DIRECTIONAL DISCRETE COSINE TRANSFORMS ARISING FROM DISCRETE COSINEAND SINE TRANSFORMS FOR DIRECTIONAL BLOCK-WISE IMAGE REPRESENTATION
Tomohiro Ichita1 , Seisuke Kyochi1 , Taizo Suzuki2 , and Yuichi Tanaka3
1: The University of Kitakyushu, Kitakyushu, Fukuoka, Japan
2: University of Tsukuba, Tsukuba, Ibaraki, Japan
3: Tokyo University of Agriculture and Technology, Koganei, Tokyo, Japan
Email: s-kyochi@kitakyu-u.ac.jp, taizo@cs.tsukuba.ac.jp, ytnk@cc.tuat.ac.jp
ABSTRACT
Directional block transforms (DBTs), such as discrete Fourier transforms, are basically less efficient for sparse image representation
than directional overlapped transforms, such as curvelet and contourlet, but have advantages in practical computation, such as lesscomputational cost, less amount of memory usage to be used, andparallel processing. In order to realize efficient DBTs, this paper proposes directional discrete cosine transforms (DDCTs) by using discrete cosine and sine transforms. The resulting transforms provide
richer directional orientations of atoms than conventional DBTs, andthus they are expected to be more efficient for image analysis andprocessing. In experiments, we evaluate DDCTs with conventional
DBTs in image recovery by a convex optimization.
Index Terms— Block transform, discrete cosine transform, directionality, image recovery, convex optimization
1. INTRODUCTION
Sparse image representation (SIR) allows us to analyze images precisely, and realize many image processing tasks, e.g., denoising,deblurring, and compressive sensing, by integrating SIR into convex optimization algorithms [1–3]. For SIR, directional frames havebeen extensively studied, such as curvelet [4], contourlet [5], dirLOT[6], and dual-tree complex wavelets/filter banks [7–11]. Recently,
more general systems, so-called dictionary [12–14], non-local frame[15], and graph wavelets/filter banks [16, 17], that explore complex
structures or non-local similarity, have been proposed.
Although those transforms can efficiently provide SIR and contribute precise image analysis and processing, they have several
problems in practical computation. First, computational complexity
is typically high, due to 2D filtering [4–6], sparse coding with iterative scheme [12–14], block matching [15], or large-scale eigenvalue
decomposition [16, 17]. Second, they require a large amount ofmemory usage to store the coefficients. Third, they require global
memory access because each support of atoms in those frames isoverlapped each other. It disturbs parallel computation.
Block transforms (BTs) are widely used in practical applications, e.g., video coding [18], because supports of atoms are identical or disjoint, and transformation can be parallelized. Moreover,they require low computational cost and a small amount of memory. Conventional directional BTs (DBTs) can be classified into
fixed type and adaptive type. The fixed class contains transforms
with directionally oriented bases, such as discrete Fourier transform(DFT) and its variants [8, 19, 20]. In the latter class, the adaptive
directional transforms apply a BT along predetermined oblique di-978-1-5090-4117-6/17/$31.00 ©2017 IEEE

4536

rections [21,22]. Applications of the latter class are relatively limited
because transform directions have to be determined from an image inadvance. For example, in image recovery, observed images are usually degraded, and thus suitable directions cannot be easily found.
For general purposes, we focus on the former fixed class. The problem on conventional fixed type DBTs is that they contain duplicatedatoms in their basis and cannot provide rich directional selectivities1 .
This degrades the efficiency of image analysis and processing.
In order to achieve DBTs with rich directional selectivities, wefirstly introduce directional discrete cosine transforms (DDCTs),
which are constructed from the discrete cosine transform (DCT) andthe discrete sine transform (DST). The resulting transform provides
richer directionally oriented atoms than conventional DBTs. Then,we extend the DDCT to the biorthogonal DDCTs (BDDCTs) for
improving the performance. DCTs and DSTs can be regarded as
modulated filter banks [23–25]. By customizing prototype filter
coefficients of DCTs and DSTs, we can design higher performancefor certain criteria, such as coding gain [26–29], while keeping richdirectional selectivity. In the experiments, the DDCT/BDDCT arecompared with the conventional DBTs in image inpainting as apractical application and we show their effectiveness.
Notations: Bold-faced lower-case and upper-case letters denote
vectors and matrices, respectively. A set of Nr [row] and Nc [column] real-valued matrices is described as RNr ×Nc . I, J, and O
are reserved for the identity, reverse identity and zero matrices, respectively. A[N ] represents an N × N square matrix. [x]i (or xi )and [X]i,j (or Xi,j ) denote the i-th element of a vector x and the(i, j)-th element of a matrix X, respectively. X(i,j) ∈ RM ×M indicates the (i, j)-th M × M subblock of X ∈ RM L1 ×M L2 . For
a given image X ∈ RNr ×Nc , vec(X) ∈ RNr Nc denotes the vectorization of X, i.e., xNr j+i = Xi,j . bvec(X) ∈ R(M L1 )(M L2 )is the block-wise vectorization of X ∈ RM L1 ×M L2 as bvec(X) =[vec(X(0,0) )⊤ vec(X(0,1) )⊤ . . . vec(X(L1 −1,L2 −1) )⊤ ]⊤ . ⊗ denotes the Kronecker product. diag(a0 , . . . , aM −1 ) and diag(A0, . . . , AM −1 ) denote diagonal and block diagonal matrices.
2. PRELIMINARIES
2.1. Conventional Block and Directional Block Transforms
DCT (type-II) [30] is one of the most standard transforms and widelyused [18]. Its transform matrix FC ∈ RM ×M is defined as:
√
(
)
2
π
1
cos(θk,n ), θk,n =
k n+
,
(1)
[FC ]k,n = αkMM2
1 We use directional selectivity as the number of orientation of atoms.

ICASSP 2017

DDCT
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(a) DCT

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0
4x4 block(b) DFT (real part)Fig. 1: Frequency partition and the corresponding atoms B
basis (M = 4).

(k1 ,k2 )inwhere the ranges of the subband index k and the time index n are
0 ≤ n, k ≤ M − 1, α0 = √12 (k = 0), and αk = 1, (k ̸= 0).
For a block in an image x = vec(X) (X ∈ RM ×M ), the 2D DCT
2
2
is given by FC ⊗ FC ∈ RM ×M . A drawback of DCT is lack of
directional selectivity in its atoms (elements of basis/frames). The
2D atom B(k1 ,k2 ) ∈ RM ×M in the DCT forms:
Bn(k11,n,k22 ) = [FC ]k1 ,n1 [FC ]k2 ,n2Scaling/Sum/Diﬀerence
Reverse permutation

2D DCT/DST
Permutation(2)

where k1 and n1 denote the vertical subband and spatial index,
respectively, and k2 and n2 denote the horizontal ones (0 ≤k1 , k2 , n1 , n2 ≤ M − 1). Clearly, it is not directionally oriented (Fig. 1(a)2 ), and thus it cannot express various directionallycomponents in images.
The DFT [19] and its variants (discrete Hartley transforms(DHTs) [20] and conjugate-symmetric Hadamard transforms [8])are the DBTs. For example, the transform matrix of DFT iskn) and whose[FW ]k,n = √1M e−jφk,n ∈ RM ×M (φk,n = 2πM
2D atoms B(k1 ,k2 ) ∈ RM ×M are specified as:

1 j (φk1 ,n1 +φk2 ,n2 )
e
. (3)
M
As shown in Fig. 1(b), the basis contains directionally-oriented
atoms. One problem on the DFT and its variants is that they contain
duplicated atoms in their basis and cannot provide rich directional
selectivity as shown in Fig. 1(b). It means that high efficiency ofimage analysis and processing cannot be achieved.
Bn(k11,n,k22 ) = [FW ]k1 ,n1 [FW ]k2 ,n2 =2.2. Primal-Dual Splitting Method
Here, we briefly review the primal-dual splitting algorithm (PDS)[31–33] as a solver of a convex optimization problem used in theexperiments in Section 4. Consider the following convex optimization problem to find x⋆ ∈ arg minx∈Rn g(x) + h(Lx), where g ∈Γ0 (Rn )3 , h ∈ Γ0 (Rm ), and L ∈ Rm×n . Then the PDS for solving
the problem is given as follows:{
xk+1 := proxγ1 g [xk − γ1 L∗ zk ],(4)
zk+1 := proxγ2 h∗ [zk + γ2 L(2xk+1 − xk )]where prox denotes the proximal operator [34], h∗ is the conjugate
function [34] of h, and L∗ is the adjoint operator of L.
3. DISCRETE DIRECTIONAL COSINE TRANSFORM
3.1. Definition of DDCT
In order to solve poor directional selectivity of the conventional
DBTs mentioned in Section 2.1, the DDCT is introduced as follows.
2 In Figs. 1 and 2, enlarged atoms are depicted for better visualization.
3 Γ (RN ) is the set of proper lower semi-continuous convex functions
0[34] on RN .

4537

0

4

8 12

1

5

9

2

6 10 143

7 11 15

13

6

0

4

8 12

1 14 17 20

1

5

9

2 15 18 21

2

6 10 14

3 16 19 22

3

7 11 154

5

13

7 11 12 13

16 20 24 28

8 23 26 29

17 21 25 29

9 24 27 30

18 22 26 30

10 25 28 31

19 23 27 31(b) B(k1 ,k2 ,1)(c) B(k1 ,k2 ,−1)(a)Fig. 2: (a) Procedure of the DDCT (M = 4). (b) and (c): Atoms
B(k1 ,k2 ,±1) in the DDCT frame.
2

2

Definition 1. The DDCT FD ∈ R2M ×M is defined as[]FC ⊗ FC
FD := P⊤
WP
1
1
FS ⊗ FS 1√ I[2M −1]
O
O
2
1
1

W=
O
I
− 2 I[(M −1)2 ] 
2 [(M −1)2 ]
1
1
O
I
I
2
2 [(M −1) ]2 [(M −1)2 ]
2

(5)

2

where FC is in (1) and P1 ∈ RM ×M is a permutation matrix. P1
places the 4M − 2 DCT and DST coefficients associated with thesubband indices k1 = 0 or k2 = 0 to the first part, and the other2(M − 1)2 coefficients associated with the subband indices k1 ̸= 0
and k2 ̸= 0 to the last (see Fig. 2(a)). FS ∈ RM ×M is defined as
√
( (
))
 1 sin π n + 1(k = 0)M
2[FS ]k,n = √.
(6)
 2 sin ( π k (n + 1 )) (k ̸= 0)
MM
2
FS is the row-wise permutated version of the type-II DST. Because the DCT (FC ) and the DST (FS ) are orthogonal matrices, the
DDCT consists a Parseval frame F⊤
D FD = I[M 2 ] .
The procedure of the DDCT is illustrated in Fig. 2(a). The
DDCT requires just two block transforms, and additions/subtractions
between two transforms with scaling operations. Its redundancy ratio is 2, which is the same as the DFT and the DTCWTs [7,8,10,11],and is less redundant than frames and dictionaries, e.g., [13, 14].
According to the above setting, the atoms of the DDCT exhibit rich directional selectivity as explained in the following. Because
the DDCT forms
a Parseval frame, its atoms are given by[]
f0 . . . f2M 2 −1 := F⊤
D . From (5), each fk is 1) an atom in the
2D DCT/DST basis, or 2) an atom arising from the sum/differenceof 2D DCT/DST atoms. Let B(k1 ,k2 ,1) , B(k1 ,k2 ,−1) ∈ RM ×M be
two directional atoms of the DDCT that correspond to the subband(k1 , k2 ) ∈ {1, . . . , M − 1} × {1, . . . , M − 1}. They can be represented as
Bn(k11,n,k22 ,±1) = [FC ]k1 ,n1 [FC ]k2 ,n2 ± [FS ]k1 ,n1 [FS ]k2 ,n22=(7)
cos (θk1 ,n1 ∓ θk2 ,n2 ) .
M
These 2D cosine functions (this is the reason why FD in (5) is
termed as DDCT) are lying along various oblique directions, as illustrated in Figs. 2(b) and (c), analogous to the DFT (3).

(a)

(b)

(a) Real-valued filter(b) Complex-valued filter(c) D1 (ω1 , ω2 )(d) D2 (ω1 , ω2 )(c)Fig. 3: Frequency responses: (a) Zoneplate, (b) (Half of) DDCT
coefficients, (c) (Half of) DDCT coefficients using DC shifting.

Whereas the DDCT satisfies the rich directionality, it suffersfrom DC leakage, as shown in (half of) DDCT coefficients of the input image (Zoneplate) in Fig. 3(b). Since DC leakage degrades theenergy compaction in the subband domain, we introduce DC shifting2
2
as follows. First, let M ∈ RM ×M be an averaging operator de1.
fined as: M = M0 ⊗M0 , where M0 ∈ RM ×M and [M0 ]k,n = M
Then, the transform matrix of the DDCT with DC shifting is definedas FD (I − M). Fig. 3(c) shows the (half of) the transformed coefficients. It is clear that DC shifting (I − M) can promote sparsity ofcoefficients.
3.2. Directional Selectivity of DDCT from the Viewpoint of 2D
Frequency Domain
Before introducing the BDDCTs, this section discusses the directional selectivity of the DDCT from the viewpoint of 2D frequency(c)(s)
domain. Let Hk (ω) and Hk (ω) (k = 1, . . . , M − 1) be thefrequency spectrum of the k-th row in DCT and DST, respectively,
which can be expressed as)
1((c)
Uk (ω) + Uk (ω)
Hk (ω) =
2
)
1 (
(s)
Hk (ω) =Uk (ω) − Uk (ω) ,
(8)
2j
∑ −1
π k n
π
j (ω+ M) , pk,n = 1 (∀k, n).
where Uk (ω) = ej 2M k M
n=0 pk,n e(c)(s)
While Hk (ω) and Hk (ω) distribute in the both positive/negative
frequency domain (Fig. 4(a)), Uk (ω) distributes in the positive frequency domain (Fig. 4(b)). From (8), we can derive)1(
Uk1 (ω1 )Uk2 (ω2 ) + Uk1 (ω1 )Uk2 (ω2 )
D1 (ω1 , ω2 ) :=
2
(c)
(c)
(s)
(s)
= Hk1 (ω1 )Hk2 (ω2 ) − Hk1 (ω1 )Hk2 (ω2 ))1(D2 (ω1 , ω2 ) :=Uk1 (ω1 )Uk2 (ω2 ) + Uk1 (ω1 )Uk2 (ω2 )
2
(c)
(c)
(s)
(s)
= Hk1 (ω1 )Hk2 (ω2 ) + Hk1 (ω1 )Hk2 (ω2 ). (9)As shown the 2D frequency spectra in Figs. 4(c) and (d), (9) ensure that a directional frequency decomposition can be realized
by the DDCT (i.e., the 2D separable DCT/DST followed by addition/subtraction).
3.3. Definition of BDDCTs
From (8), DCT and DST can be regarded as special classes of modulated filter banks [23–25]. The fact produces the possibility to enhance the DCT and the DST by finding better prototype lowpass filter coefficients pk,n . Now we define biorthogonal DCTs (BDCTs),
biorthogonal DSTs (BDSTs), and BDDCTs by extending the original DCT and DST as follows.

4538

Fig. 4: Example of frequency spectra.
M −1−1Definition 2. For given coefficients {fn }n=0
and {gn }M
n=0 ,
M ×M
M ×M
BDCT FBC ∈ R
and BDST FBS ∈ R
are definedas:√2[FBC ]k,n = pk,n αkcos(θk,n )M√
2[FBS ]k,n = pk,n αksin(θk,n )M{(fn + gn ) k : evenpk,n =.
(10)(fn − gn ) k : odd
2

2

BDDCT (denoted as FBD ∈ R2M ×M ) is defined by replacing
DCT FC and DST FS in (5) with BDCT FBC and BDST FBS .
Since both FBC and FBS are often required to be invertible, weshould consider the perfect reconstruction condition for BDCTs. For
that, we derive the following proposition.
Proposition 1. The matrices FBC and FBS can be factorized as:
FBC = FC Gc , FBS = FS Gs ,(11)
Gc = diag(f0 , . . . , fM −1 ) + Jdiag(g0 , . . . , gM −1 )Gs = diag(f0 , . . . , fM −1 ) − Jdiag(g0 , . . . , gM −1 )Gs = ΓGc Γ, Γ = diag(I, −I).
(12)
Proof. Due to the symmetry/antisymmetry of the DCT, we obtain[]pℓ ck,ℓ pM −1−ℓ ck,M −1−ℓ[]
= (fℓ + gℓ )ck,ℓ (fM −1−ℓ + gM −1−ℓ )ck,ℓ][[] fℓ gM −1−ℓ
= ck,ℓ ck,ℓgℓ fM −1−ℓ[][
] fℓ gM −1−ℓ
= ck,ℓ ck,M −1−ℓ(k : even)gℓ fM −1−ℓ[]pℓ ck,ℓ pM −1−ℓ ck,M −1−ℓ[][
] fℓ gM −1−ℓ
= ck,ℓ −ck,ℓgℓ fM −1−ℓ][[
] fℓ gM −1−ℓ
= ck,ℓ ck,M −1−ℓ(k : odd).
(13)gℓ fM −1−ℓFrom the above discussion, we can verify the factorization for the
BDCT. In a similar way, the statement on the BDST can be checked.

5

Algorithm 1 Solver for (14)

x

5:

).

(n)

t̂1

(

= prox 1 ρ∥·∥

−x

(n)

(n)
(n)
), t2 = z2 + γ2 Φ(2x(n+1) −

)
(n)
1 (n)
, t̂2 = prox 1 ι
γ t1

2
1
γ2
(n+1)
(n)
(n)
zk
= tk − γ2 t̂k (k = 1, 2).

γ2 {v}

(

)
1 (n)
.
γ t2
2

0
Magnitude Response [dB]

4:

(n)
(n)
t1 = z1 + γ2 F(2x(n)

Magnitude Response [dB](0)
1: set n = 0 and choose x(0) , z(0)1 , z2 , γ1 , γ2 .
2: while stop criterion is not satisfied do
(n)
(n)
3:
x(n+1) = proxγ1 ι(x(n) − γ1 (F⊤ z1 + Φ⊤ z2 ))
C
[0,1]
(n+1)

5

0
−5
−10
−15
−20
−25
−30
−35
−40
−45
0

−5
−10
−15
−20
−25
−30
−35
−40

0.2

0.4
0.6
Normalized Frequency

6:7:n = n + 1.
8: end while
9: Output x(n) .

0.8

1

−45
0

0.4
0.6
Normalized Frequency

0.81(b) BDST(a) DST

From the proposition, if Gc is designed to be non-singular,
BDCT FBC , BDST FBS , and BDDCT FBD are invertible.

0.2(d) House(c) Barbara(e) Mandrill

4. EXPERIMENTAL RESULTS
We evaluated the performance of the DDCT/BDDCT in image inpainting [3], i.e., missing pixel recovery, as a practical application.
First, we explain the detail algorithm used in the experiments.
The cost function for image inpainting is formulated as follows.
x⋆ = arg min ρ∥Fx∥1 + ιC[0,1] (x) + Fv (Φx),(14)

x∈RNr Nc

where ρ > 0, x = bvec(X), X ∈ RNr ×Nc , F = I[Nr Nc /M 2 ] ⊗(FBD (I − M)) or F = I[Nr Nc /M 2 ] ⊗ (FD (I − M)), Φ denotes
some degradation process, and ιA (x) is the indicator function of aset A. C[0,1] is the set of vectors whose entries are within [0, 1].
Fv ∈ Γ(RN ) is a data fidelity term on the observation v. Since we
simply evaluate the performance of the proposed method in image
inpainting, we set the data fidelity term as the ι{v} (x), where theset of {v} consists of the observation v. In order to solve (14) by
PDS, the functions g and h, and the matrix L in (4) are set as:⊤ ⊤g(x) = ιC[0,1] (x), h([z⊤1 z2 ] ) = ρ∥z1 ∥1 + ι{v} (z2 ),[ ⊤]⊤z1 = Fx, z2 = Φx, L = F
(15)
Φ⊤ .

Then, a solver of (14) can be described in Algorithm 14 . For comparison, we also used ℓ1 -norm of the coefficients obtained by the DCT,
the DFT, and the DHT in (14).
The block size was set to M = 8. The parameter Gc (orGs ) in (11) for the prototype filter of BDCT/BDST was constructedby SVD-based parameterization [26] and optimized by maximizing
the coding gain5 , which evaluates the energy compaction efficiency,
i.e., sparse representation, via subband transform. In optimization,
we used MATLAB built-in function “fminunc” (the resulting coding
gains and frequency responses (DST/BDST) are shown in Table 1
and Figs. 5(a)–(b)). The parameters γ1 and γ2 in (4), and ρ in (14)1
, and 0.5. The stopping criterion for Algowere chosen as 0.01, 12γ
1rithm 1 as ∥x(n+1) − x(n) ∥2 ≤ 0.01. We used test images shown in
Figs. 5(c)–(e), and Zoneplate (Fig. 3(a)). Each observation consists

4 For x ∈ RN , [proxγ∥·∥1 (x)]i = sign(xi ) max{|xi | − γ, 0}(soft-thresholding), proxιC(x) is the clipping operation to [0, 1], and[0,1]proxι{v} (x) = v, where v ∈ RN is an observation.
5 Since the 2M 2 × M 2 BDDCT is a 2D oversampled filterbank, we used
the generalized coding gain [27] for 2D signals whose normalized autocorrelation is given by isotropic model [28, 29] with the correlation coeffcient of
0.95.

4539

(f) DCT(g) DFT(h) DDCT(i) BDDCT

Fig. 5: (a), (b): Frequency responses of DST/BDST, (c)–(e): Original images (256 × 256) and degraded images (losing 50% pixels),(f)–(i): The (zoomed) resulting images ((h), (i): prop.).
Table 1: (Upper table) Coding gains of DDCT and BDDCT. (Lower
table) Numerical results of image recovery.
DDCT (M = 8)
11.93
Barbara (PSNR [dB]: 9.255)
DFT
DHT DDCT BDDCT
25.33 24.60
26.28
26.65
Mandrill (PSNR [dB]: 8.540)
DCTDFT
DHT DDCT BDDCT
25.13 25.40 24.77
26.10
26.25
DCT
25.56

BDDCT (M = 8)
12.52
House (PSNR [dB]: 9.366)
DFT
DHT DDCT BDDCT
24.80 24.16
25.10
25.39
Zoneplate (PSNR [dB]: 7.265)
DCT
DFT
DHT DDCT BDDCT
14.80 16.41 15.53
17.34
18.31
DCT
24.72

50% pixels chosen randomly. The matrix Φ in (14) can be obtainedby replacing some 1s of an identity matrix to 0 at the indices where
corresponding pixels are lost.
Table 1 and Figs. 5(f)–(i) show the experimental results. The
BDDCT and the DDCT provided better subjective quality for directional textures than the DCT and the DFT. Although the subjective
quality between the BDDCT and the DDCT was almost the same,the BDDCT outperformed the DDCT (and conventional DBTs) interms of the reconstruction errors (PSNR) because it was optimized
to have higher coding gain than that of the DDCT.
5. CONCLUDING REMARKSIn this paper, we proposed the DDCT and the BDDCT. The DDCT
was constructed by using DCT/DST and could provide richer directional orientations than conventional DBTs under the same block
size. Then, since the DCT and the DST could be regarded as special
realizations of modulated filter banks, we could realize the BDDCTthat improves the performance of the DDCT by optimizing coefficients in the prototype filter, while keeping the directional selectivityof atoms. In the experimental results, for images with fine textures,the DDCT and BDDCT could achieve higher subjective and numerical qualities than the conventional DBTs.

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