. A burst captures a sequence of short-exposure frames of the scene that are free of motion-blur, but with a high amount of noise in each frame and relative motion between frames. By accounting for this relative motion and using the fact that the noise is independent across frames, burst denoising attempts to aggregate these inputs and predict a single noise- and blur-free image estimate.
Recently, Mildenhall et al.  proposed an elegantly simple yet surprisingly successful approach to burst denoising. Rather than explicitly estimating inter-frame motion [12, 24, 13, 14, 18], their method produces denoised estimates at each pixel as a weighted average of observed noisy intensities in a window around that pixel’s location in all frames. These averaging weights, or kernels, are allowed to vary from pixel-to-pixel to implicitly account for motion and image discontinuities, and are predicted from the noisy input burst using a “kernel prediction network” (KPN).
However, KPNs need to produce an output that is significantly higher-dimensional than the denoised image—even for
kernels with eight frames, a KPN must predict 400 times as many kernel weights as image intensities. This comes with significant memory and computational costs, as well as difficulty in training given the many degrees of freedom in the output. As a result, KPNs have so far been used only with small kernels. This limits their denoising ability by preventing averaging across bigger spatial regions, and over frames with larger relative motion.
In this paper, we introduce an approach to predict large denoising kernels, and thus benefit from wider aggregation, while simultaneously limiting output dimensionality at each pixel, making the prediction network easier to train and compute- and memory-efficient. This approach is motivated by a long history of successful image restoration methods that have leveraged internal structure and self-similarity in natural images [5, 2, 35, 22, 40]. In burst denoising, self-similarity is particularly strong because we expect both spatial structure in the form of similar patterns that recur within a frame and across frames of the same scene, and temporal structure caused by consistency in scene and camera motion. Given the expected self-similarity and structure in the image intensities themselves, we argue that the corresponding denoising kernels must also have similar structure. Specifically, while allowing for individual per-pixel denoising kernels to be large, we assume that all the kernels for a given image span a lower-dimensional subspace.
Based on this observation, we introduce a new method for kernel-based burst denoising that achieves better accuracy and efficiency. Our contributions are:
We train a network that, given an input noisy burst, predicts both a global low-dimensional basis set of large kernels, and per-pixel coefficient vectors relative to this basis. This corresponds to a significantly lower-dimensional output than a KPN that predicts arbitrary kernels of the same size at each pixel.
Enforcing this structure on the denoising kernels acts as a form of regularization, and our experiments demonstrate that it leads to state-of-the-art denoising performance with significantly higher-quality (>1 dB PSNR) than regular KPNs.
Beyond reducing memory usage and computational burden at the output layer of our network, the structure of our output enables the final kernel filtering step to be performed much more efficiently in the Fourier domain. We show this in terms of required number of FLOPs, as well as experimentally with actual run-times.
2 Related Work
Single-image and video denoising.
Single-image denoising has been studied extensively. To overcome the ill-posed nature of the problem, classical approaches [32, 33, 42] developed regularization schemes that model the local statistics of natural images. The most successful approaches [5, 2] exploit non-local self-similarity within the image and denoise pixels by aggregating similar pixels or patches from distant regions. These methods have been extended to denoise videos [19, 25], where the search for similar patches proceeds not only within frames, but also across different frames. Recent work have improved image denoising performance using convolutional networks trained on large datasets [3, 38, 39, 22, 40, 35, 36]
Single-image denoising is a fundamentally under-constrained problem. Burst processing can reduce this ambiguity by using multiple observations (the frames of the burst) to recover a noise-free depiction of the scene. Burst denoising algorithms are now extensively used in commercial smartphone cameras  and can produce compelling results even in extreme low light scenarios [4, 21]. Like in video denoising, a significant challenge for burst processing is the robustness to inter-frame motion. Many methods explicitly estimate this motion to align and denoise the frames [12, 24, 13, 14, 18]. Current state-of-the-art burst denoising techniques [27, 18, 11, 26]
are based on deep neural networks. Many of them only require coarse registration, relying on the network to account for the small residual misalignments[27, 11, 26].
Kernel Prediction Networks.
Given a burst sequence, Mildenhall et al.  propose predicting per-pixel kernels that are then applied to the input burst to produce the denoised output. They demonstrate that KPNs outperform direct pixel synthesis networks that produce oversmooth results. Subsequent work has extended this idea to use multiple kernels of varying sizes at every pixel . KPNs have also been used in other applications, including denoising Monte Carlo renderings [1, 34, 9]
, video super-resolution and deblurring 
, frame interpolation[28, 29, 23] and video prediction [15, 6, 23, 37].
Given their high-dimensional output (per-pixel kernels, that are three-dimensional in the case of burst denoising), KPNs have significant memory and compute requirements, as well as a large number of parameters in their final layer. To ameliorate this, Marinč et al.  and Niklaus et al.  propose predicting spatially separable kernels: producing a horizontal and vertical kernel for each frame as output, and forming the spatial kernel as an outer product of these. However, this makes a strong a-priori assumption about kernel structure, and still requires constructing and filtering with different per-pixel kernels. In contrast, our approach assumes that the set of per-pixel kernels for a scene span a low-dimensional sub-space, and predicts a basis for this sub-space based on the burst input. This approach also allows us to benefit for fast filtering in the Fourier domain.
In burst denoising, we are given an input noisy burst of images , where indexes spatial locations and
the different frames in the burst. Using a heteroscedastic Gaussian noise model, which accounts for both read and shot noise, we relate this to the corresponding noise-free frames as:
where and are the read- and shot-noise parameters. Choosing the first frame as reference, our goal is to produce a single denoised image as an estimate of the first noise-free frame .
3.1 Kernel-based burst denoising
Rather than train a network to regress directly, Kernel Prediction Networks output a field of denoising kernels , one for each pixel at each frame . The kernels have a spatial support , indexed by , with separate weights for each frame. Given these predicted kernels, the denoised estimate is formed as:
A key bottleneck in this pipeline is the prediction of this dense kernel field , which requires producing numbers at every pixel of the output. Since networks with high-dimensional outputs are both expensive and require learning a large number of parameters in their last layer, KPNs have typically been used only with small kernels ( in ).
3.2 Basis Prediction Networks
Instead of directly predicting unconstrained kernels for each spatial location, we designed a network that outputs: (1) a global kernel basis , of size with ; and (2) a dimensional coefficient vector at each spatial location.
Note that we typically choose the number of basis kernels . This implies that all the kernels for a given burst lie in a low-dimensional subspace, but this subspace will be different for different bursts (i.e. the basis is burst-specific). This procedure allows us to recreate a full kernel field with far fewer predictions. Assuming a resolution image, we need only make predictions to effectively recreate a kernel field of size .
We designed our network following an encoder-decoder architecture with skip connections . Our model, however, has two decoder branches, one for the basis, the other for the coefficients (Figure 2). The encoder is shared between the two branches because the meaning of the coefficients is dependent on the predicted basis in Equation (3), so the two outputs need to be co-ordinated. This encoder takes the noisy burst and noisy parameters as input, and through multiple levels of downsampling and global average pooling at the end, yields a single global feature vector as its encoding of the image. The per-pixel coefficients are then decoded from the encoder bottleneck to the full image resolution , with channels as output. The common basis is decoded up to distinct spatial dimensions — that of the kernels — with output channels.
Since the basis branch decodes to a different spatial resolution, we need a careful treatment of the skip connections. Unlike a usual U-Net, the encoder and decoder feature size do not match. Specifically, a pixel in the basis kernel has no meaningful relation to a pixel in the input frames . Therefore, in the skip connections from the shared encoder to the basis decoder, we apply a global spatial average pooling of the encoder’s activations, and replicate the average vector to the resolution of the decoder layer. This mechanism ensures the encoder information is globally aggregated without creating nonsensical correspondences between kernel and image locations, while allowing features at multiple scales of the encoder to inform the basis decoder.
We ensure each of the reconstructed kernel has positive weights that sum to one, to represent averaging. We implement this constraint using soft-max normalizations on both the coefficient and basis decoder outputs. So every 3D kernel of the basis and every coefficient vector is normalized individually. A more detailed description of the architecture is provided in the supplementary.
Our network is trained with respect to the quality of the final denoised output —with an loss on intensities and loss on gradients. Like , we additionally use a per-frame loss to bias the network away from relying only on the reference frame. We do this with separate losses on denoised estimates from each individual frame of the input burst (formed as ). These are added to main training loss, with a weight that is decayed across training iterations.
3.3 Efficient Fourier domain filtering
Filtering by convolution with large kernels is commonly implemented in the Fourier domain, where the filtering complexity is quasilinear in image size, while the complexity of direct convolution scales with the product of image- and kernel-size. But because the kernels in KPNs vary spatially, Equation (2) does not represent a standard convolution, ruling out this acceleration.
In our case, because our kernels are defined with respect to a small set of “global” basis vectors, we can leverage Fourier-domain convolution to speed up filtering. We achieve this by combining and re-writing the expressions in Eq. (2) and Eq. (3) as:
where denotes standard spatial 2D convolution with a spatially-uniform kernel.
In other words, we first form a set of filtered versions of the input burst by standard convolution with each of the basis kernels—convolving each frame in the burst with the corresponding “slice” of the basis kernel—and then taking a spatially-varying linear combination of the filtered intensities at each pixel based on the coefficients . We can carry out these standard convolutions in the Fourier domain as:
are spatial forward and inverse Fourier transforms. This is significantly more efficient for larger kernels, especially since we need not repeat forward Fourier transform of the inputsfor different basis kernels.
|Method||Gain 1||Gain 2||Gain 4||Gain 8|
We closely follow the protocol of Mildenhall et al.  to train and evaluate our network. Our model is designed for bursts of frames with resolution . It is trained using training and validation sets constructed from the Open Images dataset  following the procedure of . We also use their test set of 73 images for evaluation.
Our default configuration uses bases with kernels of size . We train our network (as well as all ablation baselines) using the Adam optimizer  with an initial learning rate of . We drop the learning twice, by a factor of each time, whenever the validation loss saturates. Training takes around 600k iterations with batches of 24 images.
4.1 Denoising performance
Table 1 reports the PSNR of our denoised outputs on the test set. Each noise level corresponds to a sensor gain value (one stop increments of the ISO setting in a camera). Higher gain lead to noisier images. The highest noise level, denoted as , lies outside the range we trained on. We use it to evaluate our model’s extrapolation capability. In addition to our own model, we also report results for a motion-alignment-based method , several approaches based on non-local filtering [2, 5, 25], as well as the standard KPN burst denoiser —which is the current state-of-the-art. Since we did not have access to the original KPN model, we implemented a version ourselves (that we use in ablations in the next section) and also report its performance in Table 1. We find it closely matches those from ). Additionally, we train a network to directly regress the denoised pixel values from the input burst (i.e., without kernels), as well as our implementation of  with a larger kernel size of for fair comparison.
We find that our method outperforms KPN  by a significant margin, over 1 dB PSNR at all noise levels. Our implementation of  also does well, but remains inferior to our model. We show qualitative results for a subset of methods in Figure 3. Our have fewer artifacts, especially in textured regions and around thin structures like printed text.
4.2 Ablation and analysis
|Gain 1||Gain 2||Gain 4||Gain 8|
|Ours (, )||36.29||33.57||29.62||25.99|
|Common Spatial Basis||35.71||33.04||29.23||25.71|
|Per-frame Spatial Basis||36.21||33.46||29.56||25.92|
Our approach leads to better denoising quality because it enables larger kernels without vastly increasing the network’s output dimensionality and number of learnable parameters. To tease apart the contributions of kernel size and the structure of our kernel decomposition, we conduct an ablation study on our validation set. The results can be found in Table 2. The performance gap between the test and validation set results (Table 1 and 2) comes from differences in the datasets themselves.
Kernel Size. As a baseline, we consider using KPN directly with our larger kernel size of . We also consider predicting a single separable kernel at that size ( predicts separable kernels at multiple sizes, and adds them together). We find that our network outperforms the large kernel KPN variant at all noise levels—suggesting that simply increasing the kernel size is not enough. It also outperforms separable kernel prediction, suggesting that a low-dimensional subspace constraint better captures the structure of natural images than spatial separability.
For completeness, we also evaluate our basis prediction network with smaller kernels, and . Although, this leads to a drop in performance compared to our default configuration, these variants still perform better than the original KPN—suggesting our approach has a regularizing effect that benefits even smaller kernels.
Basis Size. The number of basis elements in our default configuration, , was selected from a parameter search on the validation set. We include this analysis in Table 2, reporting PSNR values for ranging from 10 to 130. We find that bases with fewer than 90 kernels lead to a drop in quality. The larger bases, , also performs very slightly worse than . We hypothesize that large bases start to have too many degrees of freedom. This increases the dimensionality of the network’s output, which negates the benefits of a subspace restriction.
Spatial vs. Spatio-temporal Basis Decomposition. Note that we define our basis as a subspace to span 3D kernels—i.e., each of our basis elements is a 3D spatio-temporal kernel. We predict a single weight at each location, which is applied to corresponding spatial kernels for all frames . However, there are other possible choices for decomposing 3D kernels, and we consider two of these in our ablation (Table 2). In both cases, we output coefficients that vary per-frame, in addition to per-location—and are interpreted as separate coefficients corresponding to a spatial basis kernel. In one case, we use a common spatial basis across all frames, with . In the other, we have a per-frame spatial basis for each frame, and . The per-frame basis increases the dimensionality of our coefficient output and leads to a slight drop in performance, likely due to a reduced regularizing effect. The common spatial basis, however, suffers a greater performance drop since it also forces kernels in all frames to share the same subspace.
We also compare qualitatively the spatio-temporal kernels produced by our default configuration with those predicted by standard KPN in Figure 4
. Our model makes better use of the temporal information, applying large weights to pixels across many frames in the burst, whereas KPN tends to overly favor the reference frame. Our network better tracks the apparent motion in the burst, shifting the kernel accordingly. And it is capable of ignoring outliers caused to excessive motion (all black kernels in Fig.4).
Fixed vs. Burst-specific Basis. Given that our network predicts both a basis and per-pixel coefficients, a natural question is whether a burst-specific kernel basis is even needed. To address this, we train a network architecture without a basis decoder to only predict coefficients for each burst, and instead learn a basis that is fixed across all bursts. We still learn the fixed basis jointly with this network, as a direct learnable tensor. Table 2 shows that using a fixed basis in this manner leads to a significant decrease in denoising quality (although still better than standard KPN).
This suggests that while a subspace restriction on kernels is useful, the ideal subspace is scene-dependent and must be predicted adaptively. We further explore this phenomenon in Table 3, where we quantify the rank of the predicted bases for individual images, and for pairs of images. Note that the rank can be lower than , since we do not effectively require the ‘basis’ vectors to be linearly independent. We find that the combined rank of basis kernels of image pairs (obtained by concatenating the two bases) is nearly twice the rank obtained from individual images—suggesting limited overlap between the basis sets of different images. We also explicitly compute the average overlap ratio across image pairs as , and find it to be around on average. This low overlap implies that different bursts do indeed require different bases, justifying our use of burst-specific bases.
|Gain 1||Gain 2||Gain 4||Gain 8|
|Ours (, )||29.9||0.30|
|Common Spatial Basis||40.8||0.49|
|Per-frame Spatial Basis||41.9||0.57|
4.3 Computational expense
Finally, we evaluate the computational expense of our approach and compare it to the different ablation settings considered in Table 2, including standard KPN. We report the total number of floating point operations (FLOPs) required for network prediction and filtering in Table 4. We find that in addition to producing higher-quality results, our approach also requires significantly fewer FLOPs than regular KPN—for kernel size . This is due to the reduced complexity of our final prediction layer, as well as the benefit of efficient filtering in the Fourier domain. Also, we find that our approach has nearly identical complexity as separable kernel prediction, while achieving higher denoising performance because it can express a more general class of kernels.
In addition to the evaluation FLOPs, Table 4 reports measured running times for the various approaches, benchmarked on a 1024 768 image on an NVIDIA 1080Ti GPU. To compute these timings, we divide the image into 128128 non-overlapping patches to form a batch and send it to the denoising network. Since regular KPN have very high memory requirements, we select the maximum batch size for each method and denoise the entire image in multiple runs. This maximizes GPU throughput. We find that our approach retains its running time advantage over KPN in practice. It is also a little faster than separable kernel prediction—likely due to the improved cache performance we get from using Fourier-domain convolutions with spatially-uniform basis kernels.
5 Conclusion and future work
In this work, we argue that local, per-pixel burst denoising kernels are highly coherent. Based on this, we present a basis prediction network that jointly infers a global, low-dimensional kernel basis and the corresponding per-pixel mixing coefficients that can be used to construct per-pixel denoising kernels. This formulation significantly reduces memory and compute requirements compared to prior kernel-predicting burst denoising methods, allowing us to substantially improve performance by using large kernels, while reducing running time.
While this work focuses on burst denoising, KPN-based methods have been applied to other image and video enhancement tasks including video super-resolution , frame interpolation [28, 29, 23], video prediction [15, 6], and video deblurring . All these tasks exhibit similar structure and will likely benefit from our approach.
There are other forms of spatiotemporal structure that can be explored to build on our work. For example, image enhancement methods have exploited self-similarity at different scales  suggesting other decompositions in scale space. Also, we assume a fixed basis size globally. Adapting this spatially to local content could yield further benefits. Finally, KPNs are still, at their heart, local filtering methods and it would be interesting to extend our work to non-local filtering methods [5, 19, 25].
Acknowledgments. ZX and AC acknowledge support from the NSF under award no. IIS-1820693, and from a gift from Adobe research.
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Appendix A Architecture
We now provide a detailed description of the architecture of our basis prediction network, which includes a shared encoder and two decoders. The shared encoder network consists of five down-sampling blocks and encodes noisy frames into a shared feature space. The coefficients decoder decodes these features into a set of per-pixel mixing coefficients. Coefficients for each output pixel are normalized with a softmax function separately. The basis decoder first reduces these features to a 1D vector, and then decodes them to an output of shape , representing a burst-specific set of basis kernels, each of shape ( and for our model). Each 3D basis element is normalized with a softmax function so that the basis kernel sums up to 1. We include regular skip connections from the encoder to the coefficient decoder, and pooled-skip connections to the basis decoder. The entire architecture is illustrated in Figure 5.
In our ablation study, we considered versions of our network that produced smaller kernels with and . For , we used one less up-sampling block in the basis decoder than for the case shown in Figure 5, and added one 33 convolutional layer with valid padding before the layer in Figure 5. For , we also used one less up-sampling block in the basis decoder, but in this case, replaced the layer in with a 22 transpose convolution layer with valid padding (i.e., one that does a 22 “full” convolution).
Our fixed basis ablation replaced the entire decoder branch with just a learned tensor, of size , to serve as the basis. However, we still retain the encoder and coefficient decoder, and the weights of these networks are learned jointly with the fixed basis tensor. Note that in this case, the denoising kernels at each pixel are formed as a linear combination of this fixed set of basis kernels, based on the coefficients predicted from the input burst by the decoder at each location (this is different from approaches that select one kernel at each location from a fixed kernel set [30, 8]). As our ablation showed, having a fixed basis set yields worse denoising performance than an adaptive basis.
Appendix B Additional Results
|Noisy Ref.||Direct||KPN ||MKPN ||Ours||GT|
|Noisy Ref.||Direct||KPN ||MKPN ||Ours||GT|
|Noisy Ref.||Direct||KPN ||MKPN ||Ours||GT|
|Noisy Ref.||Direct||KPN ||MKPN ||Ours||GT|
|Noisy Ref.||Direct||KPN ||MKPN ||Ours||GT|