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RESEARCH Open Access

Methods for depth-map filtering in view-plus-depth 3D video representationSergey Smirnov, Atanas Gotchev* and Karen Egiazarian

Abstract

View-plus-depth is a scene representation format where each pixel of a color image or video frame is augmentedby per-pixel depth represented as gray-scale image (map). In the representation, the quality of the depth mapplays a crucial role as it determines the quality of the rendered views. Among the artifacts in the received depthmap, the compression artifacts are usually most pronounced and considered most annoying. In this article, westudy the problem of post-processing of depth maps degraded by improper estimation or by block-transform-based compression. A number of post-filtering methods are studied, modified and compared for their applicabilityto the task of depth map restoration and post-filtering. The methods range from simple and trivial Gaussiansmoothing, to in-loop deblocking filter standardized in H.264 video coding standard, to more comprehensivemethods which utilize structural and color information from the accompanying color image frame. The lattergroup contains our modification of the powerful local polynomial approximation, the popular bilateral filter, and anextension of it, originally suggested for depth super-resolution. We further modify this latter approach bydeveloping an efficient implementation of it. We present experimental results demonstrating high-quality filtereddepth maps and offering practitioners options for highest-quality or better efficiency.

1 IntroductionView-plus-depth is a scene-representation format whereeach pixel of the video frame is augmented with depthvalue corresponding to the same viewpoint [1]. Thedepth is encoded as gray-scale image in a linear or loga-rithmic scale of eight or more bits of resolution. Anexample is given in Figure 1a,b. The presence of depthallows generating virtual views through so-called depthimage based rendering (DIBR) [2] and thus offers flex-ibility in the selection of viewpoint as illustrated in Fig-ure 1c. Since the depth is given explicitly, the scenerepresentation can be rescaled and maintained as toaddress parallax issues of 3D displays of different sizesand pixel densities [3]. The representation also allowsgenerating more than two virtual views which isdemanded for auto-stereoscopic displays.Another advantage of the representation is its back-

ward compatibility with conventional single-view broad-casting formats. In particular, MPEG-2 transport streamstandard used in DVB broadcasting allows transmittingauxiliary streams along with main video, which makes

possible to enrich a conventional digital video transmis-sion with depth information without hampering thecompatibility with single-view receivers.The major disadvantages of the format are the appear-

ance of dis-occluded areas in rendered views and inabil-ity to properly represent most of the semi-transparentobjects such as fog, smoke, glass-objects, thin fabrics,etc. The problems with occlusions are caused by thelack of information about what is behind a foregroundobject, when a new-perspective scene is synthesized.Such problems are tackled by occlusion filling [4] or byextending the format to multi-view multi-depth, or tolayered depth [3].Quality is an important factor for the successful utili-

zation of depth information. Depth map degraded bystrong blocky artifacts usually produces visually unac-ceptable rendered views. For successive 3D video trans-mission, efficient depth post-filtering technique shouldbe considered.Filtering of depth maps has been addressed mainly

from the point of view of increasing the resolution [5-7].In [6], a joint bilateral filtering has been suggested toupsample low-resolution depth maps. The approach hasbeen further refined in [7] by suggesting proper anti-

* Correspondence: [email protected] University of Technology, Korkeakoulunkatu 10, FI-33720, Tampere,Finland

Smirnov et al. EURASIP Journal on Advances in Signal Processing 2012, 2012:25http://asp.eurasipjournals.com/content/2012/1/25

© 2012 Smirnov et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative CommonsAttribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

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aliasing and complexity-efficient filters. In [5], a prob-abilistic framework has been suggested. For each pixelof the targeted high-resolution grid, several depthhypothesizes are built and the hypothesis with lowestcost is selected as a refined depth value. The procedureis run iteratively and bilateral filtering is employed ateach iteration to refine the cost function used for com-paring the depth hypotheses.In this article, we study the problem of post-proces-

sing of depth maps degraded by improper estimation or

by block-transform-based compression. A number ofpost-filtering methods are studied, modified, and com-pared for their applicability to the task of depth maprestoration and post-filtering. We consider methods ran-ging from simple and trivial smoothing and deblockingmethods to more comprehensive methods which utilizestructural and color information from the accompanyingcolor image frame. The present study is an extension ofthe study reported in [8]. Some of the methods includedin the comparative analysis in [8] have been further

(a)

(b) (c)

(d) (e)

Figure 1 Example of view-plus-depth image format and virtual view rendering (no occlusion filling applied for rendered images). (a)True color channel; (b) true depth channel; (c) synthesized view using true depth; (d) highly compressed depth (H.264 I-frame with QP = 51);(e) synthesized view using compressed depth from (d).


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modified and for one of them, a more efficient imple-mentation has been proposed. We present extendedexperimental results which allow evaluating the advan-tages and limitations of each method and give practi-tioners options for trading-off between highest qualityand better efficiency.

2 Depth map characteristics2.1 Properties of depth mapsDepth map is gray-scale image which encodes the dis-tance to the given scene pixels for a certain perspective.The depth is usually aligned with and ac-companies thecolor view of the same scene [9].Single view plus depth is usually a more efficient

representation of a 3D scene than two-channel stereo. Itdirectly encodes geometrical information containedotherwise in the disparity between the two views thusproviding scalability and possibility to render multipleviews for displays with different sizes [1]. Structure-wise,the depth image is piecewise smooth (as representinggradual change of depth within objects) with delineated,sharp discontinuities at object boundaries. Normally, itcontains no textures. This structure should be takeninto account when designing compression or filteringalgorithms.Having a depth map given explicitly along with color

texture, a virtual view for a desired camera position canbe synthesized using DIBR [2]. The given depth map isfirst inversely-transformed to provide the absolute dis-tance and hence the world 3D coordinates of the scenepoints. These points are projected then onto a virtualcamera plane to obtain a synthesized view. The techni-que can encounter problems with dis-occluded pixels,non-integer pixel shifts, and partly absent backgroundtextures, which problems have to be addressed in orderto successfully apply it [1].The quality of the depth image is a key factor for suc-

cessful rendering of virtual views. Distortions in thedepth channel may generate wrong objects contours orshapes in the rendered images (see, for example, Figure1d,e) and consequently hamper the visual user experi-ence manifested in headache and eye-strain, caused bywrong contours of familiar objects. At the capture stage,depth maps might be not well aligned with the corre-sponding objects. Holes and wrongly estimated depthpoints (outliers) might also exist. At the compressionstage, depth maps might suffer from blocky artifacts ifcompressed by contemporary methods such as H.264[10]. When accompanying video sequences, the consis-tency of successive depth maps in the sequence is anissue. Time-inconsistent depth sequences might causeflickering in the synthesized views as well as other 3D-specific artifacts [11].

At the capture stage, depth can be precisely estimatedin floating-point hight resolution, however, for compres-sion and transmission it is usually converted to integervalues (e.g., in 256 gray-scale gradations). Therefore, thedepth range and resolution have to be properly main-tained by suitable scaling, shifting, and quantizing,where all these transformations have to be invertible.Depth quantization is normally done in linear or loga-

rithmic scale. The latter approach allows better preser-vation of geometry details for closer objects, whilehigher geometry degradation is tolerated for objects atlonger distances. This effect corresponds to the parallax-based human stereo-vision, where the binocular depthcue losses its importance for more distanced objects andis more important and dominant for closer objects. Thesame property can be achieved if transmitting linearlyquantized inverse depth maps. This type of depth repre-sentation basically corresponds to binocular disparity(also known as horizontal parallax), including againnecessary modifications, such as scaling, shifting, andquantizing.

2.2 Depth map filtering problem formulationThis section formally formulates the problem of filteringof depth maps and specifies the notations used here-after. Consider an individual color video frame in YUV(YCbCr) or RGB color space y(x) = [yY(x), yU(x), yV(x)]or y(x) = [yR(x),yG(x),yB (x)], together with the asso-ciated per-pixel depth z(x), where x = [x1, x2] is a spatialvariable, x Î X, X being the image domain.A new, virtual view h(x) = [hY(x), hU(x), hV(x)] can be

synthesized out of the given (reference) color frame anddepth by DIBR, applying projective geometry andknowledge about the reference view camera, as dis-cussed in Section 2.1 [2]. The synthesized view is com-posed of two parts, h = hv + ho, where hv denotes thevisible pixels from the position of the virtual view cam-era and ho denotes the pixels of occluded areas. Thecorresponding domains are denoted by Xv and Xo corre-spondingly, Xv ⊂ X, Xo = X\Xv.Both y(x) and z(x) might be degraded. The degrada-

tions are modeled as additive noise contaminating theoriginal signal

yCq = yC + εC, (1)

zq = z+ ∈, (2)

where C = Y, U, V or R, G, B. Both degradations aremodeled as independent white Gaussian processes:εC(·)∼N(0, σ 2

C), ε(·)∼N(0, σ 2). Note that the varianceof color signal noise (σ 2

C) differs from the one of thedepth signal noise (s2).


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If degraded depth and reference view are used inDIBR, the result will be a lower-quality synthesized view�η. Unnatural discontinuities, e.g., blocking artifacts, inthe degraded depth image cause geometrical distortionsand distorted object boundaries in the rendered view.The goal of the filtering of degraded depth maps is tomitigate the degradation effects (caused by e.g., quanti-zation or imperfect depth estimation) in the depthimage domain, i.e., to obtain a refined depth image esti-mate z, which would be closer to the original, error-freedepth, and would improve the quality of the renderedview.

2.3 Depth map quality measuresMeasuring the quality of depth maps has to take intoaccount that depth maps are type of imagery which arenot visualized per-se, but through rendered views.In our study, we consider two types of measures:

• measures based on comparison between processedand ground truth (reference) depth;• measures based on comparison between virtualviews rendered from processed depth and fromground truth one.

Measures for the first group have the advantage ofbeing simple, while measures from the second group arecloser to subjective perception of depth. For both ofthese groups we suggest and test new measures.PSNR of Restored DepthPeak signal-to-noise ratio (PSNR) measures the ratiobetween the maximum possible power of a signal(within its range) and the power of corrupting noise.PSNR is commonly used as a measure of fidelity ofimage reconstruction. PSNR is calculated via the meansquared error (MSE):

MSE =1N

∑x

(z(x) − z(x))2, (3)

PSNR = 10log10

(MAX2

z

MSE

)(4)

where z(x) and z(x) are the reference and processedsignals; N is number of samples (pixels) and MAXz isthe maximal possible pixel value, assuming the minimalone is zero. In this metric higher value means betterquality. Applying PSNR to depth images must be donewith care and with proper rescaling, as most of depthmaps have a sub-range of the usual 8-bit range of 0 to255 and PSNR might turn to be unexpectedly high.PSNR of rendered viewPSNR is calculated to compare the quality of renderedview using processed depth versus that of using original

depth [10]. It essentially measures how close the ren-dered view is to the ‘ideal’ one. In our calculations, pix-els, dis-occluded during the rendering process, areexcluded so to make the comparison independent onthe particular hole fitting approach. For color images,we calculate PSNR independently for each color channeland then calculate the mean between three channels.Percentage of bad pixelsBad pixels percentage metric is defined in [12] to mea-sure directly the performance of stereo-matching algo-rithms.

BAD =100M

∑x

(∣∣z(x) − z(x)∣∣ > �d

),

where z is the computed depth, z is the true depthand Δd is a threshold value, (usually equal to 1). Figure2 shows thresholding results for some highly com-pressed depth maps. We include this metric to ourexperiments in an attempt to check its applicability forcomparing post-filtering methods. For this metric, lowervalue means better quality.Depth consistencyAnalysing the BAD metric, one can notice that thethresholding imposed there, does not emphasize theimportance of small or big differences. It is equallyimportant, when the error is just a quantum above thethreshold and when it is quite high.In a case of depth degraded by compression artifacts,

almost all pixels are quantized thus changing their origi-nal values and therefore causing the BAD metric toshow very low quality while the quality of the renderedviews will not be that bad.Starting from the idea that the perceptual quality of

rendered view will depend more on the amount of geo-metrical distortions than on the number of bad depthpixels, we suggest to give preference to areas where thechange between ground truth depth and compresseddepth is more abrupt. Such changes are expected tocause perceptually high geometrical distortions.Consider the gradient of the difference between true

depth and approximated depth ∇ξ = ∇(z − z). By depthconsistency we denote the percentage of pixels, havingmagnitude of that gradient higher than a pre-specifiedthreshold.

CONSIST =100N

∑(‖∇ξ‖2 > σconsist) . (5)

The measure favors non-smooth areas in the restoreddepth considered as main source of geometrical distor-tion, as illustrated in Figure 3.Gradient-Normalized RMSEAs suggested in [13], the performance of optical flowestimation algorithms can be evaluated using gradient-


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normalized RMSE metric. Such measure decreases theover-penalization of errors caused by fine textures.In our implementation, we calculate this metric for

the luminance channels of reference and rendered viewsand exclude occluded areas determined by the DIBR onthe ground truth data.

NRMSEη =

⎡⎣∑

x∈Xv

(ηY(x) − ηY(x)

)2∥∥∇ηY(x)∥∥2 + 1

⎤⎦

1/2

, (6)

where hY(x) is the luminance of the virtual image gen-erated by ground truth depth and η

Y(x) is the luminanceof virtual image generated by processed depth. For bet-ter quality, the metric shows low values.Discontinuity FalsesWe propose using a measure based on counting ofwrong occlusions in the view rendered out of processeddepth. If all occlusions between true and processed vir-tual images coincide, then depth discontinuities are pre-served correctly.

DISC =100N

#((

Xo ∪ Xo

)\(Xo ∩ Xo

)), (7)

where #(X) is cardinality (number of elements) of adomain X. The measure decreases with improving thequality of the processed depth.

3 Depth filtering approachesA number of post-processing approaches for restorationof natural images exist [14]. However, they are notdirectly applicable to range images due to differences inimage structure.In this section, we consider several existing filtering

approaches and modify them for our need. First groupof approaches works on the depth map images withusing no structural information from the available colorchannel. Gaussian smoothing and H.264 in-loopdeblocking filter [15] are the filtering approachesincluded in this group. The approaches of the secondgroup actively use available color frame to improvedepth map quality. While there is an apparent

(a) (b)

Figure 2 BAD pixels mask for “Cones” dataset(∣∣z(x) − z(x)

∣∣ > �d)caused by H.264 Intra compression with QP = 51.

(a) (b)

Figure 3 Distortions in depth “Teddy” dataset (||∇ξ||2 > sconsist) caused by H.264 Intra compression with (a) QP = 51 and the sameafter de-blocking with super-resolution filter (b).


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correlation between the color channel and the accompa-nying depth map, it is important to characterize whichcolor and structure information can help for depthprocessing.More specifically, we optimize state-of-the-art filtering

approaches, such as local polynomial approximation(LPA) [16] and bilateral filtering [17] to utilize edge-pre-serving structural information from the color channelfor refining the blocky depth maps. We suggest a newversion of the LPA approach which, according to ourexperiments, is most appropriate for depth map filtering.In addition, we suggest an accelerated implementationof the method based on hypothesis filtering as in [5],which shows superior results for the price of high com-putational cost.

3.1 LPA approachThe anisotropic LPA is a pixel-wise method for adaptivesignal estimation in noisy conditions [16]. For each pixelof the image, local sectorial neighborhood is con-structed. Sectors are fitted for different directions. In thesimplest case, instead of sectors, 1D directional esti-mates of four (by 90 degrees) or eight (by 45 degrees)different directions can be used. The length of each sec-tor, denoted as scale, is adjusted to meet the compro-mise between the exact polynomial model (low bias)and sufficient smoothing (low variance). A statistical cri-terion, denoted as intersection of confidence intervals(ICI) rule is used to find this compromise [18,19], i.e.,the optimal scale for each direction. These optimalscales in each direction determine an anisotropic star-shape neighborhood for every point of the image welladapted to the structure of the image. This neighbor-hood has been successfully utilized for shape-adaptivetransform-based color image de-noising and de-blurring[14].In the spirit of [14], we use the quantized luminance

channel yYq = yY + εY as source of structural information.The image is convolved with a set of 1D directional

polynomial kernels{ghj,θk

}, where {h}Jj=1 is the set of dif-

ferent lengths (scales) and θk = kπ

4, k = 1, 2, . . . , 8 are

the directions, thus obtaining the estimates

yhj,θk(x) =(yYq ∗ ghj,θk

)(x). The ICI rule helps to find the

optimal scale h+(x) for each direction (the notation ofdirection is omitted). This is the largest scale (in num-ber of pixels), which ensures a non-empty ICI[18,19]Ji = ∩j

i=1Ii where

Ii =[yhi(x) − σ Y

∥∥ghi∥∥ , yhi(x) + σ Y∥∥ghi∥∥]

. (8)

After finding optimal scales h+(x) for each direction atpixel x0 a star shape neighborhood �x0 is formed, as

illustrated in Figure 4a. There is a clear evidence thatthere is a relation between the adaptive neighborhoodsand the (distorted) depth, as examplified in Figure 4b.Adaptive neighborhoods are formed for every pixel inthe image domain X. Once adaptive neighborhoods arefound, one must find some modeling for depth channelbefore utilizing this structural information.Constant depth modelFor our initial implementation of LPA-ICI depth filter-ing scheme, the depth model is rather simple. Thedepth map is assumed to be constant in the neighbor-hood of the filtering pixel x0, where neighborhood isfound by the LPA-ICI procedure. This modeling isbased on the assumption that the luminance channel isnearly planar at areas where the depth is smooth(close to constant). Whenever the depth has a disconti-nuity, the luminance is most likely to have a disconti-nuity too. The constant-modeling results in simpleweighted average over the region of optimal neighbor-hood

∀x0, ∃�x0 , zq(x) ≈ const, x ∈ �x0 , (9)

z(x0) =1N

∑x∈�x0

zq(x), (10)

where N is the number of pixels inside adaptive sup-port �x0. Note, that the scheme depends on two para-meters: the noise variance of the luminance channel sY

and the positive threshold parameter Γ. The latter canbe adjusted so to control the smoothing in restoreddepth map.Linear regression depth modelIn a more sophisticated approach we apply pixelwise-planar depth assumption, stating of planarity of depthinside some neighborhood of processing pixel. This is ahigher order extension of the previous assumption.

∀x0, ∃�x0 , zq(x) ≈ Ax, x = [x, 1], x ∈ �x0 , (11)

where x is homogeneous coordinate.Based on this assumption, instead of simple averaging

in depth domain we apply plane fitting (linear regres-sion). A = dB-1, where d is a row-vector of depth valuesz(x), x ∈ �x0, B is a 3-by-N matrix of their homogeneouscoordinates in image space and B-1 is Moore-Penrosepseudoinverse of rectangular matrix. Estimated depthvalues are found with a simple linear equation:

z(x) = Ax, x = [x, 1], x ∈ �x0 . (12)

Aggregation procedureSince for each processed pixel we may have multipleestimates due to overlapping neighborhoods, we aggre-gate them as follows:


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zagg(x0) =1M

∑j=1..M

zj(x0), (13)

where M is number of estimates coming from overlap-ping regions in particular coordinate x0. A result ofdepth, filtered by LPA-ICI is given in Figure 4e.Color-driven LPA-ICILuminance channel of the color image is usually consid-ered as the most informative channel for processing andalso as the most distinguishable by the human visualsystem. That is why many image filtering mechanismsuse color transformation to extract luminance and thenprocess it in different way to compare with chrominancechannels. This also may be explained by the fact thatluminance is usually the less noisy component and thusit is most reliable. Nevertheless, for some color proces-sing tasks pixel differentiation based only on luminancechannel is not appropriate due to some colors may havethe same luminance whereas they have different visualappearance.Our hypothesis is that a color difference signal will

better differentiate color pixels, as illustrated in Figure5. L2 norm is used to form a color difference maparound pixel x0:

Cx0x =

√(Yx0 − Yx)

2 + (Ux0 − Ux)2 + (Vx0 − Vx)

2, (14)

where x0 is the currently processing pixel, and x ∈ �x0.The color difference map is used as a source of struc-

tural information, i.e., the LPA-ICI procedure is runover this map instead over the luminance channel. Dif-ferences are illustrated in Figure 5. In our implementa-tion, we calculate color-difference only for those pixelsof the neighborhood which participate in 1D directionalconvolutions. Additional computational cost of suchimplementation is about 10% of the overall LPA-ICIprocedure.For all mentioned LPA-ICI based strategies the main

adjusting parameter, capable to set proper smoothingfor varying depth degradation parameter (e.g., varyingQP in coding) is the parameter Γ.3.1.1 Comparison of LPA-ICI approaches The perfor-mance of different versions of the LPA-ICI approach arecompared in Figure 6. The ‘normalized RMSE’ (equation6) and ‘depth consistency’ (equation 5) metrics havebeen computed and averaged over a set of test images.The parameters of the filters were empirically optimizedwith ‘depth consistency’ (equation 5) as a cost measure.As it can be seen, the color-driven LPA-ICI approach

(a) (b) (c)

(d) (e)

Figure 4 Example of adaptive neighborhoods: (a) luminance channel with some of found optimal neighborhoods; (b) compressed depthwith the same neighborhoods overlaid; (c) optimal scales for one of the direction (black for small scale and white for big scale); (d) example ofhighly compressed depth map; (e) the same depth map filtered with LPA-ICI (constant depth model with aggregation).


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with plane fitting and encapsulated aggregation is thebest performing approach, while also having the moststable and consistent results. Because of the superiorperformance of color-driven LPA-ICI, we use it in the

experiments from now on. All experiments and compar-isons involving LPA-ICI presented in the following sec-tions refer to the optimized color-driven LPA-ICIimplementation.

(a) (b)

(c) (d)

Figure 5 Example of different LPA-ICI implementations: (a) luminance channel; (b) color-difference channel for central pixel of a red square;(c) LPA-ICI filtering result, optimal scales were found in luminance channel; (d) LPA-ICI filtering result, optimal scales were found in color-difference channel.

20 25 30 35 40 45 500

1

2

3

4

5

6

H.264 Quantization Parameter

Dep

th C

onsi

sten

cy (%

)

No FilteringLPA−ICI Constant ModelLPA−ICI Linear RegressionLPA−ICI Color Diff

20 25 30 35 40 45 500.02

0.03

0.04

0.05

0.06

0.07

0.08


Nor

mal

ized

RM

SE

(dB

)

No FilteringLPA−ICI Constant ModelLPA−ICI Linear RegressionLPA−ICI Color Diff

Figure 6 Comparison of different LPA-ICI approaches.


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3.2 Bilateral filterThe bilateral filter is a non-linear filter which smoothsthe image while preserves strong edges [17]. Filteredpixel value is obtained by weighted averaging of itsneighborhood combined with color weighting. For gray-scale images, filter weights are calculated based on bothspatial distance and photometric similarity, favoringnear values to distant values in both spatial domain andrange. For color images, bilateral filtering uses color dis-tance to distinguish photometric similarity between pix-els, which affects in reducing phantom colors in theresulting image. In our approach, we calculate filterweights using information from color frame in RGB,while applying filtering on depth map. Our design ofbilateral filter has been inspired by [5], as follows:

z(x) =

∑u ωs (‖x − u‖) ωc

(∥∥y(x) − y(u)∥∥)

zq(u)∑u ωs (‖x − u‖) ωc

(∣∣y(x) − y(u)∣∣) , (15)

whereωa(t) = e

−tγa , (a = s, c)

and u Î Ωx are neigh-

borhood pixels of point x. This design allows for rela-tively fast implementation by storing all possible colorand distance weights as look-up tables. Parameters gs, gcand processing window size Ωx are adjustable para-meters of the filter. Figure 7 illustrates the filtering. Thecolor channel (Figure 7a) provides the color differenceinformation, with respect to the processed pixel position(Figure 7b). It is further weighted by spatial Gaussian fil-ter to determine the weights of pixels from the depthmap taking part in estimating the current (central) pixel(Figure 7f).

3.3 Spatial-depth super resolution approachA post-processing approach was suggested aimed atincreasing the resolution of low-resolution depth images,given high-resolution color image as a reference [5]. Inour study, we study the applicability of this filter forsuppression of compression artifacts and restoration oftrue discontinuities in the depth map. The main idea ofthe filter is to process depth in probabilistic manner,constructing 3D cost volume from several depthshypothesizes. After bilateral filtering of each slice of thevolume, the hypothesis with the lowest cost is selectedas a new depth value. The procedure is applied itera-tively, calculating cost volume using the depth estimatedin previous step. The cost volume on ith iteration isconstructed to be quadratic function of the currentdepth estimate:

C(i)(x, d) = min{Lη,

(d − z(i)(x)

)2} , (16)

where Lh denotes tunable search range.

The bilateral filtering, defined as in Equation 15enforces an assumption of piecewise smoothness. Theprocedure is illustrated in Figures 8 and 9. Theapproach resembles the local depth estimation idea,where a volumetric cost volume is further aggregatedwith bilateral filter.Since cost function is discrete on d, the depth

obtained by winner-takes-all approach will be discrete aswell. To tackle this effect, the final depth estimate istaken as the minimum point of quadratic polynomialwhich approximates the cost function between three dis-crete depth candidates: d, d - 1 and d + 1

f (d) = ad2 + bd + c, (17)

dmin = − b2a

. (18)

f (dmin) is the minimum of quadratic function f(d),thus given d, f(d), f(d -1) and f(d + 1), value dmin can becalculated:

dmin = d − f (d + 1) − f (d − 1)2(f (d + 1) − f (d− 1) − 2f (d)

. (19)

After the bilateral filtering is applied to the costvolume, the depth is refined and true depth discontinu-ities might be completely recovered.In our implementation of the filter, we have suggested

two simplifications:

• we use only one iteration of the filter;• before processing we scale the depth range by fac-tor of 20, thus reducing the number of slices, andsubsequently reducing the processing time.

The main tunable parameters of the filter are theparameters of the bilateral filter gd and gc. As long asthe processing time of the filter still remains extremelyhigh, we do not perform optimization of this filterdirectly, but assume that the optimal parameters gd = fd(QP) and gc = fc(QP) found for the direct bilateral filterare optimal or nearly optimal for this filter as well.

3.4 Practical implementation of the super resolutionfilteringIn this section, we suggest several modifications to theoriginal approach to make it more memory-efficient andto improve its speed. It is straightforward to figure outthat there is no need to form cost volume in order toobtain the depth estimate for a given coordinate x atthe ith iteration. Instead, the cost function is formed forthe required neighborhood only and then filteringapplies, i.e.,


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z(i+1)(x) = argmind

[∑u∈�x

W(x, u)G(u, d)∑u∈�x

W(x, u)

],

W(x, u) = ωs (‖x − u‖) ωc(∥∥y(x) − y(u)

∥∥),

G(u, d) = min{η ∗ L,

(d − z(i)(u)

)2}.

(20)

Furthermore, the computation cost is reduced byassuming that not all depth hypotheses are applicablefor the current pixel. A safe assumption is that onlydepths within the range d Î [dmin, dmax] where dmin =min(z(u)), dmax = max(z(u)), u Î Ωx have to be checked.Additionally, depth range is scaled with the purpose to

further reduce the number of hypothesizes. This step is

especially efficient for certain types of distortions suchas compression (blocky) artifacts. For compressed depthmaps, the depth range appears to be sparse due to thequantization effect.Figure 10 illustrates histograms of depth values before

and after compression so to confirm the use of rescaledsearch range of depth hypotheses. This modificationspeeds up the procedure and relies on the subsequentquadratic interpolation to find the true minimum. Apseudo-code of the suggested procedure in Equation 20is given in following listing.Require: C, the color image; D, the depth image; X, a

spatial image domain

(a) (b)

(c) (d)

(e) (f)

Figure 7 Example of bilateral filtering: (a) color channel; (b) color-difference for central pixel (marked red); (c) color-weighed component ofbilateral product; (d) complete weights for selected window ((c) multiplied with spatial Gaussian component); (e) example of blocky depth mapfor the same scene; (f) the same block filtered with bilateral filter.


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for all x Î X dodmin = min

u∈�x

Du, dmax = maxu∈�x

Du

if dmax - dmin <gthr thenDx = Dx

else

F(x, u) =‖Cu − Cx‖ ‖u − x‖∑u ‖Cu − Cx‖ ‖u − x‖{bilateral weights}

Sbest ¬ Smax {Smax is maximum reachable valuefor S}

Figure 8 Principle of super-resolution filtering.

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Figure 9 Super-resolution filtering example. (a) Compressed depth map; (b) same depth, processed by super-resolution filter; (c) single costvolume slice, constructed from (a); (d) result of bilateral filtering for same slice.


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for d = ⌊dmin⌋ to [dmax] doS ¬ 0for all u Î Ωx doE ¬ min{(d - Du)

2, hL}S ¬ S + F(x, u) * E

end forif S <Sbest thenSbest ¬ Sdbest ¬ d

end ifend for

Dx = dbestend if

end forThe memory foot-print required by our implementa-

tion is significantly lower than the one imposed by adirect implementation. A straightforward implementa-tion would require a large memory buffer to store thecomplete cost volume in order to process it pixel-by-pixel and avoid computing (the same) color weightsacross different slices. In the proposed implementation,two memory buffers with relatively low sizes arerequired: a memory buffer which is equal to the proces-sing window size to store current color weights, and abuffer to store the cost values for the current pixelalong the ‘d’ dimension. In case of multi-thread (paralle-lized) implementation, these memory buffers are multi-plied by the number of processing threads. Moreinformation about platform-specific optimization of theproposed algorithm is given in [20].Figure 11 illustrates the performance in terms of

speed. The figure shows experiments with different

implementations of the filtering procedure. The ‘Teddy’dataset has been processed (see also Figure 12 for refer-ence). All filter versions have been implemented in Cand then compiled into MEX files to be run withinMatlab environment. The experiments have been runon a 1.3 GHz Pentium Dual-Core processor with 1 Gbof RAM under MS Windows XP operating system. Inthe figure, the vertical axis shows the execution time inseconds and the horizontal line shows the number ofslices processed (i.e., the depth dynamic range assumed).The dotted curve shows single-pass bilateral filtering. Itdoes not depend on the dynamic range, but on the win-dow size, thus it is a constant in the figure. The red lineshows the computational time for the original approachimplemented as a three step procedure for the fulldynamic range. Naturally, it is a linear function withrespect to the slices to be filtered. Our implementation(blue curve) applying a reduced dynamic range is alsolinearly depending on the number of slices, but withdramatically reduced steepness.

4 Experimental results4.1 Experimental settingIn our experiments, we consider depth maps degradedby compression. Thus degradation is characterized bythe quantization parameter (QP). For better comparisonof selected approaches, we present two types of experi-ments. In the first set of experiments, we compare theperformance of all depth filtering algorithms assumingthe true color channel is given (it has been also used inthe optimization of the tunable parameters). This showsideal filtering performance, while in practice it cannot

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be achieved due to the fact that the color data is alsodegraded by e.g., compression.In the second set of experiments, we compare the

effect of depth filtering in the case of mild quantizationof the color channel. General assumption is that colordata is transmitted with backward compatibility inmind, and hence most of the bandwidth is occupied bythe color channel. Depth maps in this scenario are heav-ily compressed, to consume not more than 10-20% ofthe total bit budget [21,22].We consider the case where both y and z are to be

coded as H.264 intra frames with some QPs, whichleads to their quantized versions yq and zq. The effect ofquantization of DCT coefficients has been studied thor-oughly in the literature and corresponding models havebeen suggested [23]. Following the degradation model inSection 2.2, we assume quantization noise terms addedto the color channels and the depth channel consideredas independent white Gaussian processes:εC(·)∼N(0, σ 2

C), ε(·)∼N(0, σ 2). While this modeling issimple, it has proven quite effective for mitigating theblocking artifacts arising from quantization of transformcoefficients [14]. In particular, it allows for establishinga direct link between the QP and the quantization noise

variance to be used for tuning deblocking filtering algo-rithms [14].Training and test datasets for our experiments (see

Figure 12) were taken from Middlebury Evaluation Test-bench [12,24,25]. In our case, we cannot tolerate holesand unknown areas in the depth datasets, since theyproduce fake discontinuities and unnatural artifacts aftercompression. We semi-manually processed 6 images tofill holes and to make their width and height be multi-ples of 16.4.1.1 Parameters optimizationEach tested algorithm has a few tunable parameterswhich could be modified according particular filteringstrategy related with a quality metric. So, to make com-parison as fair as possible, we need to tune each algo-rithm to its best, according such a strategy and withincertain range of training data.Our test approach is to find empirically optimal para-

meters for each algorithm over a set of training images.It is done separately for each quality metric. Then, foreach particular metric we evaluate it once more on theset of test images and then average. Then comparisonbetween algorithms is done for each metricindependently.

Figure 11 Execution time of different implementations of filtering approach.


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Particularly, for bilateral filtering and hypothesis(super-resolution) filtering we are optimizing the follow-ing parameters: processing window size, gs and gc. Forthe Gaussian Blurring we are optimizing parameters sand processing window size. For LPA-ICI basedapproach we are optimizing the Γ parameter.

4.2 Visual comparison resultsFigures 13 and 14 present depth images paired withconsecutive rendered frames (no occlusion filling is

applied). This approach helps to illustrate artifacts inthe depth channel as well as their effect on the renderedimages.As it is seen in the top row (a), rendering with true

depth, results in sharp and straight object contours, aswell as in continuous shapes of occlusion holes. Forsuch holes, a suitable occlusion filling approach will pro-duce good estimate.Row (b) shows unprocessed depth after strong com-

pression (H.264 with QP = 51) frame and its rendering

Figure 12 Training (upper row) and testing (lower row) datasets used in study.


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capability. Objects edges are particularly affected byblock distortions.With respect to occlusion filling, the methods behave

as follows.

• Gaussian smoothing of depth images is able toreduce number of occluded pixels, making occlusionfilling simpler. Nevertheless, this type of filtering

does not recover geometrical properties of depth,which results in incorrect contours of the renderedimages.• Internal H.264 in-loop deblocking filtering wasperformed similarly to the Gaussian smoothing, withno improvement of geometrical properties.• LPA-ICI based filtering technique performs signifi-cantly better both is sense of depth frame and

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Figure 13 Visual results for “Teddy” dataset. Left column: depth, right column: respective rendered result (no occlusion filling applied). (a)Ground truth depth; (b) depth compressed H.264 Intra with QP = 51; (c) loop-filtered depth; (d) Gaussian-filtered depth.


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rendered frame visual quality. Geometrical distor-tions are less pronounced, however, still visible inrendered channel.• Bilateral filter almost recovers the sharp edges indepth image, while has minor artifacts (for instance,see chimney of house).• Super-resolution depth filter recovers discontinu-ities as good as bilateral or even better. Resulteddepth image does not have artifacts as in the pre-vious methods. Geometrical distortions in renderedimage are not pronounced.

Among all filtering results, the latter one containsocclusions which are most similar to the occlusions ofthe original depth rendering result. Visually, super-reso-lution depth approach is considered to be the best. Thenumerically estimated results for all presentedapproaches are presented in following section.

4.3 Numerical results for ideal color channelFigure 15 summarizes averaged results over the threetest datasets: ‘venus’, ‘sawtooth’, and ‘teddy’.X-axis on all the plots represents varying QP para-

meters of the H.264 Intra coding, while each Y -axisshows a particular metric. On the most of the metricplots it is visible that there is no need to apply any kindof filtering before QP reaches some critical value. Beforethat value, the quality of the compressed depth is highenough, so no filtering could improve it.The group of structurally-constrained methods clearly

outperforms the simple methods working on the depthimage only. The two PSNR-based metrics and the BADmetric seem to be less reliable in characterizing the perfor-mance of the methods. The three remained measures,namely depth consistency, discontinuity falses and gradi-ent-normalized RMSE perform in a consistent manner.While Normalized RMSE is perhaps the measure closest

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(b)

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Figure 14 Visual results for “Teddy” dataset (continued). (a) LPA-ICI filtered depth; (b) bilateral filtered depth; (c) super-resolution (modifiedimplementation) filtered.


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to the subjective perception, we favor also the other twomeasures of this group as they are relatively simple and donot require calculation of the warped (rendered) image.

4.4 Numerical results for compressed color channelSo far, we have been working with uncompressed colorchannel. It has been involved in the optimizations and

comparisons. Our aim was to characterize the pureinfluence of the depth restoration only.In practice, when ‘color-plus-depth’ frame is com-

pressed and then transmitted over a channel, the colorframe is also compressed with a pre-specified bit-rate,aiming at maximizing visual quality of the video. Trans-mission of ‘color plus depth’ stream has also to be

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constrained within a given bit-budget. Thus, receiver-side device has to cope with compressed color and com-pressed depth.In the second experiment, we assume mild quantiza-

tion of the color image, e.g., by QP = 30. For our testimagery, the first depth QP corresponds to about 10%of the total bit-rate. ‘Depth consistency’, ‘Discontinuityfalses’ and ‘PSNR of rendered channel’ are calculated fordifferent depth maps: compressed, post-filtered withLPA-ICI filtering approach, post-processed with thebilateral filter and post-filtered with our implementationof the super-resolution approach. The resulting numbersare averaged over three dataset images. Visual results ofhypothesis filtering are presented in Figure 16 whichshows comparison between higly-compressed depth fil-tered with compressed color (second row) and same, fil-tered with ideal color (last row). The numerical results

are given in Figures 17, 18, and 19. Cases with post-pro-cessed depth are marked with color. One can see thatthe depth postprocessing clearly makes a differenceallowing to use stronger quantization of the depth chan-nel and still to achieve good quality.

5 ConclusionsIn this article, the problem of filtering of depth mapswas addressed and the case of processing of depth mapimages impaired by compression artifacts wasemphasized.Before proceeding with the actual depth processing

task, the characteristics of the representation view-plus-depth were overviewed, including methods of depthimage based rendering for virtual view generation, andformulation of the depth map filtering problem. In addi-tion, number of quality measures for evaluating the

Figure 16 Visual results of experiments with compressed color channel (testing dataset). First row: compressed color channels, secondrow: depth filtered with hypothesis filter using compressed color channel, third row: depth filtered with hypothesis filter using true colorchannel.


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depth quality were studied and new ones weresuggested.For the case of post-filtering of depth maps impaired

by compression artifacts, a number of filteringapproaches were studied, modified, optimized, and com-pared. Two groups of approaches were underlined. Inthe first group, techniques working directly on thedepth map and not taking into account the accompany-ing color frame were studied. In the second group, fil-tering techniques utilizing structural or colorinformation from the accompanying frame were consid-ered. This included the popular bilateral filter as well asits extension based on probabilistic assumptions and ori-ginally suggested for super-resolution of depth maps.Furthermore, the LPA-ICI approach was specificallymodified for the task of depth filtering and a few ver-sions of this approach were proposed. The techniquesfrom the second group have shown better performanceover all measures used. More specifically, the method

based on probabilistic assumptions showed superiorresults for the price of very high computational cost. Totackle this problem, we have suggested practical modifi-cations leading to faster and higher memory-efficientversion which adapts to the true depth range and itsstructure and is suitable for implementation on a mobileplatform. The competitive methods, i.e., LPA-ICI andbilateral filtering, should not be, however, discarded asfast implementations of those do exist as well. Theydemonstrated competitive performance and thus form ascalable set of algorithms. Practitioners can choosebetween the algorithms in the second group of methodsdepending on the requirements of their applications andavailable computational resources. The de-blocking testsdemonstrated that it is possible to tune the filteringparameters depending on the QP of the compressionengine. It is also feasible to allocate really small fractionof the total bit budget for compressing the depth, thusallowing for high-quality backward compatibility and

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channel fidelity. The price for this would be some addi-tional post-processing at the receiver side.

Competing interestsThe authors declare that they have no competing interests.

Received: 6 June 2011 Accepted: 14 February 2012Published: 14 February 2012

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doi:10.1186/1687-6180-2012-25Cite this article as: Smirnov et al.: Methods for depth-map filtering inview-plus-depth 3D video representation. EURASIP Journal on Advances inSignal Processing 2012 2012:25.

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