A) Reference view selection

In this paper, the problems of MP synthesis and lossless coding of LF images are studied by varying from small to large the number of reference views selected for view synthesis. Four view configurations are employed for selecting an increasing squared number of reference views. Since the MP structure has a squared size, the configurations are carefully designed to have a symmetric shape.

Figure 2(B1) depicts the proposed view configurations of size $f \times f,\,\; f=2,\,3,\,4,\,5,$ each used to select a specific set of $f^2$ reference views from the array of $15\times 15$ views, i.e., by selecting $f^2$ pixels from each MP. The selected views are stored using equation (1) as a reference LF image based on the corresponding RMPs of size $f \times f.$ Let us denote $R^f_{k,\ell }$ as the RMP extracted from $M_{k,\ell }$ based on the configuration $f\times f.$ The top-left part of Fig. 2(B1) depicts the case of the $2 \times 2$ configuration where each RMP stores the array of $2 \times 2$ pixels corresponding to the set of views $S_2=\{(4,\,\;4),\, (4,\,\;12),\, (12,\,\;4),\, (12,\,\;12)\}.$ Hence, $R^2_{k,\ell }$ is set using $S_2$ as follows:

(2)\begin{equation} R^2_{k,\ell} =\begin{pmatrix} M_{k,\ell}(4,\,4) & M_{k,\ell}(4,\,12) \\ M_{k,\ell}(12,\,4) & M_{k,\ell}(12,\,12) \end{pmatrix}. \end{equation}

Fig. 2. (B1) The four view configurations used to form the reference macro-pixel (RMP): the $2\times 2$ configuration selects RMPs of size $2\times 2$, the $3\times 3$ configuration selects RMPs of size $3\times 3$, the $4\times 4$ configuration selects RMPs of size $4\times 4$, the $5\times 5$ configuration selects RMPs of size $5\times 5$. (B2) The structure of the patch for synthesis of size $30\times 30$. In the case of $2\times 2$ configuration, the patch selects an array of $15\times 15$ RMPs around the current MP position. (B3) The structure of the patch for coding of size $30\times 60$ for the case of $2\times 2$ configuration. The patch collects six MPs in the causal neighborhood and two synthesized macro-pixels (SMPs) in the non-causal neighborhood of the current MP position marked with a red square. Blue denotes the position of the reference view pixels in the MP. White denotes the position of the synthesized view pixels in the MP. Gray denotes the already encoded pixel positions.

The $2\times 2$ configuration was proposed to study the MP synthesis performance when the synthesized LF image is generated based on a minimum number of reference views, and the coding performance when the synthesized LF image is affected by high distortion. While the $5\times 5$ configuration was proposed to study the MP synthesis performance when the synthesized LF image is generated based on a large number of reference views, and the coding performance when the synthesized LF image is affected by low distortion. The four proposed view configurations were found optimal after complex experiments.

B) Lossless coding of reference views

In this paper, the selected reference views are encoded lossless in the first stage as a base layer. The corresponding reference LF image is encoded by employing the pixel-wise REP-CNN method [Reference Schiopu and Munteanu5] using network models trained for each color matrix.

The tests showed that the set of reference views is too small for an MP-wise entropy codec, such as the reference codec proposed in [Reference Schiopu and Munteanu6], to take advantage of the MP structure specific to LF images and to offer a consistent improvement over the REP-CNN method [Reference Schiopu and Munteanu5]. Similar results are obtained by employing either REP-CNN [Reference Schiopu and Munteanu5] or MP-CNN [Reference Schiopu and Munteanu6]. Moreover, the base layer has a small weight in the total bitrate.

C) Macro-pixel synthesis

In the second stage, the proposed deep-learning-based MP synthesis method is employed to synthesize the entire LF image based on the selected reference views. The novelty of this paper is that the entire set of views captured by the LF image is synthesized in one step.

In this paper, the patch for synthesis is formed by collecting an array of $30 \times 30$ pixels from the reference LF image. The patch collects the RMPs found in the close neighborhood of the current MP position as an array of $n_R\times n_R = {30}/{f}\times {30}/{f}$ RMPs, which collects a set of RMPs found at a maximum of $p^f=\lfloor {n_R}/{f} \rfloor$ RMP positions from the current RMP, $R^f_{k,\ell },$ as follows:

(3)\begin{equation} \{R^f_{k-i,\,\ell-j}\}_{ i,\ j\,=\,-p^f,-p^f\,+\,1,\ldots,0,1,\ldots, p^f}. \end{equation}

Figure 2(B2) depicts the case of the $2 \times 2$ configuration where the patch for synthesis collects the neighboring RMPs found at a maximum of $p^2=7$ positions, and generates an array of $n_R\times n_R =15 \times 15$ RMPs.

If $n_R=2n,$ then the patch for synthesis collects the RMPs found between the n top-left RMP positions and the n − 1 bottom-right RMP positions. For the $4\times 4$ configuration, the patch for synthesis collects only the central part of the RMPs found at the edge of the $8 \times 8$ array of RMPs.

The proposed neural network described in Section E is employed to compute the synthesized LF image. Note that the synthesized LF image is stored similarly in a lenslet structure using equation (1) and based on the corresponding SMPs of size $15 \times 15,$ denoted by $S^f_{k,\ell },$ where $(k,\, \ell )$ is the MP's position in the matrix of microlenses, and $f\times f$ is the selected reference view configuration. Therefore, for each $M_{k,\ell }$ and a reference view configuration $f\times f$, the method generates a corresponding $(R^f_{k,\ell },\, \; S^f_{k,\ell })$ pair.

D) Lossless coding of remaining views

In the third stage, the proposed deep-learning-based method for lossless coding based on the synthesized LF image is employed to encode the fine-details of the LF image corresponding to the synthesized view positions.

The proposed deep-learning-based MP prediction method takes advantage of the prior information found in the non-causal neighborhood of the current MP position in the synthesize LF image and provides an improved MP prediction compared to our previous work from [Reference Schiopu and Munteanu6]. The MP-CNN model [Reference Schiopu and Munteanu6] cannot be employed for MP synthesis due to the specific design of forming the patch based on six MP from the causal neighborhood stored as a MP volume.

In this paper, the patch for coding depicted in Fig. 2(B3) is formed by collecting an array of $30\times 60$ pixels from eight MPs selected as follows:

1. (i) the six MPs found in the causal neighborhood of the current MP: on the n (Northern), w (Western), $nw,\, ne,\, ww,$ and nww MP positions in the LF image, i.e., $M_{k-1,\ell },\, M_{k,\ell -1},\, M_{k-1,\ell -1},\, M_{k-1,\ell +1},\, M_{k,\ell -2},$ and $M_{k-1,\ell -2};$

2. (ii) the two SMPs found in the non-causal neighborhood of the current MP: on the current MP position and e (Eastern) MP position in the synthesized LF image, i.e., $S^f_{k,\ell }$ and $S^f_{k,\ell +1}.$

The MPs found outside the image edge are filled with zeros. The proposed neural network architecture described in Section E is employed to compute the MP prediction used for lossless coding.

In our prior work [Reference Schiopu and Munteanu6], we proposed a basic reference codec built based on the CALIC [Reference Wu and Memon2] architecture and designed for lossless coding of LF images. The reference codec uses a MP-wise coding strategy and it is employed also here for encoding based on the propose MP prediction the remaining views as the enhancement layer.

In this paper, the reference codec from [Reference Schiopu and Munteanu6] was further adapted to take advantage of the specific LF structure by employing a novel causal neighborhood for generating the modeling context in the CALIC binary mode.

In the CALIC architecture [Reference Wu and Memon2], before employing the predictive coding scheme via context modeling of prediction errors, the authors proposed to check if the current pixel values can be encoded based on the neighboring pixel values using a simple binary mode routine rather than based on prediction errors. The CALIC binary mode [Reference Wu and Memon2] consists in first collecting six pixels in the following causal neighborhood $M_{k,\ell }(x-1,\, y )$, $M_{k,\ell }(x, \, y-1)$, $M_{k,\ell }(x-1,\, y-1)$, $M_{k,\ell }(x-1,\, y+1)$, $M_{k,\ell }(x-2,\, y ),$ and $M_{k,\ell }(x, \, y-2)$ of the current position $M_{k,\ell }(x,\,y)$, depicted by purple squares in Fig. 3, and then checking if it has no more than two different values, denoted by $I_1$ and $I_2$. If true, then the binary mode is triggered, otherwise the predictive coding scheme is employed. In the binary mode, the current value, denoted I, is encoded by a symbol s set as follows: $s=0,$ if $I=I_1$; $s=1,$ if $I=I_2;$ and $s=2,$ otherwise. Symbol s is encoded using a binary pattern generated based on the positions of $I_1$ and $I_2$ in the causal neighborhood, resulting in 32 modeling contexts [Reference Wu and Memon2].

Fig. 3. The proposed neighborhood for generating the binary mode context in the basic reference codec [Reference Schiopu and Munteanu6]. Blue denotes the position of the reference view pixels in the MP. White denotes the position of the synthesized view pixels in the MP. Gray denotes the already encoded pixel positions. The red square denotes the current pixel position. The purple squares denote the CALIC binary mode context. The orange squares denote the proposed binary mode context.

In this paper, we propose the use of the following causal neighborhood in the binary mode routine: $M_{k-1, \ell }(x,y)$, $M_{k, \ell -1}(x, y)$, $M_{k-1, \ell -1}(x,y)$, $M_{k-1, \ell +1}(x, y)$, $M_{k-2, \ell }(x, y)$, $M_{k, \ell -2}(x, y),$ and $M_{k-2, \ell -1}(x,y),$ depicted by orange squares in Fig. 3. One may note that the proposed neighborhood selects the pixels found on the same current position, $(x,\, y),$ in the neighboring MPs, rather than in the local neighborhood of the current MP, $M_{k,\ell }.$ Since we propose a seven pixel neighborhood, 64 modeling contexts are obtained. One can note that by increasing the number of pixels in the causal neighborhood from 6 to 7, the number of cases when the binary mode can be triggered is decreased since the constraint of finding no more than two different values becomes harder to satisfy and the proposed method will relay more on the proposed prediction.

The proposed change is applied only when all the pixels in the neighborhood are available. Our tests have showed that the proposed neighborhood has an opposite effect when employing it in the predictive coding scheme. In this case the coding contexts are not divers enough and the codding error is decoupled from the local neighborhood.

The reference codec was adapted to skip the coding of the pixels corresponding to the RMPs already encoded in base layer. For each configuration depicted in Fig. 2(B1) a binary mask is used to signal the skipped positions in the current MP, marked with blue squares in Fig. 3.

One may note that a deep-learning-based algorithm is employed at each stage of the proposed lossless coding method. Figure 1 shows that the total bitrate of the compressed LF image is obtained by concatenating the base layer, corresponding to the encoded reference views, and the enhancement layer, corresponding to the fine-details encoded for the synthesized view positions for lossless reconstruction.

E) Proposed network design

In this paper, we propose a novel neural network design which follows a multi-resolution feature extraction paradigm. The proposed architecture is depicted in Fig. 4(a) and it is inspired from the U-net architecture [Reference Ronneberger, Fischer and Brox25] (designed for biomedical image segmentation) and from the Residual Learning paradigma [Reference He, Zhang, Ren and Sun26] (designed for training time reduction).

Fig. 4. (a) The proposed network design. When switch K is set to the (B2) Synthesis branch, the MP Syntheses based on Convolutional Neural Network (MPS-CNN) model is obtained. When switch K is set to the (B3) Coding branch, the Prediction using SMPs based on Convolutional Neural Network (PSMP-CNN) model is obtained. (b) The layer structure of the Convolution Block (CB). (c) The layer structure of the Deconvolution Block (DB). (d) The layer structure of the Convolution-based Processing Block (CPB) built based on the Residual Learning paradigma.

The network is built based on the following types of blocks of layers:

1. (a) The Convolution Block (CB) is depicted in Fig. 4(b) and contains the following sequence of layers: one convolution layer with a $3\times 3$ window, N filters, and stride $s=(s_1,\,s_2);$ followed by a Batch Normalization (BN) layer and a RELU activation layer.

2. (b) The Deconvolution Block (DB) is depicted in Fig. 4(c) and contains the following sequence of layers: one deconvolution layer with a $3\times 3$ window, N filters, and stride $s=(s_1,\,s_2);$ followed by a BN layer and a RELU activation layer. Note that in proposed architecture, in the second DC block denoted by DC_2, an extra cropping layer is inserted after the deconvolution layer to remove the first line and column of the input patch and to generate an output patch having the MP size of $15 \times 15.$

3. (c) The Convolution-based Processing Block (CPB) is depicted in Fig. 4(d) and contains the equivalent of three CB blocks, where the first CB is used to decrease the resolution by setting the stride s and the other two CB blocks are used to design a modified version of the Residual Learning building block [Reference He, Zhang, Ren and Sun26].

One may note that, in this paper, we adopt the strategy of inserting a BN layer between a convolution layer and an activation layer. In the CPB block, the residual is added before applying the BN and RELU activation layers. For all convolution layers the input patch is padded in such a way that the output patch has the same size as the input.

The proposed network is employed for both MP synthesis and MP prediction. Since the patches for synthesis and coding have a different size, they are first processed by two separate branches:

1. (i) when switch K is set to the (B2) Synthesis branch, the MP Syntheses based on Convolutional Neural Network (MPS-CNN) is obtained;

2. (ii) when switch K is set to the (B3) Coding branch, the Prediction using Synthesized MPs based on Convolutional Neural Network (PSMP-CNN) is obtained.

The synthesis branch is depicted in the top-left corner of Fig. 4(a) and is processing the $30\times 30$ synthesis patch based on two CPB blocks: CPB_S1 with 32 filters, and CPB_S2 with 64 filters and stride $s = (2,\,2).$ The coding branch is depicted in the middle-left part of Fig. 4(a) and is processing the $30\times 60$ coding patch based on three CPB blocks: CPB_C1 with 32 filters, CPB_C2 with 64 filters and stride $s = (2,\,2),$ CPB_C3 with 64 filters and stride $s = (1,\,2)$. Both branches are processing the patch from initial resolution down to $15\times 15$ resolution. The remaining structure is using the U-net multi-resolution paradigma for further processing the patches at three resolutions: $15\times 15$, $8\times 8$, and $4\times 4$. In the proposed design, the last CB block contains only one filter so that it can output the MP prediction of the current MP.

One may note that the number of filters of CPB blocks is increasing up to 256 for the $4\times 4$ resolution. The number of filters in the DB blocks is set as half the number of input channels. The concatenation layers are concatenating an equal number of activation maps after a CPB block is processing the current and lower resolutions. The total number of parameters of MPS-CNN and PSMP-CNN models is around 3 million. The stochastic gradient descend optimizer is used with the Nesterov momentum activated, and the momentum parameter set to $0.9.$

In this paper, the training procedure employs the mean square error (MSE) loss function. Let $\Theta _{MPS-CNN}$ be the set of all learned parameters of the MPS-CNN model, $X_i$ the $i^{th}$ patch for synthesis in the training set, and $Y_i$ the currently predicted MP. Let $F(\cdot )$ be the function which processes $X_i$ using $\Theta _{MPS-CNN}$ to compute the MP prediction as $\hat {Y_i} = F(\Theta _{MPS-CNN},\, X_i)$. The loss function can be formulated as follows:

(4)\begin{equation} L(\Theta_{MPS-CNN}) = \frac{1}{N_P} \sum_{i=1}^{N_P} || vec(Y_i) -vec(\hat{Y_i})||^2, \end{equation}

where $N_P$ is the size of the training set. Similarly, equation (4) can be formulated for the PSMP-CNN model.

F) Overview

The workflow of the proposed method is presented in Fig. 5. The proposed method extends our previous work on MP-wise coding [Reference Schiopu and Munteanu6]. The novel contributions of this paper compared to [Reference Schiopu and Munteanu6] can be summarized as follows:

1. (1) The proposed method takes advantage of the MP synthesis technique and generates a synthesized LF image using steps B1 and B2 of the algorithm.

2. (2) The patch for coding in [Reference Schiopu and Munteanu6] is a volume of size $N\times N\times 6$ which collects six neighboring MPs, while the proposed patch for coding is a matrix of size $2N \times 4N$ which collects six neighboring MPs and two neighboring SMPs.

3. (3) The two models are completely different. MP-CNN [Reference Schiopu and Munteanu6] uses a sequence of 3D convolutions to process the patch, while PSMP-CNN was inspired from the Unet architecture to process the patches at different resolutions, and it is based on 2D convolutions.

4. (4) The reference codec from [Reference Schiopu and Munteanu6] was further adapted to take advantage of the MP structure by introducing a MP-based causal neighborhood for the CALIC binary mode.

5. (5) The compression results presented in Section IV below show that the proposed method offers an average improvement of $12.5\%$ over our MP-CNN of [Reference Schiopu and Munteanu6].

Fig. 5. The workflow of the proposed method.

The proposed method can be adapted to other LF structures. A conventional LF dataset obtained with a multi-camera setup can be re-mapped to an MP data structure by appropriate re-ordering of light rays. In general, for an MP structure of $N\times M$ pixels, one can generate the patch for synthesis of size $2N\times 2M,$ and a similar patch for coding of size $2N \times 4M.$ No other changes are necessary for the proposed network design or the MP-wise entropy codec. A similar strategy for selecting the reference views can be derived.

A) Macro-pixel synthesis results

Figure 6 shows the synthesis results for three LF images in the dataset. The pseudo-colored images show the mean absolute error over the three color channels corresponding to the randomly selected view $(7,\,7)$ in the LF image. The distribution of the residual error was truncated for absolute errors represented on more than 4 bits (i.e., larger than 15) by replacing them with the escape symbol 16. One can note that the shades of blue corresponding to absolute mean errors represented on 1–2 bits are the dominant colors of the pseudo-colored images shown in Fig. 6.

Fig. 6. Pseudo-colored images of the mean absolute error computed over the color channels for one view in the LF images. The mean absolute errors with more than 4-bit representation (i.e., larger than 15) are replaced by the escape symbol 16. The top-to-bottom rows show the synthesized view $(7,\,7)$ for each of the four view configurations: $2\times 2$, $3\times 3$, $4\times 4$, and respectively $5\times 5$.

Figure 7(a) shows the results for each LF image in the dataset, where the distortion is measured using the PSNR metric computed between the original image, LL, and the synthesized image, $\hat {LL},$ as follows:

(5)\begin{equation} PSNR = 20 \cdot \log_{10} \left( \frac{255}{\sqrt{MSE}} \right), \end{equation}

(6)\begin{equation} MSE = \frac{1}{N_{LL}}\!\!\! \sum_{i=1}^{N\cdot N_{mr}} \sum_{j=1}^{N\cdot N_{mc}} \sum_{c=1}^{N_{ch}}|| \hat{LL}(i,j,c) - LL(i,j,c)||^2, \end{equation}

$N_{LL} = (N\cdot N_{mr}) \times (N\cdot N_{mc}) \times N_{ch} = 9375 \times 6510 \times 3$.

Fig. 7. Evaluation of the one-step synthesis results of the macro-pixel synthesis. (a) Synthesis results for each image in the Test Set. (b) Rate-distortion results for the Test Set. (c) Rate-distortion results for the Red and white building image. (d) Rate-distortion results for the Sophie and Vincent 2 image.

Figure 7(b) and Table 1 show the average rate-distortion results over the Test Set of 108 images for one-step MP synthesis, where the bitrate is computed as bits per channel (bpc). Moreover, the table shows the average weight of the base layer in the total image bitrate, denoted $p_{BR},$ computed as the ration between the base layer bitrate and the total bitrate. One may note that MPS-CNN achieves an average performance of 29.82 dB when only ${4}/{225} = 1.77\%$ of the views are selected as reference views and are encoded in $2.3\%$ of the total bitrate, while an average performance of 36.19 dB is achieved when ${25}/{225}=11.11\%$ of views are encoded in $15.40\%$ of total bitrate. An improvement of around 2 dB increase is achieved with each increment of side f of the $f \times f$ view configurations.

Table 1. Lossless compression results of RMP and one-step synthesis results of SMP

Figures 7(c) and 7(d) show the rate-distortion results for two LF images from the Test Set. One may note that the image's content plays an important role in view synthesis applications and that MPS-CNN can achieve results of around 36.5 dB distortion base on only four reference views, and up to 43 dB distortion based on 25 reference views.

B) Lossless compression results

The proposed codec is denoted by MP Synthesis and Coding based on Convolutional Neural Network (MPSC-CNN) and encodes the LF images as follows:

1. (B1) REP-CNN [Reference Schiopu and Munteanu5] is employed to encode lossless the reference views;

2. (B2) MPS-CNN is employed to synthesize the LF image;

3. (B3) PSMP-CNN is employed to compute the MP prediction based on the synthesized image, and the MP-wise CALIC-based Reference codec [Reference Schiopu and Munteanu6] is employed to encode lossless the residual errors of the remaining views.

The performance of the following methods is compared:

1. (M1) the JPEG 2000 codec [Reference Skodras, Christopoulos and Ebrahimi29] based on the OpenJPEG implementation [33], the active reference software for JPEG 2000 [34], where the code runs with the “−r 1” parameter for a lossless compression setting;

2. (M2) the HEVC video codec [Reference Sullivan, Ohm, Han and Wiegand13] with all intra configuration; HEVC encodes the pseudo-video-sequence created using the spiral stacking scan pattern [Reference Helin, Astola, Rao and Tabus10,Reference Santos, Assuncao, da Silva Cruz, Tavora, Fonseca-Pinto and Faria12]; the fast x265 library [35] is used with the veryslow preset and the lossless parameter;

3. (M3) the JPEG-LS codec [Reference Weinberger, Seroussi and Sapiro1];

4. (M4) the CALIC codec [Reference Wu and Memon2] based on the authors implementation available online [36];

5. (M5) the CMS method [Reference Schiopu, Gabbouj, Gotchev and Hannuksela11] designed to encode 193 views out of the 225 views, where the remaining views are encoded using CALIC [Reference Wu and Memon2];

6. (M6) the FLIF codec [Reference Sneyers and Wuille3] based on the implementation available online [7];

7. (M7) the MP-CNN method [Reference Schiopu and Munteanu6], our preliminary work on MP-wise prediction, where the models are trained based on patches selected from the same training set;

8. (M8) the PredNN method [Reference Schiopu, Liu and Munteanu4], the first paper on pixel-wise CNN-based prediction, where the model is trained based on patches collected from the same training set; for each LF image more than 183 million patches are processed, one for each pixel and for each color matrix;

9. (M9) the REP-CNN method [Reference Schiopu and Munteanu5], the first deep-learning-based dual prediction method for pixel-wise prediction based on a similar training process as M8;

10. (M10) the proposed MPSC-CNN codec employed for the $2\times 2$ configuration of reference views;

11. (M11) the proposed MPSC-CNN codec employed for the $3\times 3$ configuration of reference views;

12. (M12) the proposed MPSC-CNN codec employed for the $4\times 4$ configuration of reference views;

13. (M13) the proposed MPSC-CNN codec employed for the $5\times 5$ configuration of reference views.

Figure 8 shows the lossless compression results for each image in the dataset using the Relative Compression (RC) metric which is used to compare the compression results of a method MX relative to our proposed method M13. The RC result for a method MX is computed as follows:

(7)\begin{equation} RC_{{\rm MX}} = \frac{Bitrate_{{\rm MX}}}{Bitrate_{{\rm M13}}}. \end{equation}

Fig. 8. Lossless compression results.

Table 2 shows the average results for the Test Set. Method M13, the proposed method based on the $5\times 5$ configuration, achieves the following average performance over the set of test images:

1. (i) $29.9\%$ improvement compared to CALIC [Reference Wu and Memon2], a traditional lossless image codec;

2. (ii) $12.5\%$ improvement compared to our previous method [Reference Schiopu and Munteanu6];

3. (iii) $9.1\%$ improvement compared to FLIF [Reference Sneyers and Wuille3], the current state-of-the-art lossless image codec; and

4. (iv) $3\%$ improvement compared to REP-CNN [Reference Schiopu and Munteanu5].

Table 2. Lossless compression results of for the Test Set (108 images)

From the complexity perspective, the proposed method requires the inference of two patches when encoding one MP. The MP-CNN model [Reference Schiopu and Munteanu6] requires the inference of a single patch when encoding one MP, however, the inference time of a 3D convolution is higher than of a 2D convolution. The two deep-learning-based pixel-wise prediction models, PredNN [Reference Schiopu, Liu and Munteanu4] and REP-CNN [Reference Schiopu and Munteanu5], require the inference of 225 patches when encoding one MP.