Threedimensional video coding is one of the main challenges restricting the widespread applications of 3D video and free viewpoint video. In this paper, a novel fractal coding algorithm with motionvectorfieldbased motion estimation for depth map sequence is proposed. We firstly add presearch restriction to rule the improper domain blocks out of the matching search process so that the number of blocks involved in the search process can be restricted to a smaller size. Some improvements for motion estimation including initial search point prediction, threshold transition condition and early termination condition are made based on the feature of fractal coding. The motionvectorfieldbased adaptive hexagon search algorithm on the basis of centerbiased distribution characteristics of depth motion vector is proposed to accelerate the search. Experimental results show that the proposed algorithm can reach optimum levels of quality and save the coding time. The PSNR of synthesized view is increased by 0.56 dB with 36.97% bit rate decrease on average compared with H.264 Full Search. And the depth encoding time is saved by up to 66.47%. Moreover, the proposed fractal depth map sequence codec outperforms the recent alternative codecs by improving the H.264/AVC, especially in much bitrate saving and encoding time reduction.
1. Introduction
W
ith the development of multimedia technology, the traditional twodimensional video has been unable to meet people's needs, while the threedimensional (3D) immersive and interactive video gives the users an unprecedented wonderful experience owing to its unique depth of field effect. 3D video is composed of texture videos and the corresponding depth map sequences. Based on the principle of binocular parallax, the users can obtain a threedimensional image, which enhances the senses of visual reality and verisimilitude. Compared with monocular video, the original data of 3D video is much huge. The transmission bandwidth is limited, which limits the widespread applications of 3D video, and therefore, it is necessary to study high efficiency 3D video compression technology.
The current 3D video compression methods mainly contain multiview video coding
[1]
and depth mapbased compression
[2]
. The multiview video uses multiple cameras to shoot the same scene from different angles. Multiview video is characterized by high complexity of coding and large amount of data. The multiview video coding needs to consider the redundancy in the same viewpoint as well as that between adjacent viewpoints. Depth map represents the information about the distance from the scene to the camera imaging plane, which quantizes the actual depth value to [0, 255] so as to obtain the grayscale map representing the depth information. In the Multiview Video plus Depth (MVD)
[3]
structure, each color image contains its corresponding depth map. This paper is to study the coding of depth map sequence.
The depth map can be represented by 3Dmesh, which can also be seen as a twodimensional image so that the depth map compression can be treated as twodimensional image compression and the existing compression standard can be applicable. In
[4]
, an early SKIPmode decision scheme and a selective predictionmode decision scheme have been developed based on the relationship between coding modes of collocated macroblocks on texture and depth frames. Depth map sequence coding aims to maximize the perceived visual quality of the synthesized virtual view instead of the depth map sequence itself, so that a structural similaritybased synthesized view distortion (SSSVD) model to relate perceptual distortion in coded depth map and synthesized view is proposed in
[5]
. It obtains better rate distortion performance and perceptual quality of synthesized views than JM reference software. In
[6]
, an efficient intra prediction algorithm for smooth regions in depth coding is proposed by considering one single prediction direction instead of multiple prediction directions. The original contextbased adaptive binary arithmetic coding (CABAC) was originally designed for lossy texture coding, and it cannot provide the best coding performance for lossless depth map coding. In
[7]
, an enhanced CABAC coding mechanism for lossless depth map coding based on the statistics of residual data is proposed.
All the above depth map sequence encoding methods are based on H.264/AVC standard
[8]
. Its basic theory is to form a prediction of the macroblock based on previouslycoded data, either from the current frame (intra prediction) or from other frames that have already been coded and transmitted (inter prediction) and then residual is transformed, quantified and entropy coded. Different from it, fractal image coding method is a novel technique in the image compression field using partial selfsimilarity of image
[9]
. Its theoretical bases are the Iterated Function System and Collage theorem
[10]
, and it is recognized as one of the most promising new generation image coding technologies at present, which is characterized by novel concepts, high compression ratio and great potential, but with high coding complexity. The depth map sequence shows a strong similarity between adjacent frames. Therefore, good compression performance can be obtained by use of fractal theory.
To obtain high efficiency encoding performance, and to decrease the complexity of depth map sequence codec as well, we propose a novel fractal depth map sequence coding algorithm with motionvectorfieldbased motion estimation. This paper is organized as follows: The theory of fractal coding is summarized in section 2. The proposed fractal depth map sequence coding algorithm is presented in section 3. The experimental results are presented in section 4. And finally the conclusions are outlined in section 5.
2. Background of Fractal Compression
Jacquin described the first practical fractal image compression algorithm based on a Partitioned Iterated Function System (PIFS). Divide the image
I
to be coded into two sizes of blocks, with the size of B×B pixels being range blocks called
R_{i}
, allowing nonoverlapping and covering the entire image; and with the size of 2B×2B pixels being domain blocks called
D_{i}
, allowing overlapping. All the domain blocks form a domain pool. The image
I
can be represented by
where
N
is the total number of range blocks.
Each range block
R_{i}
shall be approximated by a certain domain block
D_{m(i)}
with the transformation
ω_{i}
:
D_{m(i)}
→
R_{i}
, which is composed of spatial contraction transformation
γ_{i}
and grayscale modification transformation
λ_{i}
.
ω_{i}
is often a contracted affine transformation, its general form is:
γ_{i}
uses the average value of four adjacent pixels to get B×B pixel block
. The average value of four adjacent pixels is:
where
d_{h,v}
and
represent pixel values of
D_{m(i)}
and
at the pixel position (
h,v
) , respectively.
λ
_{i}
introduces grayscale adjustment parameter
s_{i}
, grayscale offset parameter
o_{i}
, and isometric transformation
t_{k}
in order to get a better approximation of
R_{i}
by further modifying the grayscale of
D_{m(i)}
. The general form of
λ_{i}
is:
where
a
,
b
,
c
and
d
determine 8 kinds of pixel rearrangements (4 kinds of rotation and 4 kinds of flip);
s_{i}
and
o_{i}
represent brightness adjustment and brightness offset respectively.
The fractal coding is aimed to look for the best matching block
D_{m(i)}
and block transformation
ω_{i}
for each
R_{i}
, so that
R'_{i}
=
ω_{i}
(
D_{m(i)}
) and
R_{i}
are as close as possible in the given distortion metric, namely
The global transformation of the image onto a close approximation of itself is then constructed from a combination of these local block transformations.
Fig. 1
shows some mappings from domain to range blocks.
Mappings from domain to range blocks
3. The Proposed Fractal Depth Map Sequence Coding Scheme
This paper proposes a fractal depth map sequence coding scheme, as shown in
Fig. 2
. Actually, it is a hybrid codec with intra prediction and inter fractal. When a depth map
F_{n}
is input to the encoder, it will be processed with range block as the basic unit, and the encoding mode will be selected between intra prediction and inter fractal. The intra prediction is to use the previously decoded adjacent pixels within the same frame to predict the current block based on spatial correlation. Compared with the previous encoders, the inter fractal adds presearch restriction and improves motion estimation. A motionvectorfieldbased adaptive hexagon search method is designed. Locate the best matching block of current range block from the reference frame according to the motion vector (
x
,
y
) , and then use the best matching block iteration to produce the currently predicted block
PRED
according to the recorded scale factor
s
and offset factor
o
. The iterative formula is as follows:
The proposed fractal depth map sequence encoder
where
r_{i}
represents pixel value of the predicted block, and
d_{i}
represents pixel value of the corresponding matching block.
In the figure, the reference frame represents the reconstructed image
F'_{n}
_{1}
of the previous depth map. A residual block
D_{n}
is generated by subtracting the predicted block
PRED
from the current range block. It is transformed and quantized as a coefficient set
X
, and these coefficients are reordered for entropy coding. Because the reconstruction process requires iteration of fractal parameters to produce the reference frame, entropy coding coefficients of fractal parameters need to be written into the code stream. The entropy coding coefficients and side information (prediction mode, quantization step size, etc.) form the compressed bit stream.
 3.1 Presearch Restriction
In the process of fractal coding, the most important step is to search in the domain pool to find the best matching block for each range block in order to minimize the matching error mean square error (MSE).
where
N
represents the number of pixels in the current block.
It can be seen that, the fractal coding process is very timeconsuming. Because the best matching block is required to be searched for every range block in the codebook
A
, while in theory, the number of candidate domain blocks is usually very large. For example, for each range block in the original image, if exhaustive search is used, the search time is O(
A
) (linearity depends on 
A
), where 
A
 represents the number of blocks in A. If 
A
 can be reduced, the fractal encoding time must be decreased. Some efforts have been made to reduce the number of blocks need to search. For example, by using a nearestneighbor search techniques
[11]
or a local search operation
[12]
or searching in the area centered on last best matching domain block
[13]
or classification of image blocks
[14]
.
In our scheme, before the matching search process for a range block, we add a presearch restriction to rule the improper domain blocks out of the matching search process based on the variance matching condition of each block so that the number of blocks involved in the matching search process can be restricted to a smaller size, thus to greatly accelerate the coding. The derivation process of the presearch restriction condition is detailed as follows.
In the fractal coding process, by setting an appropriate scale factor
s
and offset factor
o
may allow the value of
D_{i}
after affine transformation has a minimum squared distance of the value of
R_{i}
. When the partial differentiations of
s
and
o
are 0, the minimum matching error
MSE
can be obtained, in which the corresponding
s
and
o
are shown in Formulas (9) and (10).
Therefore, substitute the Equations (9) and (10) again into the Formula (8) to obtain the following MSE form different from the basic one:
Make
and it’s obvious that
Then Formula (11) can be further derived as follows：
Let
For each range block,
is already known. In order to obtain the minimum matching error MSE, the value of
m
should be as smaller as possible, which is the start point of our presearch restriction scheme.
In our presearch restriction scheme, for each range block in the matching process, if
Th
˂
m
˂1, then we need to search the next block in the codebook to find the minimum matching error MSE; otherwise it indicates that the current MSE has been small enough without requirement of the next step’s comparison and the result can be directly saved, then proceed to search for the next range block.
Th
is a threshold experimentally determined in advance and it’s set to 0.9 in our scheme.
The presearch restriction algorithm is outlined in the pseudocode as follows.
Presearch Restriction Algorithm:

● PrepareRi,i= 1,2,...,NR

● PrepareDj,j= 1,2,...,ND

● ExtendDjby spatial contraction transformation and 8 kinds of affine transformation to form a codebook

Ω = {Ωl=tkDj,k=1,...,8,j=1,...,ND,l=1,...,8ND}

● Loop Range Block: For every range blockRi,i= 1,2,...,NR, do

{

▪ Loop Domian Search: For every domain block in Ω , do

{

Computemusing formula (13)

· if0.9

go on to next domain block

· else

compute and record scale factor snd offset factor

go on to next range block

}

}
 3.2 Improvements of Motion Estimation
Currently, the fractal coding speed is very low, of which the most important reason is that the search for the best matching block is timeconsuming, and to improve the coding speed, it is necessary to improve search skills. UnsymmetricalCross MultiHexagonal Search (UMHexagonS
[15]
) is a fast motion estimation algorithm, which has been applied in the JM reference software of H.264/AVC. It greatly outperforms full search in computation with up to 90% computational reduction while still maintaining good ratedistortion performance. However it’s proposed based on the multiple prediction modes and multiple reference frames of H.264, which is significantly different from the fractal depth mapsequence coding method in this paper. Therefore, the following improvements have been made.
1. Initial search point prediction
Use the following three methods for initial search point prediction:
(1) Spatial median prediction: According to the spatial correlation, take the median of motion vectors of the left, up and rightup adjacent blocks in the current frame as the predicted motion vector, as shown in
Fig. 3
;
Spatial median prediction
(2) Origin prediction: Take (0,0) as the motion vector value;
(3) Adjacent reference frame prediction: According to the time correlation, get current motion vector from MV in the same position of the previous reference frame.
Then calculate the MSE using Formula (12) for each candidate, compare and choose the minimum MSE. The point with the minimum MSE is named as MMD (minimum matching distortion), which is used as the initial search point.
2. Threshold transition condition at the asymmetrical cross template search
The criteria of the similarity between range block and transformed domain block in fractal coding uses MSE as shown in Formula (12). We use different thresholds based on different sizes of blocks; after asymmetrical cross template search is completed, select the best matching point as a new start point for the subsequent template matching.
3. Early termination condition
Based on the characteristics of fractal coding algorithm in this paper, the early termination circumstances include two aspects:
Firstly, in the multilevel nonuniform hexagonal grid integer pixel motion search process, in addition to the early termination conditions of the algorithm itself, in order to reduce the complexity of search, if the optimal matching point is at the center of the hexagon, the search can be stopped;
Secondly, the inter fractal coding algorithm in this paper uses the hierarchical treestructure partition as shown in
Fig. 4
. Search using mode 1 at first, if the threshold condition is satisfied, then terminate the current block coding and proceed to the next block; otherwise, use mode 2 to divide the current range block into two smaller subblocks and search for each subblock. This partition goes on to mode 3 and mode 4 until the threshold condition is satisfied. When the range block has been partitioned to 4 equal subblocks as shown in mode 4, each subblock allows further partition recursively until it reaches the allowed minimum size.
Hierarchical treestructure partition
 3.3 Motionvectorfieldbased Adaptive Hexagon Search Algorithm
In the actual video sequences, most of the depth motion vectors are enclosed in the central area
[16]
. We use full search algorithm in fractal depth map sequence codec and make some statistical analysis about motion vector. As shown in
Fig. 5
(a) and
Fig. 5
(b), about 81.80% of motion vectors are found located in the central 5×5 area, i.e.
p
= ±2 pixels, using the search window
W
= ±7 . The large cross pattern (A+B+C) accounts for 74.71% of the motion vector distribution, and the small cross pattern (A+B) accounts for 68.98% of the motion vector distribution. A lot of experimental results show that more than 80% of the blocks can be regarded as stationary or quasistationary blocks, suggesting that motion vector is of centerbiased distribution characteristics
[16]
. Most of the best matching points of inter fractal encoding are located in a small scope around the initial point.
Average distributive probability of depth motion vector for the search window W = ±7
Based on this observation, we propose our motionvectorfieldbased adaptive hexagon search algorithm. Firstly, an unsymmetrical crossshaped pattern (UCSP) is used to search points around the initial point, as shown in
Fig. 6
(a), and then small crossshaped pattern (SCSP, see
Fig. 6
(b)) and large hexagonshaped pattern (LHSP) are used. In addition to the six surrounding points and the central point of the traditional LHSP, we also contains the midpoint of the two sides located above and below the search central point, as shown in
Fig. 6
(c), so that the new LHSP contains a total of 9 search points, which could improve the search accuracy with negligible increasing complexity. Furthermore, we employ a halfwaystop technology so that the small motion vector can be searched with less points, so as to accelerate the motion estimation. The specific steps are as follows:
Search patterns
Step (i):
Search by a UCSP centering the initial point. Search stops if the MMD point occurs at the center of the UCSP, and we get the motion vector, as shown in
Fig. 7
(a), where the best matching point is filled with red color. Otherwise, go to Step (ii).
The proposed motionvectorfieldbased adaptive hexagon search algorithm
Step (ii):
Take the MMD point in Step (i) as the center of the SCSP. Two or three new points need to be searched in this step. Search stops if the MMD point occurs at the center of the SCSP.
Fig. 7
(b) shows the case when two more points (filled with yellow color) are searched. Otherwise, go to Step (iii).
Step (iii):
Four additional points on a traditional LHSP centering the initial point in step (i) are checked. If the MMD point is not on the LHSP, meaning that the MMD point in Step (ii) is still the best matching point, the search stops, as shown in the in
Fig. 7
(c). Otherwise, if the MMD point is on the LHSP, go to Step (iv).
Step (iv):
Search by the new LHSP centering the MMD point in Step (iii). If the new MMD point is at the center of the new LHSP, then go to Step (v), as shown in
Fig. 7
(d). Otherwise, this step is repeated again.
Step (v):
Take the minimum MMD point in the previous step as the center, and then a SCSP is applied. Find the new MMD point, which is the final solution for the motion vector, as shown in
Fig.7
(e).
The first two steps in our proposed motion estimation scheme using two crossshaped pattern have a high probablity to find the best matching point based on centerbiased distribution characteristics of motion vector. Thus the number of points need to search is heavily reduced. In the following steps, we use large hexagonshaped pattern to avoid falling into local optimum.
4. Experimental Results and Analysis
To verify the performance of the proposed method, we have made several experiments with the Kendo, Newspaper, and Book Arrival test sequences. Experimental results obtained using our proposed fractal coding algorithm with motionvectorfieldbased motion estimation are compared with those using H.264 full search, H.264 EPZS (enhanced predictive zonal search), LCMDME proposed by Gianluca et al
[17]
, and fast MD/ME with Sobel edge detector by Zhu et al
[18]
. Since depth maps are not directly used to watch, but they are used to generate the synthetic views. Hence, we compare the quality between the synthetic views generated by the encoded depth maps. Synthetic views are generated using the View Synthesis Reference Software (VSRS) 3.0 provided by MPEG
[19]
. The views used for our experiments are listed in
Table 1
.
Color and depth sequences used for the experiments
Color and depth sequences used for the experiments
To keep coincidence with the compared methods, the color videos used for the generation of the synthetic views have been encoded using our fractal codec. The RD curves are shown in
Fig. 8
, where the PSNR is evaluated between the synthetic views generated through the original depth maps and the synthetic views created through the compressed ones
[17]
. The bitrate is calculated by the sum of the two depth sequences’ bitrate.
RD curves for different methods
It can be seen from
Fig. 8
that our proposed fractal depth map sequence coding algorithm with motionvectorfieldbased motion estimation outperforms all the other compared methods. Adjacent frames in the depth map sequence have a strong similarity, so that we can get good compression performance by inter fractal method.
The RD performance comparisons evaluated with the Bjontegaard metric
[20]
are summarized in
Table 2
. We note that the proposed method is consistently better than all the other compared methods, with 33.82% bit rate decrease and 0.45 dB PSNR increase on average than H.264 Full Search.
BDBR and BDPSNR between the proposed method and the compared five methods
BDBR and BDPSNR between the proposed method and the compared five methods
Fig. 9
shows the 10th frame of the original depth map sequence Newspaper view 2 and the decoded ones by Full Search using JM18.1
[21]
and the proposed method.
Fig. 10
shows the 10th frame of the synthesized Newspaper view 3 resulted from Full Search using JM18.1 and the proposed method.
the 10th frame of the original depth mapsequence Newspaper view 2 and the decoded ones
The 10^{th} frame of the synthesized Newspaper view 3
Fig. 11
shows the encoding time for two depth views and
Table 3
shows the percentage of average depth encoding time saved by our method compared with Full Search and LCMDME.
Comparison of depth sequences (two depth views) encoding time
Average depth encoding time saved (%) by the proposed method compared with Full Search and LCMDME
Average depth encoding time saved (%) by the proposed method compared with Full Search and LCMDME
The encoding time comparisons show our proposed method can reduce depth compression time by up to 66.47% and 42.26% compared with H.264 Full Search and LCMDME respectively. Although the encoding time is longer for our methods than the other three fast motion estimation methods, the difference is not prominent for Kendo.
5. Conclusion
In this paper, an efficient fractal depth mapsequence coding algorithm with fast motion estimation is proposed. We firstly add presearch restriction to rule the improper domain blocks out of the matching search process so that the number of blocks involved in the matching search process can be restricted to a smaller size. Some improvements for motion estimation including initial search point prediction, threshold transition condition and early termination condition are made based on the feature of fractal coding. The probability distribution characteristics of depth motion vectors are also analyzed and motionvectorfieldbased adaptive hexagon search algorithm is proposed to accelerate the search, especially for the static block or the quasi static block.
Experimental results show that the proposed algorithm can reach optimum levels of quality and save coding time at the same time. The PSNR of synthesized view is increased by 0.44 dB with 33.82% bit rate decrease on average compared with H.264 Full Search evaluated by the Bjontegaard metric. And the depth encoding time is saved by up to 66.47%. This method makes the best of the features of fractal coding and depth motion, and achieves a great improvement with considerably good results. Also, it has built a good foundation for the further research of 3D video fractal coding. Our method doesn’t consider the relation between depth map and the corresponding color video, which may be our further research direction.
BIO
Shiping Zhu received the B.S. and M.S. degrees from Xi’an University of Technology, Xi’an, China, in 1991 and 1994, and the Ph.D. degree from Harbin Institute of Technology, Harbin, China, in 1997. From 1997 to 1999, he was a Postdoctoral Fellow with Beihang University, Beijing, China. From 2000 to 2002, he was a Postdoctoral Fellow with the Brain and Cognition Research Center, Université Paul Sabatier, Toulouse, France. From 2002 to 2004, he was a Postdoctoral Fellow with the Department of Computer Science and Department of Electrical and Computer Engineering, Université de Sherbrooke, Sherbrooke, QC, Canada. Since 2005, he has been an associate professor with the Department of Measurement Control and Information Technology, School of Instrumentation Science and Optoelectronics Engineering, Beihang University, Beijing, China. His research interests include image processing and video coding, computer vision, machine vision for 3D measurement. (Email: spzhu@163.com).
Dongyu Zhao received the B.S. degree in Measurement and Control Technology and Instrumentation from Beijing Jiaotong University, Beijing, China in 2012, and she is currently pursuing the M.S. degree in Instrumentation Science and Technology at Beihang University, Beijing, China. Her research interests include image processing and video compression. (Email: dongyuzhao@msn.cn).
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