Keywords

1 Introduction

Compression artifact reduction methods aim at generating artifact-free images from lossy decoded images. To store and transfer a large amount of images and videos on the Internet, image and video compression algorithms (e.g., JPEG, H.264) are widely used [1,2,3]. However, these algorithms often introduce undesired compression artifacts, such as blocking, blurring and ringing artifacts. Thus, compression artifact reduction has attracted increasing attention and many methods have been developed in the past few decades.

Early works use manually designed filters [4, 5] and sparse coding methods [6,7,8,9] to remove compression artifacts. Recently, convolutional neural network (CNN) based approaches have been successfully applied for a lot of computer vision tasks [10,11,12,13,14,15,16,17,18,19,20], such as super-resolution [15, 16], denoising [17] and artifact reduction [18,19,20]. In particular, Dong et al. [18] firstly proposed a four-layer neural network to eliminate the JPEG compression artifacts.

Fig. 1.
figure 1

Different methodologies for video artifact reduction (a) the traditional pipeline without considering previous restored frames. (b) our Kalman filtering pipeline. Artifact reduction results between (c) Xue et al. [21] and (d) our proposed deep Kalman filtering network. (e) original frame (f) decoded frame (g) quantized prediction residual (h) distortion between the original frame and the decoded frame.

Full size image

For the video artifact reduction task, our motivations are two-fold. First, the restoration process for the current frame could benefit from the previous restored frames. A reason is that when compared to the decoded frame, the previous restored frame can provide more accurate information. Therefore temporal information (such as motion clues) from neighbouring frames is more precise and robust, and can provide the potential to further improve the performance. In addition, the dependence of previous restored frames naturally leads to a recursive pipeline for video artifact reduction. Therefore it recursively restores the current frame by potentially utilizing all past restored frames, which means we can leverage effective information propagated from previous estimations. Currently, most of the state-of-the-art deep learning approaches for compression artifact reduction are limited to remove artifacts in a single image [16,17,18,19]. Although the video artifact reduction method [21] or video super-resolution methods [22,23,24] try to combine temporal information for the restoration tasks, their methods ignore the previous restored frame and restore each frame separately as shown in Fig. 1(a). Therefore, the video artifact reduction performance can be further improved by using an appropriate dynamic filtering scheme.

Second, modern video compression algorithms may contain powerful prior information that can be utilized to restore the decoded frame. It has been observed that practical video compression standards are not optimal according to the information theory [9], therefore the resulting compression code streams still have redundancies. It is possible, at least theoretically, to improve the restoration results by exploiting the knowledge hidden in the code streams. For video compression algorithms, inter prediction is a fundamental technique used to reduce temporal redundancy. Therefore, the decoded frame consists of two components: prediction frame and quantized prediction residual. As shown in Fig. 1(g)–(h), the distortion between the original frame and the decoded frame has a strong relationship with quantized prediction residual, i.e., the region which has high distortion often corresponds to high quantized prediction residual values. Therefore, it is possible to enhance the restoration by employing this task-specific prior information.

In this paper, we propose a deep Kalman filtering network (DKFN) for video artifact reduction. The proposed approach can be used as a post-processing technique, and thus can be applied to different compression algorithms. Specifically, we model the video artifact reduction problem as a Kalman filtering procedure, which can recursively restore the decoded frame and capture information that propagated from previous restored frames. To perform Kalman filtering for decoded frames, we build two deep convolutional neural networks: prediction network and measurement network. The prediction network aims to obtain prior estimation based on the previous restored frame. At the same time, we investigate the quantized prediction residual in the coding algorithms and a novel measurement net incorporating this strong prior information is proposed for robust measurement. After that, the restored frame can be obtained by fusing the prior estimation and the measurement under the Kalman framework. Our proposed approach bridges the gap between model-based methods and learning-based methods by integrating the recursive nature of the Kalman model and highly non-linear transform ability of neural network. Therefore, our approach can restore high quality frames from a series of decoded video frames. To the best of our knowledge, we are the first to develop a new deep convolutional neural network under the Kalman filtering framework for video artifact reduction.

In summary, the main contributions of this work are two-fold. First, we formulate the video artifact reduction problem as a Kalman filtering procedure, which can recursively restore the decoded frames. In this procedure, we utilize the CNN to predict and update the state of Kalman filtering. Second, we employ the quantized prediction residual as the strong prior information for video artifact reduction through deep neural network. Experimental results show that our proposed approach outperforms the state-of-the-art methods for reducing video compression artifacts.

2 Related Work

2.1 Single Image Compression Artifact Reduction

A lot of methods have been proposed to remove the compression artifacts. Early methods [25, 26] designed new filters to reduce blocking and ringing artifacts. One of the disadvantages for these methods is that such manually designed filters cannot sufficiently handle the compression degradation and may over-smooth the decoded images. Learning methods based on sparse coding were also proposed for image artifact reduction [8, 9, 27]. Chang et al. [8] proposed to learn a sparse representation from a training image set, which is used to reduce artifacts introduced by compression. Liu et al. [9] exploited the DCT information and built a sparsity-based dual domain approach.

Recently, deep convolutional neural network based methods have been successfully utilized for the low-level computer vision tasks. Dong et al. [18] proposed artifact reduction CNN (ARCNN) to remove the artifacts from JPEG compression. Inspired by ARCNN, several methods have been proposed to reduce compression artifact by using various techniques, such as residual learning [28], skip connection [28, 29], batch normalization [17], perceptual loss [30], residual block [20] and generative adversarial network [20]. For example, Zhang et al. [17] proposed a 20-layer neural network based on batch normalization and residual learning to eliminate Gaussian noise with unknown noise level. Tai et al. [16] proposed a memory block, consisting of a recursive unit and a gate unit, to explicitly mine persistent memory through an adaptive learning process. In addition, a lot of methods [31,32,33] were proposed to learn the image prior by using CNN and achieved competitive results for the image restoration tasks.

As mentioned in [9], compression code streams still have redundancies. Therefore, it is possible to obtain a more robust estimation by exploiting the prior knowledge hidden in the encoder. However, most of the previous works do not exploit this important prior information. Although the works in [9, 19, 27] proposed to combine DCT information, it is not sufficient especially for the video artifact reduction task. In our work, we further exploit the prior information in the code streams and incorporate prediction residual into our framework for robust compression artifact reduction.

2.2 Deep Learning for Video Restoration

Due to the popularity of neural networks for image restoration, several CNN based methods [21, 22, 34,35,36] were also proposed for the video restoration tasks. For video super-resolution, Liao et al. [34] first generated an ensemble of SR draft via motion compensation, and then used a CNN model to restore the high resolution frame from all drafts. Kappeler et al. [35] estimated optical flow and selected the corresponding patches across frames to train a CNN model for video super-resolution. Based on the spatial transformation network (STN) [36], the works in [21,22,23,24] aligned the neighboring frames according to the estimated optical flow or transform parameters and increased the temporal coherence for the video SR task. Tao et al. [22] achieved sub-pixel motion compensation and resolution enhancement with high performance. Xue et al. [21] utilized a joint training strategy to optimize the motion estimation and video restoration tasks and achieved the state-of-the-art results for video artifact reduction.

Compared to the methods for single image compression artifact reduction (see Sect. 2.1), the video restoration methods exploit temporal information. However, these methods process noisy/low-resolution videos separately without considering the previous restored frames. Therefore, they cannot improve video restoration performance by utilizing more accurate temporal information. In our work, we recursively restore each frame in the videos by leveraging the previous restored frame for video artifact reduction. Although the work [37, 38] try to combine deep neural network and Kalman filter, they are not designed for the image/video enhancement tasks.

3 Methodology

We first give a brief introduction about the basic Kalman filter and then describe our formulation for video artifact reduction and the corresponding network design.

Introduction of Denotations. Let \( \mathcal {V} = \{X | X_{1}, X_{2},..., X_{t-1}, X_{t}, \ldots \}\) denote an uncompressed video sequence, where \(X_{t} \in \mathcal {R}^{mn \times 1}\) is a video frame at time step t and mn represents the spatial resolution. In order to simplify the description, we only analyze video frame with a single channel, although we consider RGB/YUV channels in our implementation. After compression, \(X^{c}_t\) is the decoded frame of \(X_{t}\). \(\hat{X}_t^-\) denotes the prior estimation and \(\hat{X}_t\) denotes the posterior estimation for restoring \(X_{t}\) from the decoded frame \(X^{c}_t\). \(R^c_t\) is the quantized prediction residual in video coding standards, such as H.264. \(R_t\) is the corresponding unquantized prediction residual.

3.1 Brief Introduction of Kalman Filter

Kalman filter [39] is an efficient recursive filter that estimates the internal state from a series of noisy measurements. In artifact reduction task, the internal state is the original image to be restored, and the noisy measurements can be considered as the images with compression artifacts.

Preliminary Formulation. The Kalman filter model assumes that the state \(X_t\) at time t is changed from the state \(X_{t-1}\) at time \(t-1\) according to

$$\begin{aligned} X_t = A_{t}X_{t-1} + w_{t-1}, \end{aligned}$$
(1)

where \(A_{t}\) is the transition matrix at time t and \(w_{t-1}\) is the process noise. The measurement \(Z_t\) of the true state \(X_t\) is defined as follows,

$$\begin{aligned} Z_t = HX_{t} + v_{t}, \end{aligned}$$
(2)

where H is the measurement matrix and \(v_{t}\) represents the measurement noise. However, the system may be non-linear in some complex scenarios. Therefore, a non-linear model for the transition process in Eq. (1) can be formulated as follows,

$$\begin{aligned} X_t = f(X_{t-1}, w_{t-1}), \end{aligned}$$
(3)

where \(f(\cdot )\) is the non-linear transition model. Linear Kalman filter corresponds to Eqs. (1) and (2). Non-linear Kalman filter corresponds to Eq. (3) and Eq. (2).

Kalman Filtering. As shown in Fig. 2(a), Kalman filtering consists of two steps, prediction and update.

In the prediction step, it calculates the prior estimation from the posterior estimation of the previous time step. For non-linear model Eqs. (3) and (2), the prediction step is accomplished by two sub-steps as follows,

$$\begin{aligned} \text {Prior state estimation: } \hat{X}^{-}_{t} = f(\hat{X}_{t-1},0), \end{aligned}$$
(4)
$$\begin{aligned} \text {Covariance estimation: } P^{-}_{t} = A_{t}P_{t-1}A^\mathrm{{T}}_{t} + Q_{t-1}, \end{aligned}$$
(5)

where \(Q_{t-1}\) is the covariance of the process noise \(w_{t-1}\) at time \(t-1\), \(P^{-}_{t}\) is a covariance matrix used for the update step. \(A_{t}\) in Eq. (5) is defined as the Jacobian matrix of \(f(\cdot )\), i.e., \(A_{t} = \frac{\partial f(\hat{X}_{t-1},0)}{\partial X}\) [40]. Prediction procedure for the linear model can be considered as a special case by setting \(f(\hat{X}_{t-1}, 0) = A_{t}\hat{X}_{t-1}\).

In the update step, Kalman filter will calculate the posterior estimate by fusing the prior estimate from the prediction step and the measurement. Details about the update step can be found in Sect. 3.6.

An overview of Kalman filtering is shown in Fig. 2(a). First, the prediction step uses the estimated state \(\hat{X}_{t-1}\) at time \(t-1\) to obtain a prior state estimation \(\hat{X}^-_{t}\) at time t. Then the prior state estimation \(\hat{X}^-_{t}\) and the measurement \(Z_t\) are used by the update step to obtain the posterior estimation of the state, denoted by \(\hat{X}_{t}\). These two steps are performed recursively.

Fig. 2.
figure 2

(a) Basic Kalman model. (b) Overview of the proposed deep Kalman filtering network for video artifact reduction. \(X_{t}^{c}\) is the decoded frame at time t. \(\hat{X}_{t-1}\) represents the restored frame from \(t-1\). The prediction network generates prior estimate \(\hat{X}^-_t\) for original frame based on \(X_{t}^{c}\) and \(\hat{X}_{t-1}\). The measurement network uses the decoded frame \(X_{t}^{c}\) and the quantized prediction residual \(R^c_t\) to obtain an initial measurement \(Z_{t}\). After that, we can build the posterior estimate by fusing the prior estimate \(\hat{X}_{t}^{-}\) and the measurement \(Z_{t}\).

Full size image

3.2 Overview of Our Deep Kalman Filtering Network

Figure 2(b) shows an overview of the proposed deep Kalman filtering network. In this framework, the previous restored frame \(\hat{X}_{t-1}\) and the decoded frame \(X^c_{t}\) are used for obtaining a prior estimate \(\hat{X}^-_{t}\). Then the measurement \(Z_{t}\) obtained from the measurement network and the prior estimate \(\hat{X}^{-}_{t}\) are used by the update step for obtaining the posterior estimation \(\hat{X}_{t}\).

The proposed DKFN framework follows the Kalman filtering procedure by using the prediction and update steps for obtaining the predicted state \(\hat{X}_t\). The main differences to the original Kalman filtering are as follows.

First, in the prior estimation sub-step of the prediction step, we use the temporal mapping sub-network as the non-linear function \(f(\cdot )\) in Eq. (3) to obtain the prior state estimation. Specifically, the temporal mapping sub-network takes the previous restored frame \(\hat{X}_{t-1}\) and the decoded frame \(X^c_t\) as the input and generates the prior estimate \(\hat{X}^-_{t}\) for the current frame. Details are given in Sect. 3.3.

Second, in the covariance estimation sub-step, the transition matrix \(A_{t}\) in Eq. (5) is approximated by a linearization sub-network. In conventional non-linear Kalman filter, the Jacobian matrix of the non-linear function \(f(\cdot )\) is used to obtain \(A_{t}\). For our CNN implementation of \(f(\cdot )\), however, it is too complex to compute, especially when we need to compute the Jacobian matrix for each pixel location. Therefore, we approximate the calculation of Jacobian matrix by using a linearization sub-network. Details are given in Sect. 3.4.

Third, in the update procedure, we use a measurement network to generate the measurement. In comparison, the conventional Kalman filter might directly use the decoded frame with compression artifacts as the measurement. Details are given in Sect. 3.5.

3.3 Temporal Mapping Network

Mathematical Formulation. Based on the temporal characteristic of video sequences, the temporal mapping sub-network is used for implementing the non-linear function \(f(\cdot )\) in the prior state estimation as follows:

$$\begin{aligned} \hat{X}_t^- = \mathcal {F}(\hat{X}_{t-1},X^c_{t};\theta _{f}), \end{aligned}$$
(6)

where \(\theta _{f}\) are the trainable parameters. Equation(6) indicates that the prior estimation of the current frame \(X_{t}\) is related with its estimated temporal neighbouring frame \(\hat{X}_{t-1}\) and its decoded frame \(X^{c}_{t} \) at the current time step. This formulation is based on the following assumptions. First, temporal evolution characteristic can provide a strong motion clue to predict \(X_{t}\) based on the previous frame \(\hat{X}_{t-1}\). Second, due to the existence of complex motion scenarios with occlusion, it is necessary to exploit the information from the decoded frame \(X^{c}_{t}\) to build a more accurate estimation for \(X_{t}\). Based on this assumption, our formulation in Eq. (6) adds the decoded frame \(X^c_t\) as the extra input to the transition function \(f(\cdot )\) defined in Eq. (4).

Network Implementation. The temporal mapping sub-network architecture is shown in Fig. 3(a). Specifically, each residual block contains two convolutional layers with the pre-activation structure [41]. We use convolutional layers with \(3 \times 3\) kernels and 64 output channels in the whole network except the last layer. The last layer is a simple convolutional neural network with one feature map without non-linear transform. Generalized divisive normalization (GDN) and inverse GDN (IGDN) [42] are used because they are well-suited for Gaussianizing data from natural images. More training details are discussed in Sect. 3.7.

Fig. 3.
figure 3

Network architecture of the proposed (a) Temporal Mapping Network (b) Linearization network. (c) Measurement Network. For better illustration, we omit the matrix conversion process of \(X_{t}, \hat{X}^{c}_{t}\) and \(R^c_{t}\) from \(\mathcal {R}^{mn \times 1}\) to \(\mathcal {R}^{m \times n}\). Here ‘Conv,3\(\,\times \,\)3,64’ represents the convolution operation with the 3\(\,\times \,\)3 kernel and 64 feature maps. ‘Reshape, \(m\times n\)’ is the operation that reshapes one matrix to \(m \times n\). \(\bigoplus \) and \(\bigotimes \) represent element-wise addition and matrix multiplication.

Full size image

3.4 Linearization Network

Linearization network aims to learn a linear transition matrix \(A_{t}\) for the covariance estimation in Eq. (5) adaptively for different image regions. It is non-trivial to calculate the Jacobian matrix of transition function \(\mathcal {F}(\cdot )\) and linearize it through Taylor series. Therefore, we use a simple neural network to learn a transition matrix. Specifically, given the prior estimation \(\hat{X}_{t}^{-}\), previous restored frame \(\hat{X}_{t-1}\) and decoded frame \(X_{t}^{c}\), the problem is expressed by the following way,

$$\begin{aligned} \hat{X}_{t}^{-} = \mathcal {F}(\hat{X}_{t-1},X^c_{t};\theta _{f}) \approx \tilde{A}_t \hat{X}_{t-1}, \text { where } \tilde{A}_t=\mathcal {G}(\hat{X}_{t-1},X_{t}^{c}; \theta _{m}), \end{aligned}$$
(7)

\(\mathcal {G}(\hat{X}_{t-1},X_{t}^{c}; \theta _{m})\) is the linearization network with the parameters \(\theta _m\). \(\tilde{A}_t\) is the output of this network. The network architecture is shown in Fig. 3(b). Given \(\hat{X}_{t-1}, X_{t}^{c} \in \mathcal {R}^{mn \times 1}\), the network will generate a transition matrix \(\tilde{A}_t \in \mathcal {R}^{mn \times mn}\) as an approximation to \(A_t\) in Eq. (5).

3.5 Measurement Network

Since prediction based coding is one of the most important techniques for video coding standards (such as MPEG2, H.264 or H.265), we take this coding approach into consideration when designing the measurement network. In prediction based coding, the decoded frame \(X^{c}_{t}\) can be decoupled into two components, i.e., \(X^{c}_t = X^{p}_{t} + R^{c}_{t}\), where \(X^{p}_{t}\) and \(R^{c}_{t}\) represent the prediction frame and the quantized prediction residual, respectively. Note that quantization is only used for the prediction residual and the distortion in compression only comes from \(R^{c}_{t}\). In addition, for non-predictive video codecs, such as JPEG2000, we use an existing warp operation [23, 36] from the previous decoded frame to the current decoded frame, and the difference between them is considered as the prediction residual \(R^{c}_{t}\). For most of video codecs (e.g.,H.264 and H.265), we can directly utilize the quantized prediction residual in the code streams.

We obtain the measurement using the quantized prediction residual \(R^{c}_{t}\) as follows,

$$\begin{aligned} Z_{t} = X^p_t + \hat{R}_{t},\text { where } \hat{R}_{t} = \mathcal {M}(X_{t}^{c}, R^{c}_{t}; \theta _z), \end{aligned}$$
(8)

where \(\hat{R}_{t}\) is the restored residual to remove the effect of quantization so that \(Z_{t}\) is close to the original image \(X_t\). We use a deep neural network as shown in Fig. 3(c) (with same architecture as Fig. 3(a)) for the function \( \mathcal {M}(\cdot )\). This network takes the decoded frame \(X_{t}^{c}\) and the quantized prediction residual \(R^{c}_{t}\) as the input and estimates the restored residual \(\hat{R}_{t}\). There are two advantages of our formulation for measurement. On one hand, instead of utilizing the decoded frame \(X_{t}^c\) as the measurement, our measurement formulation avoids explicitly modeling the complex relationship between original frames and decoded frames. On the other hand, most of the existing artifact reduction methods can be embedded into our model as the measurement method, which provides a flexible framework to obtain a more accurate measurement value.

3.6 Update Step

Given the prior state estimation \(\hat{X}^{-}_{t}\) from the temporal mapping network (Sect. 3.3), the transition matrix \(\tilde{A}_{t}\) obtained from the linearization network (Sect. 3.4), and the measurement \( Z_{t}\) obtained from the measurement network (Sect. 3.5), we can use the following stepsFootnote 1 to obtain the posterior estimation of the restored image:

$$\begin{aligned} P^{-}_{t} = \tilde{A}_{t}P_{t-1}\tilde{A}^\mathrm{{T}}_{t} + Q_{t-1}, \end{aligned}$$
(9)
$$\begin{aligned} K_{t} = P^{-}_{t}H^\mathrm{{T}}(H P^{-}_{t}H^\mathrm{{T}} + U_{t})^{-1}, \end{aligned}$$
(10)
$$\begin{aligned} \hat{X}_{t} = \hat{X}^{-}_{t}+K_{t}(Z_{t}-H\hat{X}^{-}_{t}), \end{aligned}$$
(11)
$$\begin{aligned} P_{t} = (I-K_{t}H)P^{-}_{t}, \end{aligned}$$
(12)

where \(\hat{X}_{t}\) represents the posterior estimation for the image \(X_t\). \(P^{-}_{t}\) and \(P_{t}\) are the estimated state covariance matrixs for the prior estimation and the posterior estimation respectively. \(K_{t}\) is the Kalman gain at time t. H is the measurement matrix defined in Eq. (2) and is assumed to be an identity matrix in this work. \(Q_{t-1}\) and \(U_{t}\) are the process noise covariance matrix and the measurement noise covariance matrix respectively. We assume \(Q_{t-1}\) and \(U_{t}\) to be constant over time. For more details about the update procedure of Kalman filtering, please refer to [40].

Discussion. Our approach can solve the error accumulation problem of the recursive pipeline through the adaptive Kalman gain. For example, when the errors accumulate in the previous restored frames, the degree of reliability for prior estimation (i.e., information from the previous frame) will be decreased and the final result will depend more on the measurement (i.e., the current frame).

3.7 Training Strategy

There are three sets of trainable parameters \(\theta _{f}\), \(\theta _{m}\) and \(\theta _{z}\) in our approach. First, the parameters \(\theta _{f}\) in the temporal mapping network are optimized as follows,

$$\begin{aligned} \mathcal {L}_{f}(\theta _{f}) = || X_{t} - \mathcal {F}(\hat{X}_{t-1}, X_{t}^{c}; \theta _{f}) ||^{2}_{2}, \end{aligned}$$
(13)

Note that in the minimization procedure for Eq. (13), the restored frame of the previous one \(\hat{X}_{t-1}\) is required. This leads to the chicken-and-egg problem. A straightforward method is to feed several frames of a clip into the network and train all the input frames in the iteration. However, this strategy increases GPU memory consumption significantly and simultaneous training multi-frames for a video clip is non-trivial. Alternatively, we adopt an on-line update strategy. Specifically, the estimation results \(\hat{X}_{t}\) in each iteration will be saved in a buffer. In the following iterations, \(\hat{X}_{t}\) in the buffer will be used to provide more accurate temporal information when estimating \(X_{t+1}\). Therefore, each training sample in the buffer will be updated in an epoch. We only need to optimize one frame for a video clip in each iteration, which is more efficient.

After that, the parameters \(\theta _{f}\) are fixed and we can optimize the linearization network \(\mathcal {G}(\theta _m)\) by using the following loss function:

$$\begin{aligned} \mathcal {L}_{m}(\theta _{m}) = || \hat{X}_{t}^{-} - \mathcal {G}(\hat{X}_{t-1},X_{t}^{c}; \theta _{m}) \hat{X}_{t-1}||^{2}_{2}, \end{aligned}$$
(14)

Note that we use a small patch size \((4 \times 4)\) to reduce the computational cost when optimizing \(\theta _{m}\).

Then, we will train the measurement net and optimize \(\theta _{z}\) based on the following loss function,

$$\begin{aligned} \mathcal {L}_{z}(\theta _{z}) = || {X}_{t} - (\mathcal {M}({X}_{t}^{c},R_{t}^{c}; \theta _{z})+ X_{t}^{p})||^{2}_{2}, \end{aligned}$$
(15)

Finally, we fine-tune the whole deep Kalman filtering network based on the loss \(\mathcal {L}\) defined as follows,

$$\begin{aligned} \mathcal {L}(\theta ) = ||X_t -\hat{X}_{t}||_2^2, \end{aligned}$$
(16)

\(\theta \) are the trainable parameters in the deep Kalman filtering network.

4 Experiments

To demonstrate the effectiveness of the proposed model for video artifact reduction, we perform the experiments on the benchmark dataset Vimeo-90K [21]. Our approach is implemented by using the Tensorflow [43] platform. It takes 22 h to train the whole model by using two Titan X GPUs.

Fig. 4.
figure 4

Quantitative (PSNR/SSIM) and visual comparison of JPEG2000 artifact reduction on the Vimeo dataset for q = 20.

Full size image

4.1 Experimental Setup

Dataset. The Vimeo-90K dataset [21] is recently built for evaluating different video processing tasks, such as video denoising, video super-resolution (SR) and video artifact reduction. It consists of 4,278 videos with 89,800 independent clips that are different from each other in content. All frames have the resolutio of \(448\times 256\). For video compression artifact reduction, we follow [21] to use 64,612 clips for training and 7,824 clips for performance evaluation. In this section, PSNR and SSIM [44] are utilized as the evaluation metrics.

To demonstrate the effectiveness of the proposed method, we generate compressed/decoded frames through two coding settings, i.e., codec HEVC (x265) with quantization parameter \(qp=32\) and \(qp=37\) and codec JPEG2000 with quality \(q=20\) and \(q=40\).

Implementation Details. For model training, we use the Adam solver [45] with the initial learning rate of 0.001, \(\beta _{1}=0.9\) and \(\beta _{2}=0.999\). The learning rate is divided by 10 after every 20 epochs. We apply gradient clip with global norm 0.001 to stabilize the training process. The mini-batch size is set to 32. We use the method in [46] for weight initialization. Our approach takes 0.15 s to restore a color image with the size of 448 \(\times \) 256.

We first train the temporal mapping network using the loss \(\mathcal {L}_f\) in Eq. (13). After 40 epochs, we fix the parameters \(\theta _{f}\) and train the linearization network by using the loss \(\mathcal {L}_m\). Then we train the measurement network using the loss \(\mathcal {L}_z\) in Eq. (15). After 40 epochs, the training loss will become stable. Finally, we fine-tune the whole model. In the following experiments, we train different models for different codecs or quality levels.

Fig. 5.
figure 5

Quantitative (PSNR/SSIM) and visual comparison of different methods for HEVC artifact reduction on the Vimeo dataset at qp = 37.

Full size image

4.2 Experimental Results

Comparison with the State-of-the-Art Methods. To demonstrate the effectiveness of our approach, we compare it with several recent image and video artifact reduction methods: ARCNN [18], DnCNN [17], V-BM4D [47] and Toflow [21]. In addition, modern video codecs already have a default artifact reduction scheme. For example, HEVC utilizes loop filter [1] (HEVC-LF) to reduce the blocking artifacts. This technique is also included for comparison.

For ARCNN [18] and DnCNN [17], we use the code provided by the authors and train their models on the Vimeo training dataset. For V-BM4D and Toflow, we directly cited their results in [21]. The results of HEVC-LF are generated by enabling loop filter and SAO [1] in HEVC codec (x265). For fair comparison with the existing approaches, we follow [21] and only evaluate the \(4^{th}\) frame of each clip in the Vimeo dataset. The quantitative results are reported in Tables 1 and 2. As we can see, our proposed approach outperforms the state-of-the-art methods by more than 0.6db in term of PSNR.

Table 1. Average PSNR/SSIM results on the Vimeo dataset for JPEG2000 artifact reduction (q = 20,40).
Full size table
Table 2. Average PSNR/SSIM results on the Vimeo test sequences for HEVC artifact reduction (qp = 32,37).
Full size table

Qualitative comparisons of ARCNN [18], Toflow [21], HEVC-LF [1] and ours are shown in Figs. 4 and 5. In these figures, the blocking artifacts exist in JPEG2000/HEVC decoded frame, our proposed method successfully removes these artifacts while other methods still have observable artifacts. For example, the equipment (the fourth row in Fig. 4) and the railing (the fourth row in Fig. 5) both have complex texture and structure, our method can well recover these complex regions while other baseline methods may fail.

Table 3. Ablation study of the proposed deep Kalman filtering method on the Vimeo-90k dataset. The results with or without using the prediction residual (PR) in the measurement network (MN) are reported in the first two rows. The results with or without using the recursive filtering (RF) scheme in the temporal network (TM) are reported in the \(3^{rd}\) and \(4^{th}\) rows. Our full model is MN+PR+TM+RF (the \(5^{th}\) row).
Full size table

Ablation Study of Measurement Network (MN).In this subsection, we investigate the effectiveness of the proposed measurement network. Note that the output of our measurement network itself can be readily used as the artifact reduction result. So the results in this subsection are obtained without using the temporal mapping network. In order to validate that prediction residual can serve as important prior information for improving the performance, we train another model with the same architecture but without using prediction residual (PR) as the input. Therefore, it generates restored frames by only using the decoded frames as the input. Quantitative results on the Vimeo-90k dataset are listed in Table 3. When compared with our simplified model without prediction residual(see the \(1^{st}\) row), our simplified model with prediction residual (MN+PR, see the \(2^{nd}\) row) can boost the performance by 0.34 dB in term of PSNR. It demonstrates that incorporating strong prior information can improve the restoration performance.

Ablation Study on the Temporal Mapping Network (TM). We further evaluate the effectiveness of the temporal mapping network. Note that the output of our temporal mapping network itself can be also readily used for the video artifact reduction. So the results in this subsection are obtained without using the measurement network. For comparison, we train another model, which utilizes the same network architecture as our temporal mapping network but the input is the concatenation of \(X^{c}_t\) and \( X^{c}_{t-1}\). Namely, it restores the current frame without considering previous restored frames. The quantitative results are reported in Table 3. When compared with our simplified model without using recursive filtering (RF) (see the \(3^{rd}\) row), our simplified model with recursive filtering (TM+RF, see the \(4^{th}\) row) can significantly improve the quality of restored frame by 0.41dB in term of PSNR. A possiable explanation is our recursive filtering scheme can effectively leverage information from previous restored frames, which provides more accurate pixel information.

It is worth mentioning that the result in the \(5^{th}\) row is the best as we combine the outputs from both the measurement network and the temporal mapping network through the Kalman update process.

Table 4. Average PSNR/SSIM results evaluated on two new datasets for video artifact reduction (JPEG2000, q = 20) for cross dataset validation.
Full size table

Cross Dataset Validation. The results on the HEVC standard sequences (Class D) and the MPI Sintel Flow dataset in Table 4 show that our approach performs better than the state-of-the-art methods.

Comparison with the RNN Based Approach. We use the recurrent network to completely replace the Kalman filter in Fig. 2. Specifically, the same CNN architecture is used to extract the features from the distorted frames at each time step and a convolutional gated recurrent unit (GRU) module is used to restore the original image based on these features. The result of our work is 37.93 dB, which outperforms the recurrent network based method (37.10 dB). One possible explanation is that it is difficult to train the recurrent network, while our pipeline makes it easier to learn the network by using the domain knowledge of prediction residual and combining both measurement and prior estimation.

5 Conclusions

In this paper, we have proposed a deep Kalman filtering network for video artifact reduction. We model the video compression artifact reduction task as a Kalman filtering procedure and update the state function by learning deep neural networks. Our framework can take advantage of both the recursive nature of Kalman filtering and representation learning ability of neural network. Experimental results have demonstrated the superiority of our deep Kalman filtering network over the state-of-the-art methods. Our methodology can also be extended to solve other low-level computer vision tasks, such as video super-resolution or denoising, which will be studied in the future.