1 Introduction
Image denoising is a longstanding problem finding applications in fields ranging from astronomical imaging to handheld photography. With the development of digital photography and smartphone technology, it has recently become possible to take highquality photos using relatively inexpensive equipment. However, there are still major differences between the imaging capabilities of handheld devices such as smartphones and professional equipment such as DSLRs. One of the most important factors for taking noisefree photos is to collect as much light as possible. Professional cameras contain several hardware solutions such as large aperture lenses, sensors with large photosites and highquality A/D converters for increased light collection [1]. However, for the sake of compactness, smartphones contain smaller and less expensive variants of these hardware elements which can result in noisy imaging, especially in low light conditions. To combat this problem, image denoising algorithms are implemented as a software solution in most smartphones [2].
Classical methods for image denoising developed in the early 1990s like antisotropic diffusion [3] and total variation denoising [4] use analytical priors and nonlinear optimization to recover a clear image. More recently, plain neural networks [5]
, convolutional neural networks
[6] and neural networks with autoencoder architectures were used for single image denoising [7, 8, 9]. The combination of the last two, a convolutional autoencoder, showed promising results in medical image denoising [10]. The same architecture was further improved by addition of skip connections [11] placed between the encoder and the decoder. This architecture was used both in single image denoising [12, 13] as well as in denoising of the images captured in quick succession, i.e. burst image denoising [14].Burst image denoising methods operate on a set of successive, rapidly taken images to compute a single, noisefree result. Since the noise is usually randomly distributed and a set of images obtained by the same camera have similar characteristics, it is reasonable to expect that burst image denoising tends to work better than single image denoising in most cases. Burst denoising methods commonly first align the successive images as a preprocessing step, and then fuse and denoise the aligned images [15, 2, 1, 16]. Many of the stateoftheart approaches in burst image denoising are based on fully convolutional neural network architectures [1, 14, 16, 17].
Neural network based image denoising methods commonly operate on the whole image and predict the denoised image directly. Another approach is to design a network that can learn to predict spatially variant kernels for each pixel in the input. These perpixel
kernels are then convolved over the input to produce the final output. This approach worked very well in video frame interpolation
[18], denoising of renderings [19, 20] and burst image denoising [14].In this paper we propose a novel neural network based method burst image denoising method. Our method is conceptually similar to [14] but it operates in a fundamentally different way. Specifically, instead of predicting a single, fixedsize kernel for each pixel, our method predicts kernels of multiple sizes for each pixel taking into account the spatially varying image structures. This allows our method to adapt to the properties of image structures with different properties (e.g. textured vs. nontextured, homogeneous areas) and successfully denoise a wide range of images without having to manually tweak the kernel size depending on the characteristics of the image.
2 Proposed method
2.1 MultiKernel Prediction Network
Our goal is to perform denoising on burst images corrupted by noise due to low light or shot exposure photography. Given a noisy set of input burst images, kernels are predicted for each pixel using a deep neural network. These kernels are then convolved with their respective input pixels in the respective burst image which are then averaged to reconstruct the denoised image. [14] performed this denoising by predicting kernels of a predetermined size as given in (1):
(1) 
However, images consist of heterogeneous patches where every pixel in a patch is different compared to its surrounding pixels. Flat regions like sky in an image would provide for a more certain prediction while densely cluttered leaves in an image would provide for more uncertainties in the prediction. The predicted kernels should be able accommodate to the different regions and pixels in the image. Naturally, kernels of predetermined size cannot successfully adapt to the diversity of pixels in the images.
Our method, MultiKernel Prediction Networks (MKPN) is a direct extension of the [14] and [18]. To the best of our knowledge this is the first attempt to utilize multiple kernels of different sizes predicted by a neural network for image denoising. Although [21] combine different kernels, they assume a fixed size of the kernel and do not use a deep learning based approach. MKPN predicts kernels of different sizes for each pixel in the image belonging to the set of burst images instead of predicting fixedsize kernels. The predicted kernels of different sizes are then convolved with each pixel in the input image from the set of noisy burst images which are then averaged to obtain the final reconstruction as described in (2):
(2) 
Here is the length of the input burst, is the set of kernel sizes, is a kernel with size for pixel located at and is a patch with size in centered at . is the th image from the input burst. Kernels of different sizes extract and accumulate information from image structures with different characteristics.
Ideally, MKPN will work well when many kernels of different sizes are used. However, convolving each kernel with its corresponding patch significantly increases the required amount of computations. In order to reduce the amount of computations, we apply two enhancements:
Separable Kernel Estimation
: We approximate the 2D kernels by a linear combination of separable 1D kernels [22, 18]. In this way the number of learnable parameters is reduced from to for each kernel of size , significantly reducing the computation cost.Kernel Fusion: Instead of convolving each kernel separately with each image in the burst, MKPN performs inplace kernel addition as described in (3), before the convolution operation:
(3) 
The accumulated kernels are then convolved with the corresponding image patches in a single operation, as shown in (4):
(4) 
This essentially brings the number of convolution operations equal to that of [14]. The computational cost of the inplace additions is negligible. Note that model compression [23, 24] and efficient representations [25] can further reduce the computational costs.
2.2 Network Architecture
Fig. 1 shows an overview of the MKPN architecture. MKPN has a typical UNet [26] shape resembling the architectures of [14, 18]
. The network consists of convolutional layers, ReLU activation functions, average pooling layers and billinear upsampling layers. The convolutional layers use
filters with zero padding and stride one, and are always followed by a ReLU activation function. The average pooling layers have a pool size of
and stride two, which in effect decreases the spatial resolution by a factor of two. On the contrary, the billinear upsampling layer increases the spatial resolution by a factor of two.A series of convolutional and average pooling layers encode the input features into a latent representation. Series of bilinear upsampling and convolutional layers decode these features to predict the per pixel kernels. After each upsampling layer, the high resolution features from the encoder side of the architecture are concatenated to the decoder side.
The channel dimension of the layers of the last convolution block is , where is the sum of the predefined sizes of different kernels and is the number of images in the input burst. For example, if the selected kernel sizes are and , then . In addition, another billinear upsampling layer is employed which scales up the learned linear combinations of 1D per pixel kernels of various sizes to match the input shape. Outer products of 1D kernels are computed to produce the 2D kernels. These kernels are applied on the input burst using local convolution as shown in Fig. 2. Lastly, the deconvolved burst images are averaged to produce the final output.
2.3 Training Parameters
All experimental models were trained in endtoend fashion using backpropagation.
Total loss consists of basic and annealing loss as devised in [14]. Basic loss is composed of mean squared error on pixel intensities and L1 loss on gradient intensities between the denoised image and the ground truth :
(5) 
The basic loss tries to make the average of all estimations close to the ground truth . However, using only the basic loss can lead to convergence at an undesirable local minima that does not utilize all the images in the burst effectively [14]. Therefore, we add a second loss term annealing loss to (5) that attempts to make each estimation close to the ground truth independently. The total loss is obtained as follows:
(6) 
where is the number of images in the burst, is the set of kernel sizes, and
are the hyperparameters controlling the weight decay,
is the training step, and . Please note that during training we discard inplace addition of kernels to enable better convergence. However, once the network is well trained, the inplace addition can help speed up inference.3 Experiments
We followed the same procedure for dataset generation and noise estimation as in [14]. We trained our models and evaluated their performance on synthetically generated datasets from the Open Images dataset [27].
3.1 Data Generation
The images from the Open Images dataset were downsampled in each dimension using a box filter to reduce noise and compression artifacts. Random patches of size were sampled from the images and those were used for both creating the ground truth and the remaining burst images, where is the total number of images in the burst. These burst images are offset from the first image by and , where and are the offsets of image in horizontal and vertical directions, respectively. The offsets simulate misalignments between consecutive frames caused by hand movements that may occur during handheld photography. Values for
are sampled with probability
from a 2D uniform integer distribution between , otherwise from a 2D uniform integer distribution between , where .The burst images are also considered to be noisy, and hence a signal dependent Gaussian noise is added to the burst:
(7) 
where is the noisy measurement of true intensity at pixel . Read and shot noise parameters and are sampled uniformly from and , respectively. These ranges were selected from the real observed data. Synthetic train dataset is generated on the fly, while the test datasets are pregenerated using different gains (noise levels) that correspond to a fixed set of read and shot noise parameters. The selected values simulate the light sensitivities that correspond to the ISO settings on a real camera. The read and shot noise for each gain is as given:

Gain : , ,

Gain : , ,

Gain : , ,

Gain : , .
3.2 Noise Estimation
The camera noise is estimated from the first image in a burst. The noise estimate helps the model denoise beyond the noise levels of the training data [14]. It is defined as:
(8) 
where is the intensity of pixel in the first image of a burst. In real data the noise parameters and are available in the DNG raw image format [28]. This noise estimate is of the same dimension as the burst images and is appended to the end of the burst.
In all our experiments, we set the parameters as follows , , and .
3.3 Results and Discussion
In the experiments we performed using MKPN, we define the sizes of the kernels in advance – . The kernel sizes and the number of kernels however, are not limited to these and can be used in any combination to produce the best results for a given dataset. The two enhancements are utilized in the endtoend training of MKPN, as explained in Section 2.1. We evaluate the performance of MKPN by comparing it to stateoftheart models for burst image denoising:

KPN [14] is a kernel prediction network that outputs perpixel kernels of fixed size (set here to ) which are then convolved with the input burst images to obtain the output.
Each model was trained for 1M iterations and evaluated on four precomputed test datasets, each one generated with a different noise level. Quantitative analysis is performed using objective quality metrics such as PSNR and SSIM [29] as shown in Table 1. For a qualitative evaluation, we visually inspect the images of MKPN and stateoftheart models as shown in Fig. 3 and Fig. 4.
Gain 1  Gain 2  Gain 4  Gain 8  

Model  PSNR  SSIM  PSNR  SSIM  PSNR  SSIM  PSNR  SSIM 
MKPN  35.10  0.925  32.22  0.878  28.61  0.781  25.78  0.692 
KPN [14]  34.28  0.914  31.46  0.862  27.86  0.756  24.90  0.654 
KPN  34.96  0.922  32.04  0.871  28.44  0.770  25.66  0.686 
KPN  34.71  0.918  31.77  0.870  28.15  0.772  25.34  0.683 
KPN  34.67  0.919  31.78  0.871  28.15  0.773  25.26  0.679 
KPN  34.57  0.917  31.63  0.867  28.07  0.769  25.34  0.681 
KPN  34.54  0.918  31.65  0.866  28.10  0.765  25.37  0.679 
KPN  34.30  0.906  31.36  0.855  27.76  0.754  24.88  0.658 
KPN  33.65  0.897  30.79  0.844  27.26  0.735  24.55  0.632 
KPN  31.66  0.837  28.50  0.759  24.32  0.607  21.30  0.486 
Table 1 compares the performance of different models and shows the average values for the four test datasets with different noise levels. It is to be noted that KPN performs almost the same as KPN [14] with a reduced number of computations due to the separable kernels. Among the stateoftheart models, we find that KPN shows superior performance – even better than KPN [14]. The results in Table 1 also indicate that the results, in terms of PSNR and SSIM, get better as the size of the kernel increases. This can be attributed to the increased information that the kernels can extract from the surrounding region of the pixel and use it for better denoising. MKPN, on the other hand, combines the different kernels and outperforms all stateoftheart models – KPN [14] and the derived KPN on all test datasets for each noise level.
In Fig. 3 and Fig. 4, we inspect the denoised images from the test dataset to reveal the properties of different kernel sizes and the advantages of MKPN over KPN. The first row displays the full image while the second row shows an enlarged part marked by the green rectangle in the corresponding images in the first row. The order of images in the columns is as follows – noisy input burst, denoising results of KPN [14], KPN , MKPN and the ground truth. In Fig. 3, we find that the reconstructed image of KPN [14] is better than that of KPN . Although, in Table 1, KPN performed superior to other stateoftheart models, the denoised images from KPN are oversmoothed due to the large size of the kernels. Hence, using a large kernel results in suboptimal visual quality of the denoised image. MKPN restores the structure of fur in Fig. 3 with more detail than other stateoftheart models. In Fig. 4, KPN [14] fails to smooth out the background well enough. This is due to the small kernel trying to greedily denoise the local region, giving it an unrealistic texture. KPN , on the other hand, denoises the background efficiently, but fails to recover the sharp edges. With MKPN, the legs of the grasshopper are reconstructed sharply while having a smooth background.
While KPN models with small kernels are able to reconstruct detailed features more successfully, their performance on flat, homogeneous areas is worse compared to the KPN models using larger kernels. On the other hand, KPN models with larger kernels tend to oversmooth the detailed features. MKPN manages to alleviate the shortcomings of the KPN models relying on fixed kernel sizes.
4 Conclusion
Burst image denoising, with its inherent challenges, remains to be an open problem. In this work, we propose MKPN, a DNN based method for denoising of burst images captured by handheld cameras. The novelty of this method lies in predicting kernels of different sizes and performing kernel fusion by inplace addition before the convolution operation. MKPN effectively combines the best behavior of small and large kernels – it manages to denoise flat areas as well as preserve the detailed image structures. Kernel fusion ensures that MKPN is able to extract different information from the different kernels without compromising on computational efficiency. We empirically show that MKPN outperforms stateoftheart models quantitatively and provides visually pleasing results.
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