block dct to wavelet transcoding in transform domain
TRANSCRIPT
SIViP (2012) 6:179–195DOI 10.1007/s11760-011-0259-z
ORIGINAL PAPER
Block DCT to wavelet transcoding in transform domain
Viswanath Kapinaiah · Jayanta Mukherjee ·Prabir Kumar Biswas
Received: 17 March 2009 / Revised: 15 August 2011 / Accepted: 16 August 2011 / Published online: 8 September 2011© Springer-Verlag London Limited 2011
Abstract The Discrete Cosine Transform (DCT) to wave-let transcoding provides input for several wavelet-basedpost-processing techniques of the DCT-coded image/videosignals. Transcoding in domain transform avoids inversetransform and retransform operations and saves computa-tion. In this paper, we propose a new technique for transcod-ing the DCT blocks to wavelet coefficients directly in thetransform domain. We perform filtering, IDCT and down-sampling operations in a single combined step. The proposedtechnique achieves the same computational result as that of aspatial domain technique. The transcoding matrices used inthe proposed technique are found to satisfy certain symmetricand sparse properties, which are exploited to reduce the com-putational cost. As the number of zeros in the DCT coeffi-cients is significantly higher compared to the spatial domain,computational cost reduces significantly. Also, with the pro-posed technique, it is possible to speedup the operation byignoring some elements in the filtering matrices whose mag-nitudes are smaller than a threshold value. We demonstratethe application of the proposed transcoding for deblockingof the DCT-coded images in wavelet domain.
V. Kapinaiah (B)Department of Telecommunication Engineering,SIT, Tumkur, Indiae-mail: [email protected]; [email protected]
J. MukherjeeDepartment of Computer Science and Engineering,IIT, Kharagpur, Indiae-mail: [email protected]
P. K. BiswasDepartment of Electronics and Electrical Communication Engineering,IIT, Kharagpur, Indiae-mail: [email protected]
Keywords Block DCT · DWT · Transcoding ·Transform domain filtering · Deblocking
1 Introduction
The Discrete Cosine Transform (DCT) [2] and the DiscreteWavelet Transform (DWT) [1,14,29,30] are the two impor-tant transforms used in image/video processing applications.The DWT has been adopted by the compression standardJPEG2000 [22]. The DWT is also used in the compressionalgorithms such as Set Partitioning In Hierarchical Trees(SPIHT) [25] and Embedded Zerotree Wavelet (EZW) [26].On the other hand, the DCT is used in the JPEG [37],MPEG-2 and H.263 standards due its nice decorrelation andenergy compaction properties [23]. The DCT-based codersare still widely used and are shared by a wide range ofreceivers accommodating heterogeneous services together.One of the reasons for its popularity is that the implemen-tation of the DCT hardware (or software) is less expensivethan that of the DWT [42]. Besides, efficient algorithms inthe DCT domain make data processing advantageous for sev-eral image processing applications. Transcoding of waveletcoefficients to DCT coefficients and vice versa empowers theinter-operability between the services based on these trans-forms. There are other motivations behind this transcodingoperation:
• To achieve efficient bandwidth and buffer utilization bywavelet coding as well as adding flexibility to the DCT-based services [10,21,35].
• To provide an interface with this heterogeneous trans-coding to several wavelet-based post-processing tech-niques when the inputs are in the form of DCT-codedimage/video signals.
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If the transcoding is provided as a network service betweenthe service provider and consumer, one unit is expected toprovide service for many users. The design of transcoderin the transform domain is of importance as it avoids fulldecompression and recompression operations. The basic ideaof transform domain processing (also called as compresseddomain processing) is to convert spatial domain operationsinto their equivalent in the transform domain. In this case,computations are carried out solely with the coefficients rep-resenting an image without converting them into the spatialrepresentation.
Interestingly, very little work has been done in the areaof DCT to wavelet transcoding and vice versa. This may bedue to the apparent absence of a direct relationship betweenthe DCT and the DWT. However, there are a few techniquesadvanced toward this. The authors in [10] proposed a progres-sive transcoder, which computes the DCT coefficients froma coarsely decoded image by inverting the wavelet coeffi-cients. They rearranged the computed DCT coefficients toobtain the final DCT blocks. In [31,36], the authors usedthe DCT domain doubling [6] for transcoding the lower-frequency wavelet subband. This method resulted in a poorreconstruction of images even though it reduces computa-tional complexity to a large extent. To improve the qualityof reconstruction, wavelet synthesis operation is performedin the block DCT domain [32,33,35]. In [35], the authorsproposed new algorithms for transcoding the DWT coeffi-cients into DCT coefficients by adopting filtering techniquesin the DCT domain. They proposed an equivalent computa-tion of the spatial domain operations of the IDWT followedby the DCT, using the convolution–multiplication propertiesof DCTs [16]. The approach proposed in [32,33] distinctlydiffers from the previous approach [35] by adopting a differ-ent filtering technique [12,17,43] in the block DCT space. Inthis case, Type-II DCTs of convolution matrices are directlyused in the computation. This computation is efficiently per-formed by coupling the upsampling operation, as requiredfor synthesizing the wavelet coefficients. It may be noted thatin [35], the Type-I DCTs (refer [16]) of convolution matriceswere used in the process of filtering. However, it is observedthat both the approaches are computationally equivalent andprovide same results.
Chong and Kim [4] reported a transcoding techniquefor converting the DCT coefficients to wavelet coefficientsusing the transform domain filtering (TDF) [13]. This workpre-computes the product of the matrices corresponding tothe IDCT with the convolution (filtering) matrices. Aftercomputing the filtered blocks, the wavelet coefficients areobtained by downsampling the filtered blocks in the spatialdomain. In this technique [4], contributions of the adjacentblocks are also taken care of to compute the result. In wavelettranscoding, only half the samples are to be computed. Sincethis technique performs downsampling in the spatial domain
after filtering, redundant samples are also computed in thetranscoding. These samples are subsequently discarded in thedownsampling stage. Hence, the computational efficiency ofthis approach depends only on the sparseness of the inputDCT blocks.
In this paper, we propose algorithms for obtaining theDWT subbands from the DCT blocks directly in the trans-form domain. This work is the reverse transcoding operationof the works reported in [32,33,35]. The technique is basedon the combined step of filtering [34] and downsamplingdirectly in the block DCT domain. The combined oper-ation of filtering, IDCT and downsampling significantlyreduces the computational requirements. While filtering,both approaches based on the convolution–multiplicationproperty [16] and linear property of convolution [43] in theDCT domain are explored for the given task. In this trans-coding technique, the filtering is performed by the waveletanalysis filter bank. It is observed that the resulting trans-coding matrices are sparse and require less number of oper-ations. The proposed technique achieves the same results asthose obtained from a spatial domain transcoder, where DCTblocks are first transformed into spatial domain, followed bythe wavelet analysis using filter bank. The effect of quantiza-tion on the transcoded subbands is also extensively studied.By exploiting the sparseness in the block DCT coefficients,the proposed techniques further reduce the cost of transcod-ing (by trading off the quality of transcoded images).
The paper is organized as follows. In the next section,we briefly explain the wavelet transforms and arrangementof the filter coefficients for perfect reconstruction. Section 3reviews definition of DCT/IDCT and some of the propertiesof the DCTs. This section also describes operations such asdownsampling, composition/decomposition and symmetricextension in the block DCT domain. Section 4 describes fil-tering in the DCT domain followed by its application towardthe proposed transcoding. Section 5 presents computationalcomplexity of the proposed transcoding techniques. Subse-quently, in Sect. 6, results of the proposed techniques and spa-tial transcoding technique are presented. It also demonstratesthe application of the proposed technique for deblocking ofthe DCT-coded images in the wavelet domain.
2 The DWT and arrangement of filter coefficients
Wavelet transforms are computed using two-channel anal-ysis and synthesis filter banks. A simple one-stage imple-mentation of the DWT is illustrated in Fig. 1a, and it can beperformed using FIR filters. A one-stage DWT requires twoanalysis filters, h(n) and g(n) followed by downsamplingoperation. Each pair of filters correspond to a lowpass and ahighpass filter, respectively. If the combination of these fourfilters satisfies certain properties as explained in the next sec-
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2
2
2
2
)(0 nx )(~0 nx
)(1 nd
)(1 na)' (nh
)' (ng)(ng
)(nh
(a) 1-D DWT
(b) 2-D DWT on Image
Fig. 1 Forward and inverse DWT
Table 1 The alignment of ‘9/7’ filter coefficients
Lowpass Highpass
Index h(n) h′(n) Index g(n) Index g′(n)
0 0.6029 1.1151 −1 1.1151 1 0.6029
±1 0.2669 0.5913 −2, 0 −0.5913 0, 2 −0.2669
±2 −0.0782 −0.0575 −3, 1 −0.0575 −1, 3 −0.0782
±3 −0.0169 −0.0913 −4, 2 0.0913 −2, 4 0.0169
±4 0.0267 0 0 −3, 5 0.0267
tion, the original signal/image can be reconstructed withoutany loss of information. The most well-known filter bankin this category is the Daubechies 9/7 filter bank [3]. Thefilter coefficients are as shown in Table 1. To achieve perfectreconstruction, the filter coefficients are arranged in a spe-cific order as shown in Table 1. From the arrangement, it canbe observed that the filter h is centered at zero (0), while gis centered at −1. As a result, the downsampling operationeffectively retains the even indexed samples from the low-pass filtered output and the odd indexed samples from thehighpass filtered output sequence [22].
For perfect reconstruction with odd length symmetric fil-ters (5/3 and 9/7 filters used in JPEG2000), the downsamplingoperation retains the even indexed samples from the lowpassoutput and the odd indexed samples from the highpass outputsequence [22]. The detailed analysis on the properties of suchfilters for implementing the DWT and the IDWT with perfectreconstruction can be found in [15,28]. For multilevel DWT,the analysis operations are carried out iteratively to get asmany levels of decomposition as the resolution of the sig-nal/image permits. The multiresolution IDWT is computedby iterating or cascading the single stage filter bank to obtaina multiple stage synthesis filter banks. In the present work,
the wavelet decomposition and reconstruction operations arereferred simply as analysis and synthesis, respectively.
2.1 Downsampling during analysis
The downsampling of a discrete signal x(n) can be performedby two ways depending on the parity of index. Given an inputx(n), n = 0, 1, 2, . . . , N − 1, the downsampled sequencex̂e(m), m = 0, 1, 2, . . . , N/2 − 1 is obtained by retainingthe even index samples of x(n) as follows:
x̂e(m) = x(n) for m = 2n (1)
Similarly, the downsampled sequence x̂o(m), m = 0, 1, 2,
. . . , N/2 − 1, can be obtained by dropping the even indexsamples and retaining the odd index samples of x(n), as fol-lows:
x̂o(m) = x(n) for m = 2n + 1 (2)
To perform these operations in this work, Eq. (1) is used fordownsampling of lowpass filtered signal. On the other hand,Eq. (2) is used for downsampling of highpass filtered signal.This effectively shifts the result after filtering by one pixeland also matches the index with lowpass and highpass filteredsignals in the analysis.
After performing 2-D one-level DWT with an image, foursubbands LL, HL, LH and HH are obtained as depicted inFig. 1b. The first letter (L or H) corresponds to applyingeither a lowpass (L) or highpass (H) filter to the rows, andthe second letter refers to similar approach to the columns.The same filter bank can be used (by giving LL as the input tofilter bank) to produce a multilevel DWT. Symmetric exten-sion of signal at the border is used before computing theforward DWT. The proposed transcoder in this paper com-
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putes wavelet coefficients by the composite operation of fil-tering and downsampling from the DCT blocks directly inthe transform domain.
3 Types of DCTs and convolution multiplicationproperty
In this section, we briefly review different types of DCT andsymmetric convolution in the DCT domain. For simplicity,initially we restrict our discussion to 1-D. The concepts aretrivially extended to 2-D.
Let h(n), 0 ≤ n ≤ N be a sequence of length N + 1. ItsN -point Type-I DCT, C1e{h(n)} is defined as:
H (N )I (k) =
√2
Nβ(k)
N∑n=0
h(n) cos
(nπk
N
), 0 ≤ k ≤ N
(3)
where β(k) in Eq. (3) is 12 for k = 0 and N , otherwise its
value is 1.Let x(n), n = 0, 1, 2, . . . , N − 1, be a sequence of
length N . Its N -point Type-II DCT, C2e{x(n)} is defined as:
X (N )I I (k) =
√2
Nα(k)
N−1∑n=0
x(n)
× cos
((2n + 1)πk
2N
), 0 ≤ k ≤ N − 1 (4)
Similarly, the Type-II N -point IDCT, C−12e {X (N )(k)} is
defined as:
C−12e {X (N )
I I (k)} = x(n)
=√
2
Nα(k)
N−1∑k=0
X (N )I I (k)
× cos
((2n + 1)πk
2N
), 0 ≤ n ≤ N − 1
(5)
where α(k) in Eqs. (4 and 5) is√
12 for k = 0, otherwise its
value is 1.
3.1 Convolution–multiplication in the DCT domain
In [16], Martucci discussed how convolution–multiplicationproperties hold for trigonometric transforms with symmetricconvolution. For the Type-I and Type-II DCTs, convolution–multiplication properties can be stated as follows:
C2e {x(n)� h(n)} = √2N .C2e {x(n)} C1e {h(n)} (6)
where the operator � denotes symmetric convolution. Notehere that the N th coefficient of the Type-II DCT (for x(n))is zero; hence, only N multiplications are involved in the
computation of Eq. (6). For convenience, hereafter the usualDCT X will be considered as Type-II DCT, X I I .
3.2 Downsampling from the DCT of signals
In this section, we show how to perform the Type-II IDCTfollowed by the downsampling of a signal in a single com-bined step. The signal, x̂e(m), m = 0, 1, 2, . . . , N/2 − 1,(refer Eq. 1) as obtained through even indexed samplesof x(n), n = 0, 1, . . . , N − 1 using IDCT of X (N )(k),
k = 0, 1, . . . , N − 1 is given by:
x̂e(m) =√
2
Nα(k)
N−1∑k=0
X (N )(k)
× cos
((m + 1
4 )πk
N/2
), 0 ≤ m ≤ N
2− 1 (7)
The signal x̂o(m), m = 0, 1, . . . , N/2 − 1, (refer Eq. 2)as obtained through odd indexed samples of x(n), n =0, 1, . . . , N −1 using IDCT of X (N )(k), k = 0, 1, . . . , N −1is given by:
x̂o(m) =√
2
Nα(k)
N−1∑k=0
X (N )(k)
× cos
((m + 3
4 )πk
N/2
), 0 ≤ m ≤ N
2− 1 (8)
Let the downsampling basis matrices in the Eqs. (7 and 8)be denoted as B(N/2,N )
E and B(N/2,N )O , respectively. Let
C (N ,N ) be the N -point Type-II DCT basis matrix (in Eq. 4).It may be noted that the rows of basis matrices BE and BO
are the even indexed and odd indexed rows of the N -pointType-II IDCT basis matrix C (N ,N )t
, respectively. Here, thedownsampled signal ( N
2 samples) is computed directly fromN -point Type-II DCT block using the Eqs. (7 and 8). Thisreduces the cost involved in the computation of downsampledblocks from N -point DCT blocks.
Same concepts can be easily extended to 2-D also. Forexample, to calculate downsampled 4 × 4 block from an8×8 Type-II DCT block, the conversion matrices B(4,8)
E and
B(4,8)O (Eqs. 7 and 8) are as follows:
B(4,8)E
=
⎡⎢⎢⎣
0.3536 0.4904 0.4619 0.4157 0.3536 0.2778 0.1913 0.09750.3536 0.2778 −0.1913 −0.4904 −0.3536 0.0975 0.4619 0.41570.3536 −0.0975 −0.4619 0.2778 0.3536 −0.4157 −0.1913 0.49040.3536 −0.4157 0.1913 0.0975 −0.3536 0.4904 −0.4619 0.2778
⎤⎥⎥⎦
(9)B(4,8)
O
=
⎡⎢⎢⎣
0.3536 0.4157 0.1913 −0.0975 −0.3536 −0.4904 −0.4619 −0.27780.3536 0.0975 −0.4619 −0.2778 0.3536 0.4157 −0.1913 −0.49040.3536 −0.2778 −0.1913 0.4904 −0.3536 −0.0975 0.4619 −0.41570.3536 −0.4904 0.4619 −0.4157 0.3536 −0.2778 0.1913 −0.0975
⎤⎥⎥⎦ .
(10)
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Let us denote the transpose of the matrix X by Xt . Let X (8,8)
be an 8×8 Type-II DCT block. Its downsampled block x̂(4,4)
for LL subband transcoding is computed as:
x̂(4,4) = B(4,8)E X (8,8) B(4,8)t
E (11)
Similarly, x̂(4,4) for HH subband transcoding is computed as:
x̂(4,4) = B(4,8)O X (8,8) B(4,8)t
O (12)
For convenience in this work, the Type-II DCT and Type-IIIDCT are referred to as simply DCT and IDCT.
3.3 Composition and decomposition of DCT blocks
Let X (N )i , 0 ≤ i ≤ M −1, be the i-th N -point DCT block of
a sequence x(n), n = 0, 1, . . . , M × N − 1. In [11], Jiangand Feng showed that a block DCT transformation is nothingbut orthonormal expansion of the sequence {x(n)} with a setof M × N basis vectors. Each basis vector is derived from thebasis vectors of N -point DCT. Hence, there exists an invert-ible linear transformation from M blocks of N -point DCTto the usual M N -point DCT. In other words, for a sequenceof N -point DCT blocks {X (N )
i }, i = 0, 1, . . . , M − 1, thereexists a matrix A(M,N ) of size M N × M N such that:
X (M N ) = A(M,N )
⎡⎢⎢⎢⎢⎣
X (N )0
X (N )1...
X (N )M−1
⎤⎥⎥⎥⎥⎦ (13)
where X (M N ) denote the corresponding composite M N -point DCT of the signal. Note here that the conversion matri-ces and their inverses are sparse [11]. For detailed analysisof composition and decomposition of DCT blocks, one mayrefer the discussion made in [18].
Let b0, b1, . . . , bM−1 be the submatrices of A−1(M,N ), each
of size N × M N defined as follows:
A−1(M,N ) �
⎡⎢⎢⎢⎢⎣
b(N ,M N )0
b(N ,M N )1
...
b(N ,M N )M−1
⎤⎥⎥⎥⎥⎦ . (14)
In the decomposing process, i thN -point DCT block, {X (N )i },
i = 0, 1, . . . , M−1 can be computed directly by multiplyingthe corresponding submatrix of A−1
(M,N ) as follows:
X (N )i = b(N ,M N )
i X (M N ), i = 0, 1, . . . , M − 1. (15)
Note here that the output DCT blocks without the boundaryartifact can be directly computed during filtering operationusing the Eq. (15). This reduces computation involved indecomposing the larger DCT block after filtering operation.As the conversion matrices and their inverses are sparse [11],
it requires less number of multiplications and additions of thetwo sparse matrices compared to those of full matrix multi-plication. The analysis in 1-D can be easily extended to 2-Dby successive application of Eqs. (13) and (15) to all rowsand then to the resulting columns.
3.4 Symmetric extension
The digital filtering techniques are based on convolutionoperation. The convolution of finite-length signals with FIRfilters introduces border distortions. To deal with border dis-tortions, the border should be treated differently from otherparts of the signal (image). When filter coefficients fall out-side signal (image) during filtering, the values are extendedoutside the boundary. There are different ways of extend-ing the signal (image) such as zero-padding, symmetrization,periodic-padding and so on. For details of the extension meth-ods in wavelet transforms, one may refer to the Chapter 8 ofthe book entitled “Wavelets and Filter Banks,” by Strang andNguyen [28]. From a practical viewpoint, often it is pref-erable to use simple schemes based on signal extension onthe boundaries. Out of the available signal (image) extensionmodes, symmetric extension is the most popular technique.This method assumes that signals or images can be recov-ered outside their original support by symmetric boundaryvalue replication. Practically, this involves the computationof a few extra coefficients at the boundary [8].
In [33], the authors showed how to obtain the symmetri-cally extended blocks at the boundary in the DCT domain.The symmetrically extended coefficients are computed fromthe DCT block X (N ,N ) as follows:
X (N ,N )s = �X (N ,N )�t (16)
where � � D({(−1)m}N−1m=0) and D(.) is the diagonal matrix.
Note here that multiplication by � is costless. This impliesthat the symmetrically extended signal (image) DCT blockcan be obtained by changing the sign of a few DCT coeffi-cients as provided by the Eq. (16). Symmetrical extensionof DCT blocks eliminates the distortion at the border bycomputing a few extra samples outside the boundary. Beforetranscoding, the DCT blocks are symmetrically extended atborder in the block DCT space. The proposed transcodingtechnique produces subband coefficients in the N/2 × N/2block space. The exact size of the decomposed wavelet coef-ficient band can be obtained by retaining the central W × Wcoefficients.
4 Filtering with symmetric convolution
The convolution multiplication property as expressed inEq. (6) has a particular significance in its application to fil-tering in the Type-II block DCT domain. Given Type-I DCT
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Type-IIDCT of Signal
Type-IDCT of Filter
Point-wiseMultiplication
Type-II DCT of
Filtered Signal
Fig. 2 DCT domain filtering
of the impulse response of a filter and an input in the Type-IIDCT space, one can easily compute the filtered output inthe same Type-II DCT space using Eq. (6). The block dia-gram of the DCT domain filtering is illustrated in Fig. 2.Filtering operation in the transform domain is equivalent tosymmetric convolution in the spatial domain of an image. Toperform symmetric convolution in 2-D, specifications for thefirst quadrant of the spatial domain are only required. Onemay refer [16,34] for more details on the filtering in the DCTdomain using symmetric convolution.
4.1 Block DCT to wavelet transcoding (BWT)
In this section, we describe the wavelet subband transcoding,making use of composition and decomposition of the DCTblocks and downsampling Eqs. (1 and 2) along with filteringoperation. For the sake of convenience, the present discussionis restricted to 1-D. To eliminate the boundary effect in caseof large filter length (K ≥ N , where 2K +1 = FL ), filteringis to be performed on the larger blocks. As explained earlier,one can perform the composition of the DCT blocks to getlarger block using Eq. (13) in the transform domain directly.After filtering, output DCT blocks are extracted by comput-ing the blocks, which are not affected by the discontinuity onthe boundary in the decomposition process (refer to Eq. 15).Before performing filtering, the DCT blocks at both ends aresymmetrically extended so as to obtain same results as thatof DWT in the block space. For example, if the number ofDCT blocks is four, then two symmetrically extended blocksare padded at the boundary in the DCT domain, so that thetotal number of DCT blocks become six. In this case, sub-band transcoding produces four output blocks of size N/2.The length of the subband becomes W = 16.
In the proposed approach, three adjacent Type-II DCTblocks of size N (= 8, for a 4 × 4 block transcoding) arecomposed to get one larger Type-II DCT block, X (3N ). Theresulting DCT block will be subjected to filtering operation.Let H (3N )
I be the Type-I DCT response of the analysis filter.The output filtered block is computed by point-wise multi-plication of the composed DCT block X (3N ) and the Type-IDCT H (3N )
I . The larger DCT block is decomposed to get
DCT blocks of smaller size. After decomposition, the blockthat is not affected by the boundary effect is retained as theoutput DCT block. To obtain the transcoded wavelet coef-ficient blocks, the output DCT block is downsampled fol-lowing the Eqs. (1 and 2). For filtering with the larger blockof size 3N , the filter Type-I DCT coefficients are arrangeddiagonally in a matrix as:
D(3N ,3N ) = √2 × 3 × N
×
⎡⎢⎢⎢⎢⎣
H (3N )I (0) 0 0 . . . 0
0 H (3N )I (1) 0 . . . 0
......
.... . .
...
0 0 0 . . . H (3N )I (3N −1)
⎤⎥⎥⎥⎥⎦.
(17)
Let D(3N ,3N )L and D(3N ,3N )
H be the diagonal matrices obtainedfrom the Type-I DCTs of analysis filters, h(n) and g(n),respectively. To obtain transcoding matrices, the filteringoperation is combined with composition and decompositionprocesses as follows:
T (3N ,3N )L = A−1
(3,N )D(3N ,3N )L A(3,N ) (18)
T (3N ,3N )H = A−1
(3,N )D(3N ,3N )H A(3,N ) (19)
where A(3,N ) is the DCT block of composition matrix (refer
to Eq. 13). It may be noted that the matrices T (3N ,3N )L and
T (3N ,3N )H produce three DCT blocks of which only the cen-
ter block is retained as output. The cost of filtering can bereduced by considering only the submatrix of A−1
(3,N ), whichyield the output central DCT block (refer to Eq. 15). Fortranscoding, filtering operation is also combined with thedownsampling operation as follows:
T (N/2,3N )L = B(N/2,N )
E b(N ,3N )1 H (3N ,3N )
L A(3,N ) (20)
T (N/2,3N )H = B(N/2,N )
O b(N ,3N )1 H (3N ,3N )
H A(3,N ) (21)
where b(N ,3N )1 is the submatrix of A−1
(3,2N ), and the matri-
ces T (N/2,3N )L , and T (N/2,3N )
H are of size N/2 × 3N . Fora given analysis, filter bank h(n) and g(n), the transcodingmatrices T (N/2,3N )
L and T (N/2,3N )H are pre-computed. A typ-
ical examples of T (N/2,3N )L and T (N/2,3N )
H computed usingthe 5/3 wavelet analysis filters are given in Eqs. (42 and 43)(refer to “Appendix”).
Let X (N )i be the input in the block DCT space. The 1-D
for lowpass subband transcoding is expressed as follows:
a(N/2)i = T (N/2,3N )
L
⎡⎢⎣
X (N )i−1
X (N )i
X (N )i+1
⎤⎥⎦ (22)
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Similarly, 1-D highpass subband transcoding is expressed as:
d(N/2)i = T (N/2,3N )
H
⎡⎢⎣
X (N )i−1
X (N )i
X (N )i+1
⎤⎥⎦ (23)
The 2-D transcoding can be implemented by simply apply-ing 1-D transcoding to the input DCT block columns andthen to rows of the resulting blocks. Each wavelet subbandtranscoding is expressed as follows:
z(N ,N/2)i, j = T1
⎡⎢⎣
X (N ,N )i−1, j
X (N ,N )i, j
X (N ,N )i+1, j
⎤⎥⎦
w(N/2,N/2)i, j = [z(N ,N/2)
i, j−1 z(N ,N/2)i, j z(N ,N/2)
i, j+1 ]T t2
(24)
where z(N ,N/2)i j represents the 1-D transcoded output matrix
in the vertical direction for the DCT blocks, [X (N ,N )i−1, j , X (N ,N )
i, j ,
X (N ,N )i+1, j ]t . Also, T1 and T2 are the corresponding transcod-
ing matrices T (N/2,3N )L or T (N/2,3N )
H depending on the typeof subband. For example, LL subband transcoded blocks arecomputed using T1 = T2 = TL . It may be noted that theoutput blocks are in the spatial domain, which is an essen-tial requirement of the DCT to wavelet transcoding. We referthis algorithm as Block DCT to Wavelet Transcoding (BWT).Specifically for the JPEG compression technique, a value ofN = 8 is used so that the downsampled coefficient blocks(wavelet subbands) are of size N/2 = 4.
4.2 Separable linear filtering in the DCT domain
Like all unitary orthogonal transforms, the DCTs are distrib-utive to matrix multiplications [17]. With this property, onemay perform the filtering in the DCT domain also. For blocksof size N × N , the linear filtering can be expressed in theDCT domain as follows:
Y (N ,N )i j =
1∑m=−1
1∑n=−1
V (N ,N )m X (N ,N )
i+m, j+n H (N ,N )t
n . (25)
Note that the filtering matrices V (N ,N )m (DCT of vm) and
H (N ,N )n (DCT of hm) can be pre-computed for given filter
coefficients v(k) and h(l). The 2-D linear filtering in theDCT domain (Eq. 25) can be performed separately in thevertical and horizontal directions as follows:
Z (N ,N )i j = V (N ,N )
−1 X (N ,N )i−1, j + V (N ,N )
0 X (N ,N )i, j
+V (N ,N )1 X (N ,N )
i+1, j (26)
Y (N ,N )i j = Z (N ,N )
i−1, j H (N ,N )t
−1 + Z (N ,N )i, j H (N ,N )t
0
+Z (N ,N )i+1, j H (N ,N )t
1 (27)
where Z (N ,N )i j represents the 1-D linear filtered output matrix
for the DCT input matrix X (N ,N )i, j in the horizontal direc-
tion. The DCT domain 2-D linear filtering computation asin Eqs. (26 and 27) for N = 8 requires 3 ∗ 83 + 3 ∗ 83 =3, 072 multiplications and 3 ∗ 7 ∗ 82 + 3 ∗ 7 ∗ 82 + 2 ∗ 82 =2, 816 additions for one 8 × 8 DCT block. For blocks of size8 × 8 (i.e. JPEG-based encoding), the DCT domain 2-D lin-ear symmetric filtering (Eqs. 26 and 27) can be efficiently per-formed using any of the techniques described in [12,17,43].
4.3 Block DCT to wavelet transcoding with linearconvolution (BWTL)
In this section, we explain the wavelet subband transcodingusing filtering in the DCT domain (Eqs. 26 and 27). Here, thefiltering matrices in the DCT domain V (N ,N )
m and H (N ,N )n are
computed using the wavelet analysis filters g(n) and h(n),respectively. The filtering matrices computed from g(n) areused in the lowpass (L) analysis. On the other hand, the fil-tering matrices computed from h(n) are used in the highpass(H) analysis.
Interestingly, Eq. (7) is used for downsampling the band,which requires filtering with lowpass analysis filter (bandstarts with L), while for the highpass analysis (band startswith H), Eq. (8) is used for downsampling. The transcodedwavelet subband coefficient blocks (in the spatial domain)can be obtained by filtering the DCT blocks of the imagewith the transcoding matrices computed using the analysisfilter bank. Here, the filter matrices are combined with basismatrices (Eqs. 7 and 8) to compute the downsampled blocksdirectly. With the DCT matrices of the analysis lowpass fil-ter V (N ,N )
m and the input DCT blocks X (N )i , the Eq. (26) is
expressed for transcoding as follows:
a(N/2)i =
1∑m=−1
B(N/2,N )E V (N ,N )
m X (N )i+m (28)
where X (N )i+m are the input DCT blocks, and a(N/2)
i are ofthe transcoded approximation of the subband (L) coefficientblock.
Similarly for the DCT matrices computed from analysishighpass filter, H (N ,N )
n and the input DCT blocks, the trans-coded detail subband (H) coefficient block d(N/2)
i is com-puted (using Eq. 26) as follows:
d(N/2)i =
1∑n=−1
B(N/2,N )O H (N ,N )
n X (N )i+m (29)
Let the combined product matrices, B(N/2,N )E V (N ,N )
m
and B(N/2,N )O H (N ,N )
n are denoted as transcoding matrices
R(N/2,N )m and S(N/2,N )
n , respectively. The Eqs. (28 and 29)
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186 SIViP (2012) 6:179–195
are modified as follows:
a(N/2)i =
1∑m=−1
R(N/2,N )m X (N )
i+m (30)
d(N/2)i =
1∑n=−1
S(N/2,N )n X (N )
i+m . (31)
Note that the matrices R(N/2,N )m and S(N/2,N )
n can also be pre-computed for a given wavelet analysis filter bank, h(k) andg(k). The expressions in Eqs. (30 and 31) provide transcodedwavelet subband coefficient blocks of size N/2.
The concepts are easily extended to 2-D by the applicationof the 1-D transcoding to all rows and then to the resultingcolumns. In 2-D transcoding, approximation coefficient (LL)subband is computed as follows:
a(N/2,N/2)i j =
1∑m=−1
1∑n=−1
R(N/2,N )m X (N ,N )
i+m, j+n R(N/2,N )t
n (32)
where the transcoding matrices, R(N/2,N )m and R(N/2,N )
n , areof size N/2× N . Similarly, transcoded detail coefficient (HLor LH or HH) subband blocks are computed using the corre-sponding R(N/2,N )
m and S(N/2,N )n for vertical and horizontal
directions. We refer this technique as Block DCT to WaveletTranscoding with Linear filtering (BWTL).
For transcoding, the matrices R(N/2,N )m and S(N/2,N )
n asin the Eqs. (30 and 31) are computed using N = 4 using‘9/7’ analysis filters [1] (refer Table 2). The matrices R(4,8)
m s,computed using a 9-tap analysis lowpass filter h(n), are asfollows:
R(4,8)−1
=
⎡⎢⎢⎣
0.0702 −0.0963 0.0992 −0.1195 0.1374 −0.1219 0.0743 −0.02760.0035 −0.0028 −0.0027 0.0096 −0.0154 0.0178 −0.0156 0.0091
0 0 0 0 0 0 0 00 0 0 0 0 0 0 0
⎤⎥⎥⎦
(33)
R(4,8)0
=
⎡⎢⎢⎣
0.2833 0.3806 0.3400 0.2751 0.1499 0.0108 −0.0459 −0.02650.3501 0.2673 −0.1792 −0.4557 −0.2719 0.0644 0.0840 0.01410.3441 −0.0818 −0.4516 0.2747 0.2779 −0.1913 −0.0335 0.00850.3777 −0.4423 0.2264 0.0486 −0.2562 0.2236 −0.0736 0.0115
⎤⎥⎥⎦
(34)
R(4,8)1
=
⎡⎢⎢⎣
0 0 0 0 0 0 0 00 0 0 0 0 0 0 0
0.0094 0.0131 0.0123 0.0111 0.0094 0.0074 0.0051 0.0026−0.0242 −0.0380 −0.0445 −0.0440 −0.0311 −0.0108 0.0052 0.0082
⎤⎥⎥⎦
(35)
Similarly, Sns matrices, computed using a 7-tap analysishighpass filter g(n), are as follows:
S(4,8)−1
Table 2 JPEG2000 ‘9/7’ filter coefficients
n Analysis Synthesis
h(n) g(n) h ′(n) g ′(n)
0 0.6029 −1.1151 1.1151 −0.6029
±1 0.2669 0.5913 0.5913 0.2669
±2 −0.0782 0.0575 −0.0575 0.0782
±3 −0.0169 −0.0913 −0.0913 −0.0169
±4 0.0267 0 0 −0.0267
=
⎡⎢⎢⎣
−0.0120 0.0098 0.0091 −0.0328 0.0526 −0.0607 0.0532 −0.03100 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 0
⎤⎥⎥⎦
(36)S(4,8)
0
=
⎡⎢⎢⎣
0.0119 0.0052 −0.0377 0.0233 0.3823 0.8306 0.9079 0.54030 −0.0011 0.0692 0.1598 −0.4349 −0.7557 0.3981 1.0085
0.0322 −0.0417 0.0708 −0.3200 0.4672 0.1519 −0.9436 0.8460−0.1971 0.2939 −0.3708 0.5241 −0.6559 0.6321 −0.4425 0.2043
⎤⎥⎥⎦
(37)S(4,8)
1
=
⎡⎢⎢⎣
0 0 0 0 0 0 0 00 0 0 0 0 0 0 0
−0.0323 −0.0448 −0.0422 −0.0380 −0.0323 −0.0254 −0.0175 −0.00890.1971 0.2885 0.3016 0.2850 0.2210 0.1272 0.0444 0.0037
⎤⎥⎥⎦
(38)
It may be noted here that the transcoding matrices, R(4,8)m and
S(4,8)n , are of size 4 × 8 and require less number of computa-
tions for multiplication with the input 8×8 DCT blocks. Thecomplexity further reduces in case of sparse DCT blocks.
4.4 Multilevel DWT transcoding
Multilevel transcoding requires the computation of lower-level subbands from the transcoded approximation at the pre-vious level. This can be accomplished in two ways: (i) byperforming the DCT on the first level transcoded LL coef-ficients, followed by the proposed transcoding, and (ii) bya hybrid approach wherein conventional DWT computationcan be followed from the first level onwards. The process isiterated to obtain the lower-level subband coefficients. Fortranscoding in this work, hybrid approach is employed.
5 Computational complexity
In this section, we discuss the costs associated with thetransform domain and the spatial domain approaches. Letthe cost of a single multiplication and a single addition bedenoted as M and A, respectively. The total cost for an oper-ation requiring a number of multiplications and b numberof additions is denoted as aM + bA. Combined cost (CC)
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SIViP (2012) 6:179–195 187
measure [19] can be obtained by considering the cost ofa multiplication operation three times of the cost of addi-tion. Here, as the filter matrices are pre-computed, their costis excluded from the complexity. In computing the cost ofmatrix multiplication, it is considered that an element x in thematrix is a zero element, if |x | < Threshold and two elementsx and y become non-distinct, if ||x | − |y|| < Threshold.The value of threshold used in determining the complexity isempirically taken as 10−6. One may obtain the performanceas same as linear filtering using this threshold.
It may be noted that the technique reported in [4] com-putes all spatial domain samples in the filtering process.They are downsampled in the spatial domain to obtain thewavelet coefficients. Due to this operation, the redundantsamples are also computed. As a consequence, the com-plexity of filtering depends heavily on the sparseness of theDCT data. The proposed approaches are distinctly differentfrom the transcoding approach reported in [4], by computingonly the downsampled subband coefficients. This reduces thecomplexity to half when compared to the technique presentedin [4].
In the complexity analysis, cost of the 8 × 8 DCT compu-tation is based on the results reported by Wu and Man [39].Here, the computational costs are 88M and 466A, and perpixel cost is obtained by dividing the block size (64 pixels)as 1.37M and 7.28A. It may be noted that the smaller com-putational costs are shown in bold numerals.
5.1 Computational cost in the spatial domain
Costs associated with spatial domain are due to computa-tions involved in: (i) Block IDCT and (ii) The DWT of image(after IDCT). As mentioned earlier, computation of an 8 × 8IDCT block requires 1.37M + 7.28A operations per pixel.Implementation of the DWT depends on the length of anal-ysis filters, FL . Here, computation of the wavelet subbandusing filter bank (FB) algorithm requires FLM + (FL −1)Aoperations per pixel [24], and lifting scheme (LS) requires2(FL + 2) number of multiplications and additions [7].
We also refer spatial domain transcoding using filter bankand lifting scheme for DWT implementation as ST-FB andST-LS, respectively. If the filters are of odd length and thedifference between them is small, the complexity is com-puted by average of these two lengths. For example, using9/7 wavelet filters, average of costs associated with a filterlength of 9 and 7 is taken as the computational cost. In thiscase, per pixel combined costs are 42.39 and 31.39 for ST-FBand ST-LS approaches, respectively.
5.2 Computational cost of the proposed BWT
Costs associated with this method are due to computationsinvolved in multiplying the DCT domain filtering matrices
(refer to Eqs. 20 and 21) with the input DCT coefficientblocks of size N × N (for the JPEG-based encoding N = 8).Using 5/3 synthesis filters, the BWT costs are as follows witha threshold of 10−6:
• Computation of approximation wavelet coefficient blockof size 4 × 4 requires multiplication of TL with a matrixcontaining three DCT blocks row-wise and then column-wise. This requires 8(45M+ 44A) operations. Transcod-ing of the LL subband requires 360M+352A operations.The per pixel cost is 5.625M + 5.5A operations. Thecombined cost of transcoding the LL subband is 22.375operations per pixel.
• Computation of wavelet diagonal coefficient block withsize 4×4 requires multiplication of TH with a matrix con-taining three DCT blocks row-wise and then column-wise.This requires 8(39M+36A) operations. The transcodingof HH subband requires 312M + 288A operations. Theper pixel cost is 4.875M+4.5A operations. The combinedcost of the HH subband transcoding is 19.125 operationsper pixel.
In 2-D, average subband transcoding cost (/pixel) for sepa-rable 5/3 wavelet filters by the proposed BWT method are5.25M + 5A operations. The combined cost for per pixeltranscoding using BWT is 20.75 operations. Computationalcosts for 9/7 filters are computed in a similar way.
5.3 Computational cost of the proposed BWTL
Costs associated with the BWTL approach are due to thecomputations involved in multiplying the matrices Rm andSn (Eqs. 33–38) with three input DCT blocks of size N × N(for the JPEG-based encoding N = 8).
For example, cost of transcoding of 4 × 4 of LL subbandblock using 5/3 analysis lowpass filter requires multiplicationof R−1, R0 and R1 with three adjacent DCT blocks row-wiseand then column-wise.
• Multiplication of R−1 with a DCT block of size 8 × 8requires 8(8M + 6A) operations.
• Multiplication of R0 with a DCT block of size 8 × 8requires 8(32M + 24A) operations.
• Multiplication of R1 with a DCT block of size 8 × 8requires 8(8M + 6A) operations.
• Addition of three blocks requires 32A(= 2×16A) oper-ations.
An 8×8 block transcoding using 5-tap lowpass filter requiresa total of 384M + 320A operations. Dividing by the blocksize, resulting transcoding cost becomes 6M+5Aoperationsper pixel. For computing a 4 × 4 block of the approximation(LL) subband, the combined cost is 23 operations per pixel.
Similarly, the HH subband transcoding using Sns (com-puted from 7-tap highpass filter) requires 8(40M + 30A)
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Table 3 Transcodingcomplexity (/pixel) withnon-sparse data for various filterlength (FL )
FL BWT BWTL ST-FB ST-LS
M A CC M A CC M A CC M A CC
3 4.87 4.5 19.12 5.0 4.0 19.0 4.37 9.28 22.39 3.87 9.78 21.39
5 5.87 5.5 23.12 6.0 5.0 23.0 6.37 11.28 30.39 4.87 10.78 25.39
7 6.87 6.5 27.12 7.0 6.0 27.0 8.37 13.28 38.39 5.87 11.78 29.39
9 7.87 7.5 31.12 8.0 7.0 31.0 10.37 15.28 46.39 6.87 12.78 33.39
11 8.87 8.5 35.12 9.0 8.0 35.0 12.37 17.28 54.39 7.87 13.78 37.39
13 9.87 9.5 39.12 10.0 9.0 39.0 14.37 19.28 62.39 8.87 14.78 41.39
15 10.87 10.5 43.12 11.0 10.0 43.0 16.37 21.28 70.39 9.87 15.78 45.39
17 11.87 11.5 47.12 12.0 11.0 47.0 18.37 23.28 78.39 10.87 16.78 49.39
Table 4 Transcodingcomplexity (/pixel) with sparsedata for various filter length(FL )
FL BWT BWTL ST-FB ST-LS
M A CC M A CC M A CC M A CC
3 2.4375 2.25 9.56 2.5 2.0 9.5 4.37 9.28 22.39 3.87 9.78 21.39
5 2.8125 2.75 11.18 3.0 2.5 11.5 6.37 11.28 30.39 4.87 10.78 25.39
7 3.4375 3.25 13.56 3.5 3.0 13.5 8.37 13.28 38.39 5.87 11.78 29.39
9 3.9375 3.75 15.56 4.0 3.5 15.5 10.37 15.28 46.39 6.87 12.78 33.39
11 4.4375 4.25 17.56 4.5 4.0 17.5 12.37 17.28 54.39 7.87 13.78 37.39
13 4.9375 4.75 19.56 5.0 4.5 19.5 14.37 19.28 62.39 8.87 14.78 41.39
15 5.4375 5.25 21.56 5.5 5.0 21.5 16.37 21.28 70.39 9.87 15.78 45.39
17 5.9375 5.75 23.56 6.0 5.5 23.5 18.37 23.28 78.39 10.87 16.78 49.39
operations. In this case, a total of 320M + 256A operationsare required for transcoding. Dividing by the DCT block size,the transcoding cost becomes 5M+4A operations per pixel.For computing a 4 × 4 block of the detail (HH) subband, thecombined cost is 19 operations per pixel.
5.4 Sparseness of DCT coefficients
Sparseness is an important factor that reduces complexity ofthe DCT domain processing. In case of DCT-based compres-sion schemes, sparse data are very common. Actually, a highpercentage of coefficients in the DCT blocks have zero valuesafter quantization at low bit rate compression. The authorsin [12,17,43] considered a DCT block as sparse if its 4 × 4upper quadrant contains non-zero elements. In such cases, thecost of transcoding reduces to 50% of the original complex-ity. The transcoding costs are provided in Tables 3 and 4 fornon-sparse and sparse data input with filters of various kernelsizes. The transcoding complexity (/pixel) using non-sparsedata input with 9/7 filters are 29.12, 29, 42.39 and 31.39 oper-ations for the proposed BWT, BWTL, ST-FB and ST-LS,respectively. The complexity using sparse data input reducesto 14.56 and 14.5 for the proposed BWT for BWTL tech-niques. On the other hand, the complexity remains unchangedin the spatial domain techniques (Table 4). This implies thatthe proposed approaches are efficient compared to spatial
domain ST-FB and ST-LS approaches. One may note that thespatial domain transcoding does not benefit from the sparse-ness of input DCT blocks, whereas the proposed approachescan exploit the sparseness of input DCT blocks.
In case of complete sparse data (only top left corner 16coefficients), a computation saving of 64.83 and 54.07%for a 7-tap filter and 66.59 and 53.58% for a 9-tap filteris achieved by the proposed BWTL approach compared tothe spatial domain ST-FB and ST-LS approaches, respec-tively. Similarly, for a 17-tap filter, computation saving of70.02 and 52.42% can be achieved by the proposed BWTLapproach compared to the ST-FB and ST-LS spatial domainapproaches, respectively. Similarly, for the BWT with a 7-tapfilter, a computation saving of 64.68 and 53.86% is achievedwhen compared to the ST-FB and ST-LS approaches, respec-tively. Note here that the transform domain transcodingachieves the result as same as the spatial domain techniques.
6 Transcoding results
For transcoding demonstration, we used ‘lena’, ‘baboon’,‘peppers’ and ‘girl’ images in the experimentation. The per-formance evaluation is based on the symmetric wavelet ‘9/7’filters [1] as presented in Table 2. The transcoding matricesRm and Sn (Eqs. 33–38) computed using the 9/7 analysis fil-ters are used by the BWTL algorithm. The matrices TL and
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SIViP (2012) 6:179–195 189
Table 5 Wavelet to DCTtranscoding: PSNR (dB)performances for non-sparsedata
Image Approximation, LL Horizontal, HL Vertical, LH Diagonal, HH
BWTL BWT BWTL BWT BWTL BWT BWTL BWT
Lena 306.60 300.15 320.09 300.71 321.88 300.58 324.92 318.77
Baboon 306.55 300.00 319.00 300.26 321.36 300.41 322.33 309.61
Peppers 306.74 300.24 320.01 300.67 321.77 300.58 324.87 315.47
Girl 312.08 305.50 324.88 305.92 327.04 305.84 329.15 319.61
Table 6 Wavelet to DCTtranscoding: PSNR (dB)performances for sparse data,comparison with waveletsubbands of original image
Image Approximation, LL Horizontal, HL Vertical, LH Diagonal, HH
BWTL BWT BWTL BWT BWTL BWT BWTL BWT
Lena 40.18 40.18 36.20 36.26 32.20 32.26 32.76 32.79
Baboon 30.09 30.09 20.91 20.92 25.48 25.41 21.65 21.58
Peppers 37.11 37.11 30.83 30.86 30.38 30.43 28.42 28.41
Girl 40.10 40.10 32.51 32.57 34.06 34.13 32.90 32.91
Table 7 PSNR (dB)performance of transcoding forsparse data obtained by differentquantization of DCT blocks
Image Approximation, LL Horizontal, HL Vertical, LH Diagonal, HH
Q1 Q2 Q3 Q1 Q2 Q3 Q1 Q2 Q3 Q1 Q2 Q3
Lena 33.21 32.33 29.07 33.64 33.47 32.11 31.31 31.00 29.62 30.94 31.95 31.91
Baboon 27.99 27.65 24.12 21.52 21.52 20.46 24.89 24.74 24.09 21.14 21.12 21.23
Peppers 33.34 32.47 29.00 31.46 31.27 29.41 31.19 30.65 29.18 28.12 28.10 28.03
Girl 32.92 32.18 28.95 31.57 31.42 30.34 32.41 32.21 30.83 32.14 32.11 32.24
TH (refer to Eqs. 20 and 21) computed using the 9/7 analysisfilters are used by the BWT algorithm. Wavelet subbands arecomputed using the proposed BWT and BWTL approachesusing the corresponding filtering matrices.
In the spatial domain approach, the IDCT is first per-formed on the DCT blocks to obtain the spatial domainimage. Then, the image is wavelet transformed to obtainLL, HL, LH, HH subband coefficients. For comparing thequality of the transcoded subbands, Peak-Signal to NoiseRatio (PSNR) is used. The PSNR values are computedusing the subbands obtained by the spatial domain approachas reference. Table 5 shows the PSNR (dB) performancesusing 9/7 filters for non-sparse input DCT blocks. Onemay observe that the PSNR values are significantly higher(above 300 dB). This implies that the proposed approachachieves the results, which are equivalent to the spatialdomain approach. In case of sparse data transcoding, the sub-bands are also compared with the subbands obtained fromtranscoding of the original images. Table 6 shows the com-parative results of complete sparse data transcoding with thesubbands obtained from original images. It may be notedhere that in sparse data representation, only top left corner4 × 4 coefficients out of 8 × 8 are considered without anyquantization.
Further, to study the effect of DCT quantization, three dif-ferent quantization tables Q1, Q2 and Q3 [27] are selected.The DCT blocks of images are obtained with these quantiza-tion tables. The subbands are computed using the quantizedDCT blocks as input to the proposed techniques. Table 7shows the PSNR performances of quantized DCT blocksobtained from these JPEG quantization tables. Here, alsoin computing the PSNR values, the wavelet subbands of theoriginal image are used as reference. It may be noted thatthe transcoded HH subbands are least affected for differentquantization values. This is due to the fact that most of thehigh frequency coefficients after quantization become zeros.Hence, they do not change the quality of the transcoded sub-bands. Also note here that the transform domain transcodingtechnique achieves the result as same as the spatial domaintechniques.
The BWT operates on the three input coefficient blocksat the same time, whereas the BWTL operates on the threeinput blocks separately and the results are added thereafter.Interestingly, the transcoding matrices (Rn and Sn) obtainedin the BWTL technique can be represented as the subma-trices of TL and TH matrices (refer to “Appendix”). Thesematrices become equivalent after multiplying the normaliz-ing factors of
√2L N (or
√2M N , here L(= M) = 3) in com-
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puting the composite matrices. Hence, both the approachesare almost computationally equivalent and produce similarperformances. The reverse transcoding (wavelet to DCT)techniques WBDT and WBDTL as proposed in [32,33,35]are also computationally equivalent. The cost of transcodingdepends on the length of the filter.
6.1 Deblocking in the wavelet domain
The block DCT coding is the most popular technique used byimage compression standards, such as JPEG, H.261/263 andMPEG. However, block-based coding technique yields dis-continuities across block boundaries. These discontinuitiesare called blocking artifacts, which deteriorate image qual-ity to a great extent, especially at low bit rates. This kind ofdegradation is highly undesirable and affects the judgementof an observer. The most popular strategy for alleviating theblocking artifacts is to use post-processing techniques at thedecoder side. This strategy is of practical interest since itonly requires the decoded image and hence is fully com-patible with the image coding standards. An ideal deblock-ing algorithm should remove visible blocking artifacts whilemaintaining original image content as much as possible.Recently, wavelet-based deblocking algorithms [9,27,40,41]have gained more attention. This is due to the ability of wave-let-based deblocking algorithm to suppress blocking arti-facts while preserving true edges and textural information.Blocking artifacts are clearly visible in the high frequencysubbands and can be easily suppressed. In such scenario,the proposed transcoder provides subband coefficients fordeblocking the image in wavelet domain directly from theDCT blocks.
The images with blocking artifacts are obtained using thethree quantization tables [27,44] Q1, Q2 and Q3 as givenby Eqs. (39, 40 and 41).
Q1 =
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
20 24 28 32 36 80 98 14424 24 28 34 52 70 128 18428 28 32 48 74 114 156 19032 34 48 58 112 128 174 19636 52 74 112 136 162 206 22480 70 114 128 162 208 242 20098 128 156 174 206 242 240 206
144 184 190 196 224 200 206 208
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(39)
Q2 =
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
50 60 70 70 90 120 255 25560 60 70 96 130 255 255 25570 70 80 120 200 255 255 25570 96 120 145 255 255 255 25590 130 200 255 255 255 255 255
120 255 255 255 255 255 255 255255 255 255 255 255 255 255 255255 255 255 255 255 255 255 255
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(40)
Q3 =
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
110 130 150 192 255 255 255 255130 150 192 255 255 255 255 255150 192 255 255 255 255 255 255192 255 255 255 255 255 255 255255 255 255 255 255 255 255 255255 255 255 255 255 255 255 255255 255 255 255 255 255 255 255255 255 255 255 255 255 255 255
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(41)
For deblocking through transcoding, the wavelet subbandsare computed by the proposed BWTL algorithm. Here, thematrices Rn and Sn are computed using 5/3 filters (samefilters are used by the author in [27]). The PSNR valuesare computed using the reconstructed images obtained afterdeblocking via transform domain (BWTL-DB-IDWT) andspatial domain (IDCT-DWT-DB-IDWT) approaches. ThePSNR performances are provided in Table 8 for three quan-tization tables using both approaches. One may observe thatthe resulting PSNR values obtained are the same as the spatialdomain technique.
In this work, for judging the image quality reconstructedfrom the block DCT space and to take into account of vis-ible blocking and blurring artifacts, a no reference metricsuggested by Wang et al. [38] is used. In [20], this metrichas been referred to as the JPEG Quality-Metric (JPQM).The computation of this metric is described in [38], andthe MATLAB code for computing JPQM measure is avail-able at the Website http://anchovy.ece.utexas.edu/z~wang/research/nr_jpeg_quality/index.html. It may be noted that foran image with good visual quality, the JPQM value shouldbe close to 10. Table 8 provides the JPQM values for recon-structed images from their compressed counterparts usingthree different quantization tables. The deblocking resultsobtained for ‘lena’ image are shown in Figs. 3, 4 and 5 forthree quantization tables Q1–Q3. From the results, it is evi-dent that deblocking significantly improves the quality of theprocessed images. Interestingly, the JPQM values obtainedat various compression levels by the proposed transformdomain technique are quite close to those obtained from thespatial domain technique. As the proposed technique oper-ates in the block DCT space, presence of blocking artifactsis natural in these images. But the spatial domain techniquehas a distinct edge over this limitation. The deblocking algo-rithm using the transcoder is capable of reducing this gap toa great extent.
6.2 Thresholding of elements in the trancoder matrix
As reported in the work [5,34], numerical accuracy is notthe primary objective in many applications. When pre-viewing the result of a convolution/filtering operation, fastresponse time is also a desirable feature for an algorithm.With the proposed technique, it is possible to significantlyspeedup the operation by zeroing out all elements in the
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Table 8 Image deblocking:performance comparison
DB Deblocking
Image (JPQM) Quantization Test image BWTL-DB-IDWT IDCT-DWT-DB-IDWT
PSNR (dB) JPQM PSNR (dB) JPQM PSNR (dB) JPQM
Q1 (0.43 bpp) 33.65 6.04 33.966 7.34 33.966 8.51
Lena (10.88) Q2 (0.24 bpp) 30.70 3.01 31.210 7.16 31.206 8.19
Q3 (0.15 bpp) 27.38 −1.12 27.940 5.35 27.936 6.80
Q1 (0.43 bpp) 26.08 6.64 26.263 7.61 26.264 7.79
Baboon (9.06) Q2 (0.24 bpp) 23.39 4.50 23.602 6.87 23.602 6.82
Q3 (0.15 bpp) 21.17 0.79 21.416 5.32 21.415 5.76
Q1 (0.43 bpp) 32.60 6.49 32.759 7.47 32.760 8.68
Peppers (11.35) Q2 (0.24 bpp) 30.42 3.37 30.812 7.36 30.811 8.21
Q3 (0.15 bpp) 27.22 −0.39 27.726 5.65 27.724 6.99
Q1 (0.43 bpp) 32.88 6.20 33.168 7.42 33.168 8.31
Girl (10.83) Q2 (0.24 bpp) 30.43 2.94 30.837 6.84 30.832 7.74Q3 (0.15 bpp) 27.29 −1.66 27.761 5.80 27.752 6.33
Fig. 3 Reduction in blockingartifacts with Q1 (0.43 bpp):a test image PSNR = 33.65 dB,JPQM = 6.04 b processed imagein the spatial domain (using[27]) PSNR = 33.966 dB,JPQM = 8.51 c processed imagein through transcoding(proposed) PSNR = 33.966 dB,JPQM = 7.34
transcoding matrices whose magnitude is smaller than somespecified threshold [34]. In this way, numerical accuracy istraded-off for speed, while the visual quality is gracefullydegraded. Table 9 presents the transcoding performancesfor LL subband with various threshold values using 5/3filters.
In the present context, speedup refers to how much thetranscoding algorithm (after thresholding) is faster than thecorresponding algorithm with full elements in the transcodermatrix. The speedup factor is computed by dividing thecomputational operations at threshold value of 10−4 fromrespective multiplications and additions at other thresholds.To demonstrate the reduction in the cost by thresholding,transcoding matrices (Eqs.44 and 45) for a threshold value
of 10−2) are provided in “Appendix”. In Fig. 6, the trade-offis demonstrated by plotting obtained speedup for CC ver-sus the PSNR values for the approximate and the spatiallytranscoded results of the image lena.
As explained in [34], thresholding of the elements in thetranscoding matrix also introduces noise into the computedsubbands. For a noise-free input image, the DCT domaintranscoding is very sensitive to the threshold values, whereasthe transcoding technique is insensitive to the threshold valuefor a noisy image. If the noise in the image is more than thenoise introduced by the transcoding matrices, then the oper-ation is insensitive to threshold value. This makes the thres-holding scheme more appealing as it saves the computationto a great extent.
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Fig. 4 Reduction in blockingartifacts with Q2 (0.24 bpp):a test image PSNR = 30.70 dB,JPQM = 3.01 b processed imagein the spatial domain (using[27]) PSNR = 31.206 dB,JPQM = 8.19 c processed imagein through transcoding(proposed) PSNR = 31.21 dB,JPQM = 7.16
Fig. 5 Reduction in blockingartifacts with Q3 (0.15 bpp):a test image PSNR = 27.38 dB,JPQM =−1.12 b processedimage in the spatial domain(using [27]) PSNR = 27.936 dB,JPQM = 6.8 c processed imagein through transcoding(proposed) PSNR = 27.94 dB,JPQM = 5.35
Table 9 Proposed BWT performance variation with threshold
Threshold Transcoding of the LL subband Cost Speedup for CC
Lena Mandrill Pepper Girl M A CC
JPQM PSNR (dB) JPQM PSNR (dB) JPQM PSNR (dB) JPQM PSNR (dB)
10−2 4.40 17.75 5.16 17.61 4.60 17.80 6.31 23.28 17 34 85 2.17
10−3 8.94 77.88 7.51 77.67 9.67 77.91 9.25 83.25 39 44 161 1.15
10−4 8.94 314.13 7.51 313.86 9.67 314.23 9.25 319.50 47 44 185 1.00
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1 1.2 1.4 1.6 1.8 2 2.20
50
100
150
200
250
300
350
Speedup
PS
NR
dB
Fig. 6 Speedup factor versus image quality PSNR (dB)
7 Conclusion
We have proposed a transcoding technique for computingthe wavelet coefficients from the DCT blocks. The approachis based on filtering and downsampling of the DCT blocksin transform domain. We have developed transcoding in theblock DCT space so that the JPEG-based tools can oper-ate directly on them. We demonstrated the application oftranscoder for image deblocking in the wavelet domain. Theresults show that the proposed method achieves the samePSNR values as obtained by a spatial domain technique. Per-formance of the techniques with the sparse input DCT blocksare also observed. The effect of quantization on the transcod-ed subbands is also studied. By exploiting the sparseness inthe block DCT coefficients, the proposed techniques reducethe cost of transcoding (by trading off the quality of trans-coded images). The proposed approach is based on simplelinear operation such as matrix multiplication. This makesthe algorithm more convenient for efficient hardware imple-mentations.
The deblocking of DCT-coded images in the waveletdomain is also demonstrated using the proposed transcodingtechnique. The method uses a suitable deblocking algorithmin the wavelet domain. The results show that the proposedmethod achieves same PSNR values compared to the spatialdomain approach. The proposed algorithm can be used by theDWT-based post-processing techniques when the input dataare available as block DCT coefficients. Thus, the proposedtechnique is useful in heterogenous transcoding.
Acknowledgments Authors are thankful to anonymous Reviewersfor their excellent comments/suggestions for improving the manuscript.
Appendix
Typical examples of T (N/2,3N )L and T (N/2,3N )
H (elements|x | > 10−4) computed using 5/3 analysis filters with N = 8are given below:
T (N/2,3N )t
L =
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
0.0090 0 0 0−0.0144 0 0 00.0187 0 0 0
−0.0237 0 0 00.0271 0 0 0
−0.0267 0 0 00.0216 0 0 0
−0.0121 0 0 00.0632 0.0722 0.0722 0.08120.0892 0.0587 −0.0206 −0.10040.0854 −0.0431 −0.1041 0.05490.0712 −0.1119 0.0634 0.01170.0451 −0.0722 0.0722 −0.06320.0150 0.0146 −0.0624 0.0665
−0.0061 0.0374 −0.0155 −0.0325−0.0099 0.0094 0.0111 0.0038
0 0 0 −0.00900 0 0 −0.01250 0 0 −0.01180 0 0 −0.01060 0 0 −0.00900 0 0 −0.00710 0 0 −0.00490 0 0 −0.0025
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(42)
T (N/2,3N )t
H =
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
0 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 −0.0180
−0.0032 −0.0008 0.0022 0.0288−0.0057 0.0138 0.0057 −0.03740.0061 0.0175 −0.0309 0.04740.0361 −0.0361 0.0361 −0.05410.0692 −0.0587 0.0138 0.05340.0805 0.0333 −0.0805 −0.04310.0545 0.0963 0.0816 0.0241
0 0 0 0.01800 0 0 0.02500 0 0 0.02360 0 0 0.02120 0 0 0.01800 0 0 0.01420 0 0 0.00980 0 0 0.0050
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(43)
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One may observe that the matrices are sparse and itreduces the cost of transcoding.
Similarly, T (N/2,3N )L and T (N/2,3N )
H computed using 5/3analysis filters with N = 8 for elements |x | > 10−2 aregiven below:
T (4,12)t
L =
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
0 0 0 0−0.0144 0 0 00.0187 0 0 0
−0.0237 0 0 00.0271 0 0 0
−0.0267 0 0 00.0216 0 0 0
−0.0121 0 0 00.0632 0.0722 0.0722 0.08120.0892 0.0587 −0.0206 −0.10040.0854 −0.0431 −0.1041 0.05490.0712 −0.1119 0.0634 0.01170.0451 −0.0722 0.0722 −0.06320.0150 0.0146 −0.0624 0.0665
0 0.0374 −0.0155 −0.03250 0 0.0111 00 0 0 00 0 0 −0.01250 0 0 −0.01180 0 0 −0.01060 0 0 00 0 0 00 0 0 00 0 0 0
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(44)
T (4,12)t
H =
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
0 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 −0.01800 0 0 0.02880 0.0138 0 −0.03740 0.0175 −0.0309 0.0474
0.0361 −0.0361 0.0361 −0.05410.0692 −0.0587 0.0138 0.05340.0805 0.0333 −0.0805 −0.04310.0545 0.0963 0.0816 0.0241
0 0 0 0.01800 0 0 0.02500 0 0 0.02360 0 0 0.02120 0 0 0.01800 0 0 0.01420 0 0 00 0 0 0
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(45)
One may observe that the thresholding of elements in filtermatrices further reduces the cost of transcoding.
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