot microscope parameters specifically for particular
ROI types and imaging tasks.
While our paper primarily focuses on addressing the computationalchallenges of E2E optimization of highly non-local optical encoders, we also demonstrate proof-ofconcept that an SLM-based programmable microscope can implement such an engineered non-localoptical encoder, paving the way to implementing ROI-type and task specific optical encoders ina single physical microscope.
In contrast, 3D snapshot imaging has classically been performedusing fixed optical encoders based on microlens arrays [8, 9, 10, 11, 12, 13, 14, 15], pseudorandomdiffusers/masks [16, 17], and designed or optimized diffusers/masks [18, 5, 19], with some of thesemethods using deep learning for reconstruction [11, 15].
Problem statement We define an optical encoder as M ϕ parameterized by ϕ and a computationaldecoder as R θ parameterized by θ .
We wish to develop a reconstruction network architecture fordecoding non-local encodings in 2D images c produced by M ϕ and also to enable the end-to-endoptimization of ϕ to produce non-local encodings.
We’ll consider two applications to investigatethese optimization problems: (1) lensless photography (Figure 2), where we optimize only thereconstruction network R θ to reconstruct images of natural scenes from camera images capturedby the DiffuserCam [25], and (2) 3D snapshot microscopy using an SLM-based programmablemicroscope (Figure 1A), where we perform E2E optimization of both the reconstruction network R θand the optical encoding M ϕ based on the SLM parameters in order to image 3D volumes v andreconstruct 3D volumes ˆv .
While we focus on solving the computational challenges involved in the
E2E optimization of large highly non-local optical encoders, we also demonstrate a proof-of-conceptimplementation of our designed optical encoder using a spatial light modulator based programmablemicroscope.
1.1 Our contributions
We developed a large-scale, parallel, multi-GPU differentiable wave optics simulation of aprogrammable microscope, based on a 4 f optical model with a phase mask ( ϕ ) implementedusing a spatial light modulator (SLM), described further in Appendix A.1.
SLMs ϕ canhave over 10 6 optimizable parameters, which we can feasibly simulate and optimize toproduce PSFs with 170 × to 7372 × more unique voxels than previous attempts at deeplearning PSF optimization using single GPUs [6, 3] (Appendix A.5).2.
We collected a large dataset of high resolution 3D confocal volumes of zebrafish larvae forthe purpose of ROI-type specific end-to-end optimization of optical encoders.3.
We introduce an efficient FourierNet reconstruction network architecture for decoding fromnon-local optical encoders using very large global convolutions implemented via Fourierconvolutions.
We show that our networks outperform the state-of-the-art deep decoders for DiffuserCam
Our method enables, for the first time, direct end-to-end optimization of highly non-local optical encoders in the space of spatial light modulator (SLM) pixels with over 10 6 parame- ters.
In simulation, we demonstrate the potential for significant improvements in imaging resulting from ROI-type specific optimization of optical encoders.
1.2 Prior work
Neural network architectures for computational imaging have all used convolution layers withsmall filters.
End-to-end optimization of optical encoders have largely been performed with UNetbased architectures [1, 2, 3, 4, 5, 7], with one method using a ResNet-based architecture [6].
Suchoptimization has always led to local optical encodings.
End-to-end optimization has never beenattempted for large-field of view 3D snapshot microscopy due to the difficulty of simulating andreconstructing from non-local encoders.
However, small filter convolutional deep networks have beenused in a non-end-to-end manner to reconstruct volumes from 3D snapshot microscopes designedusing microlens arrays [15, 26].
For photography and MRI, using deep learning combined withunrolling iterations of traditional deconvolution algorithms provides the benefits of fast amortizedoptimization and higher quality reconstructions due to learning of structural priors [27, 28, 25, 26].
In our work, we demonstrate that convolution layers with large filters implemented efficiently in the
Fourier domain enable the end-to-end learning of highly non-local optical encoders for 3D snapshotmicroscopy.
Large convolution filters have been shown to be helpful for other computer visionapplications such as semantic segmentation and salient object detection [29, 30], and the Fourierdomain parameterization of small filters has been described previously [31].
A pioneering strategy in 3D snapshot microscopy has been light field microscopy [8], which employsa microlens array at the microscope’s image plane to create subimages encoding both amplitudeand phase of light [8, 32].
A variety of microlens-array-based light field microscopes have beenused to perform whole-brain imaging [8, 11, 14, 22, 13, 12, 9, 33, 19].
[19] optimizes the placementof microlenses, but not in an end-to-end manner.
Despite variation in design, microlens-basedmicroscopes have, to various degrees, four main limitations that can be improved: 1) blocking orscattering of light between microlenses, causing light inefficiency, 2) not making use of all pixels onthe camera to encode a 3D volume, leading to inefficient compression and suboptimal reconstructionsˆv , 3) aliasing at some planes with generally nonuniform axial resolution, and 4) a fixed opticalencoding scheme.
An alternative to using microlenses is to implement a coded detection strategy using a phase mask ordiffuser to spread light broadly across the camera sensor [16, 18, 34, 5, 10, 17, 19].
The designedphase masks can be implemented either by manufacturing a custom optical element or using aprogrammable SLM [2, 3, 4, 5, 19, 6].
Using a programmable element allows different microscopeparameters to be used for different sample and ROI types.
Methods
[Figure] We show our network architecture and an overview of autoencoder training both the microscopeparameters ϕ and reconstruction network parameters θ in Figure 1A.
The programmable microscopeis simulated by a differentiable implementation of a wave-optics model of light propagation.
Wehave selected a programmable microscope design based on pupil-plane phase modulation with aprogrammable spatial light modulator, for which imaging is well-approximated by a computationally- efficient convolution [35].
A detailed description of our simulation is provided in Appendix A.1.
Forlensless photography, there is no optical simulation because the images have been collected on a realcamera.
2.1 FourierNet for decoding non-local optical encoders
Because images created by optical encoders can potentially encode signals from the incoming lightfield to any location in the camera image in a non-local manner, it is essential that a reconstructionnetwork have global context.
Existing multi-scale architectures such as the UNet [36] can achieveglobal context, but at the expense of many computation layers with small convolution kernels whichwe believe have a local information bias which is inappropriate for computational optics.
In this paper,we introduce relatively shallow architectures for computational imaging, which rely on convolutionlayers with very large filters, implemented efficiently in the Fourier domain [31].
FourierNet We propose a simple three layer convolutional network architecture with very largeglobal convolutions at the very first layer, followed by two standard local convolutional layers (Figure1B).
We define a global convolution as a convolution with kernel size equal to the input image.
Sucha convolution achieves global context in a single step but is computationally expensive.
Globalconvolutions are implemented more efficiently in the Fourier domain, yielding a speed up of twoorders of magnitude.
Due to the use of Fourier convolutions to enable global context, we call ourarchitecture FourierNet.
In contrast to a typical UNet which can contain many tens of convolutionlayers, the FourierNet is only three layers deep, which requires backpropagation through fewer layerscompared to a typical UNet with the same receptive field.
Fourier domain convolutions It is well-known that large kernel convolutions can be implementedmore efficiently in the Fourier domain [37, 38, 39].
A naive implementation of global convolutionrequires O ( N 2 ) operations, where N is number of pixels in both the image and the kernel.
Analternative global convolution implementation is to Fourier transform F the input x and convolutionkernel w , perform element-wise multiplication in Fourier space, and finally inverse Fourier transform
F − 1 , requiring only O ( N log N ) operations [38, 39].
Following [31], we store and optimize theweights in Fourier space W .
This over-parameterization costs 8 × the memory of an equivalent realvalued large filter but saves the computational cost of Fourier transforming the real-valued weights(Appendix A.6).
Thus a Fourier convolution is defined:
[Equation]
For image and kernel sizes of 256 × 256 pixels, our implementation leads to nearly 500 × speedup:standard PyTorch convolution takes 2860ms, while Fourier convolution takes 5.92ms on a TITAN X.
We also show how we can naturally extend our Fourier convolutions and FourierNet to a multiscaleversion using cropping in the Fourier domain in Appendix A.2.
2.2 Physics-based autoencoder for simultaneous engineering of microscope encoder and
reconstruction network decoder
We describe the imaging process as the following transformation from the 3D light intensity volumev to the 2D image formed on the camera c :
[Equation]
where M ϕ denotes the microscope parameterized by a 2D phase mask, ϕ .
This phase mask ϕdescribes the 3D-to-2D encoding of this microscope model completely.
A Poisson distributionwith mean rate µ c describes the physics of photon detection at the camera, but sampling from thisdistribution is not differentiable.
We approximate the noise distribution with a rectified Gaussian.
We include details on M ϕ in Appendix A.1 [35].
Jointly training reconstruction networks andmicroscope parameters involves image simulation, reconstruction, then gradient backpropagation toupdate the reconstruction network and microscope parameters.
Our parallelization strategy enablesoptimization of phase masks with millions of parameters in a feasible amount of time and memoryper GPU.
This also enables us to produce PSFs with multiple orders of magnitude more uniquevoxels than previous attempts (Appendix A.4).
Details on parallelization, planewise reconstructionnetworks, and planewise sparse gradients are provided in Appendix A.4, Figure 12.
We use the the normalized mean squared error (NMSE) as the basis for our loss function.
Since wecare more about the high spatial frequency content of the image which contains mostly cells, and lessabout the low spatial frequencies which contain mostly background, we implemented a two part lossfunction.
L HNMSE measures the NMSE between the high pass filtered volume and high pass filteredreconstruction.
This is combined with the unfiltered NMSE L NMSE between the original volume andits reconstruction.
Formally, our loss function L ( v , ˆv ) is defined:
[Equation]
where H ( · ) denotes high pass filtering and E ( · ) denotes the mean over pixels and sample volumes.
Both loss terms are normalized as shown to reduce variance in L , which can otherwise cause largemagnitude fluctuations based on the brightness variation across training volumes and cause traininginstability.
For our experiments, we set the weight β for the L NMSE term to 0.1.
Results
DiffuserCam Lensless Mirflickr Dataset We also test reconstruction performance on experimentalcomputational photography data 1 from [25] (Figure 2).
This is a dataset constructed by displaying
RGB color natural images from the MIRFlickr dataset on a monitor and then capturing diffusedimages by the DiffuserCam lensless camera.
The dataset contains 24,000 pairs of DiffuserCam andground truth images.
The goal of the dataset is to learn to reconstruct the ground truth images fromthe diffused images.
As in [25], we train on 23,000 paired diffused and ground truth images, and teston 999 held-out pairs of images.
Larval Zebrafish Snapshot Microscopy Dataset
We show all our results for snapshot microscopyusing our simulation of a snapshot microscope, and our engineered optical encoders have not beenexperimentally tested on a programmable microscope.
We simulate snapshot imaging using highresolution confocal imaging volumes of zebrafish.
These are volumes of transgenic larval zebrafishwhole brains expressing nuclear-restricted GCaMP6 calcium indicator in all neurons.
These imagesare representative of brain-wide activity imaging.
We train on 58 different zebrafish volumes (whichwe augment heavily) and test on 10 held-out volumes.
For all experiments, we downsample the highresolution confocal data to (1.0 µ m z, 1.625 µ m y, 1.625 µ m x).
We created 4 datasets from thesescanned volumes corresponding to imaging different fields of view or regions of interest for ourexperiments.
Full specifications for these datasets are in Table 6 (Appendix A.4).
For Figure 3, werestrict the field of view to (200 µ m z, 416 µ m y, 416 µ m x) with a tall cylinder cutout of diameter193 µ m and height 200 µ m and image with 256 × 256 pixels on the simulated camera sensor.
Figure4 and Table 3 show our larger experiments with 512 × 512 simulated camera pixels, with a field ofview of (250 µ m z, 832 µ m y, 832 µ m x).
3.1 FourierNets outperform state-of-the-art for reconstructing natural images captured byDiffuserCam lensless camera
[Table]
We compare our FourierNet architecture to the best learned method from [25] using an unrolled
ADMM and a denoising UNet, as well as to a vanilla UNet from [25].
Architecture details arein Appendix A.6, A.7.
We can see that FourierNet visually outperforms the methods from [25] g r o u n d t r u t h
D i ff u s e r
C a m i m a g e
L e - A D
MM - U ( M S E +
L P I P S )
U N e t ( M S E +
L P I P S )
F o u r i e r
N e t ( (cid:31) (cid:30) (cid:29) (cid:28) )
( M S E +
L P I P S )            in Figure 2, and quantitatively in Table 1 across all quality metrics.
Unlike [25], we find thattraining FourierNets with the MSE loss alone provides visually similar results as shown in Figure 13(Appendix A.7).
Microscope
Reconstruction
[Table]
FourierNet2D
[Table] FourierNet2D [Table] We compare optimizing microscope parameters ϕ with two neural networks: 1) using our FourierNetwith 2D convolutions (FourierNet2D) and 2) using a vanilla UNet with 2D convolutions (UNet2D).
Training is a two step process in which a planewise reconstruction network with fewer parameters and2D convolutions is used during optimization of ϕ , then once ϕ is fixed a more powerful reconstructionnetwork with 3D convolutions is used. We find this scheme achieves better reconstruction qualitybecause the reconstruction network does not need to constantly adapt to a changing optical encoding(Appendix A.6). We train these microscopes and reconstruction networks on ROIs which are tallcylindrical cutouts of zebrafish with diameter 193 µ m and height 200 µ m. Sample volumes areimaged on a camera with 256 × 256 pixels (Figure 3). FourierNet2D has 2 convolution layers(with 99.8% of its kernel parameters in the initial Fourier convolution layer), while UNet2D has 32convolution layers (kernel parameters approximately uniformly distributed per layer).
FourierNet2Dand
UNet2D both have ∼ 4 × 10 7 parameters; FourierNet3D has ∼ 6 × 10 7 parameters vs. ∼ 10 8for UNet3D. Architecture details are in Appendix A.6, A.8.
Simulated camera images (Figure 3) show that the UNet microscope does not make sufficient useof camera pixels, producing only a single view of the volume. We believe this is due to a localinformation prior in the small kernels of UNets. The FourierNet microscope uses more camera pixelsand performs better than the UNet microscope for reconstruction (Figure 3, quantified in Table 2). Times in Table 1 are for only the forward pass on a single GPU; one training iteration on 8 GPUs takes∼ 0.4 seconds for FourierNet3D and ∼ 0.8 seconds for UNet3D (Appendix A.6).
Both reconstruction x y z x x y 50 μm 100 μm
FourierNet ground truth
UNet z
y x x y m a x p r
o j e c t
i o n x z m a x
p
r
o j e c t
i o n x y m a x p
r
o j e c t
i o n 3.00 s 0.01 s 0.
s l
i g h t p r o p a g a t i o n s i m
u l a t e d c a m e r a
i m a g e r e c o n s t r u c t e d v o l u m e networks must reconstruct from images that have a compressed encoding of 3D information, but the
FourierNet2D is clearly more effective than the UNet2D at optimizing this encoding.
3.3 FourierNets outperform UNets for 3D snapshot microscopy volume reconstruction
We can determine which architecture is better for volume reconstruction by choosing fixed microscopeparameters and varying the architecture.
In Table 2, we compare results using a FourierNet with3D convolutions (FourierNet3D) and a vanilla UNet with 3D convolutions (UNet3D).
Architecturedetails are in Appendix A.6, A.8.
Reconstruction results in Table 2 compare normalized MSE L HNMSE between the high pass filtered volume and high pass filtered reconstruction, the multiscale structural similarity MS − SSIMbetween the true volume and its reconstruction, and finally the peak signal-to-noise ratio PSNR .
We also visualize reconstruction results for a volume in the head of a zebrafish in Figure 3.
The
UNet3D reconstruction networks (using either microscope) fall significantly behind the FourierNet3Dreconstruction network in all metrics. Microscope parameters optimized for [Figure] [Table]
Tested on
[Figure] [Table] [Figure] [Table]
[Figure] [Table] [Figure] .4 Engineered optical encoders can be optimized for a region of interest
To explore the effect of ROI size on optimized ϕ and the resulting reconstruction performance, weoptimized ϕ s for three different regions of interest: 1) Type A, ROIs with a short cylinder cutout of386 µ m diameter and 25 µ m height, 2) Type B, ROIs with a tall cylinder cutout of 386 µ m diameterand 250 µ m height, and 3) Type C, ROIs without any cutout of dimension (250 µ m z × 832 µ m y ×832 µ m x) (Table 3).
All ROI types were imaged with 512 × 512 pixels on the simulated camera.
Wethen tested reconstruction performance on all combinations of optimized ϕ and ROI type, as shownin Table 3 and visualized in Figure 4 for Type B. We include architecture details in Appendix A.6,
A.9.
We see in Table 3 that for all types, highest performance is achieved using the phase mask optimizedfor that particular type.
These results show that there is potentially a large benefit to optimizingtype-specific optical encoders, which can be easily implemented in a programmable microscope.
3.5 Engineered optical encoders can be implemented on a programmable microscope
We demonstrate that the optical encoders engineered by simulations can be implemented on realhardware using a prototype programmable microscope.
Our engineered phase masks were displayedon a spatial light modulator (SLM) located in the pupil plane.
We observe a good qualitative matchbetween
the features of the simulated and experimental optical encoders (point spread functions) in
Figure 5, technical details in Appendix A.10.
There is also a good qualitative match between thesimulated camera images generated by both optical encoders.
These results demonstrate the potentialfor programmable microscopy with spatial light modulators.
[Figure]
Discussion
Summary We have presented the FourierNet architecture as a deep network based decoder forcomputational imaging with highly non-local optical encoders.
We demonstrated the superiorityof FourierNets for lensless photography on the DiffuserCam dataset, and also for the end-to-endoptimization of highly non-local optical encodings (where UNets fail) for 3D snapshot microscopy.
FourierNets are many orders of magnitude faster than traditional iterative reconstruction algorithms[25] and generate higher quality reconstructions by learning image priors.
Generally, our global
Fourier-domain convolution architecture could be applicable to other problems where global integration of features is necessary, though we have focused on computational imaging applications wherephysics-based optical encoders induce global mixing of information.
Our work solves two importantcomputational problems preventing E2E optimization of large highly non-local optical encoders: thecomputational complexity of simulating large non-local encoders, and the effective decoding fromsuch encoders.
Our contributions are primarily computational, however we also demonstrate thepotential for implementing 3D snapshot microscopy with an E2E optimized non-local optical encoderusing an SLM-based programmable microscope.
And in simulation, we demonstrated the potentialfor region of interest specific optimization of optical encodings for 3D snapshot microscopy, whichcan be efficiently implemented in a programmable microscope.
Limitations While we have fully demonstrated that our framework now enables computational E2Eoptimization of large non-local optical encoders, we do not focus on their hardware implementation.
The specific claims regarding the potential benefits of ROI-specific optimization of optical encodersare only evaluated in simulation, and have only been evaluated for whole brain larval zebrafish data.
Reproducibility We train on 8 Quadro RTX 8000 GPUs for the largest experiments, and havedescribed our pre-processing, training, and testing procedures in Appendix A.4, A.6, A.8.
We willalso release our simulation software, training scripts, and our datasets upon publication.
Broader Impact End-to-end optimization of optics can improve image quality but also enablemultiplexed imaging not currently possible with standard optics.
Our optical modeling library aswell as our neural network architectures will be open-sourced and could enable new experiments inneuroscience through whole-brain imaging at an order of magnitude greater temporal resolution.
Ourtraining procedure does require long optimization periods with many GPUs, which poses a carbonfootprint and barrier to usage compared to conventional microscopy.
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