Low Complex Image Resizing Algorithm using Fixed-point Integer

Transformation

James McAvoy, Ehsan Rahimi and Chris Joslin

Department of Systems and Computer Engineering, Carleton University, 1125 Colonel By Dr., Ottawa, ON, Canada

Keywords:

Resizing Algorithm, Image Halving and Doubling, DCT Transformation, Fixed-point Integer Transformation,

Subband Approximation, Low-complexity.

Abstract:

This paper proposes an efﬁcient image resizing algorithm, including both halving and doubling, in the DCT

domain. The proposed image resizing algorithm works on a 4 by 4 DCT block framework with a lower

complexity compared to the similar previous methods. Compared to the images that were halved or doubled

through the bilinear interpolation, the proposed algorithm produces images with similar or higher PSNR or

SSIM values at the signiﬁcantly lower computational cost. The test results also conﬁrm that our approach

improves the current frequency domain resizing algorithms through the ﬁxed-point integer transformation

which reduces the computational cost by more than 60% with negligible dB loss.

1 INTRODUCTION

Image resizing algorithms are often required to reduce

the memory space or the bandwidth required to store

or transmit videos or images. Usually, systems re-

size videos or images in the spatial domain by deci-

mation or interpolation of pixels; however, it is more

beneﬁcial to resize them in the compressed domain,

which avoids the high computational overhead asso-

ciated with decompression and compression operati-

ons. Recently, video data is often stored in the com-

pressed format based on blocks of 4 ×4 discrete co-

sine transform (DCT) coefﬁcients. Video compres-

sion standards such as H.264/AVC (ITU, 2012; ISO,

2012) and H.265/HEVC employ 4 ×4 ﬁxed-point ap-

proximation of the popular DCT to transform frames

from the spatial to the frequency domain.

In the early days of developing resizing algorithms

in the Discrete Cosine Transform (DCT) domain, re-

searchers devoted effort to image

1

halving and dou-

bling problem. Some of the early contributors in this

ﬁeld were Chang and Messerchmitt (Chang and Mes-

serchmitt, 1995), and Merhav and Bhaskaran (Mer-

hav and Bhaskaran, 1997) who developed resizing al-

gorithms that exploit linear, distributive and unitary

transform properties of DCT. Although image quality

was similar and often times superior than resizing in

1

Image and video frames will be used interchangeably

in this paper.

the spatial domain, the computational complexity was

almost same as spatial domain resizing techniques.

Dugad and Ahuja (Dugad and Ahuja, 2001) pro-

posed a simple fast computation algorithm for the

image halving and doubling by exploiting the low-

frequency DCT coefﬁcients. Later, Mukherjee and

Mitra (Mukherjee and Mitra, 2002) proposed some

modiﬁcations to the Dugad and Ahuja’s algorithm

that improved the image quality and increased the

computational cost. It is worth mentioning that

both algorithms use subband approximation of DCT

coefﬁcients while performing image resizing opera-

tions in the frequency domain. Jiang and Feng (Ji-

ang and Feng, 2002) formulated spatial relationships

of the DCT coefﬁcients between a block and sub-

blocks. Their approach decomposes and recompo-

ses blocks of DCT coefﬁcients. Mukherjee and Mitra

(Mukherjee and Mitra, 2005) also created several re-

sizing algorithms based on decomposition and re-

composition combined with subband approximation

such as IHAC, IDAD, LMDS, and LMUS algorithms.

Depending on the ordering of these operations, one

can vary the computational cost and the ﬁnal image

quality for the resizing algorithm.

Among all image resizing algorithms presented

recently(Mukherjee and Mitra, 2005; Mukhopadhyay

and Mitra, 2004; Nam et al., 2010; Aggarwal and

Singh, 2015; Hung and Siu, 2014; Meher et al., 2014),

two of these algorithms are the base of others and of

McAvoy, J., Rahimi, E. and Joslin, C.

Low Complex Image Resizing Algorithm using Fixed-point Integer Transformation.

DOI: 10.5220/0006616901430149

In Proceedings of the 13th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISIGRAPP 2018) - Volume 4: VISAPP, pages

143-149

ISBN: 978-989-758-290-5

Copyright © 2018 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

143

interest to us in this article. The ﬁrst is the image

halving algorithm presented by Mukherjee and Mitra

in (Mukherjee and Mitra, 2005), i.e. Image hal-

ving through approximation followed by composition

(IHAC) and the second is its reverse doubling algo-

rithm, Image doubling through decomposition follo-

wed by approximation. Both of these algorithms in-

clude conversion matrix to compose and decompose

blocks of DCT coefﬁcients when resizing images in

the DCT domain. These matrices consist of entries

that are the approximation of irrational numbers. Alt-

hough conversion matrices containe several zero en-

tries, matrix multiplication with blocks of DCT coef-

ﬁcients contributes most of the overall computational

cost; however, both these algorithms have been shown

to provide greater efﬁciency compared to resizing in

the spatial domain (Mukherjee and Mitra, 2005; Muk-

hopadhyay and Mitra, 2004).

This paper aims to enhance the Mukherjee and

Mitra image and doubling algorithms by deriving a

ﬁxed-point integer approximation of the conversion

matrices. Indeed, the proposed ﬁxed-point integer ap-

proximation algorithm is very similar to the approach

used in H.264/AVC standard body in order to deﬁne

the default inverse transform process. H.264/AVC

deviated from previous video compression standards

by employing a 4 ×4 ﬁxed-point integer transform

instead of the popular 8 ×8 DCT. This way, as repor-

ted by Malvar et al. (Hallapuro et al., 2002; Malvar

et al., 2003), a reduction in the complexity with the

negligible impact on image quality can be achieved.

In this paper, we describe how IHAC and IDDA

algorithms that are enhanced by deriving a ﬁxed-point

integer conversion matrix from the ﬂoating-point con-

version matrix, in Section 2. The image quality per-

formance is assessed in Section 3 and then, the com-

putational cost of the proposed algorithms is exami-

ned in Section 4.

2 PROPOSED METHOD

Image halving is an operation that takes an image of

size N ×N and outputs an image of N/2×N/2, where

image doubling is the inverse operation to resize an

image with the resolution of 2N ×2N.

Supposing that b denotes a 8 ×8 block in the spa-

tial domain containing four adjacent blocks b

k

which

k ∈ {0,1,2,3} as illustrated in Fig. 1. Therefore, b

k

denotes a 4 ×4 block in the spatial domain whose

DCT coefﬁcients are encoded as 4 ×4 block B

k

in the

compressed domain. Generally, to half an image one

needs to convert four adjacent DCT blocks B

1

, B

2

, B

3

and B

4

DCT blocks to a single 4 ×4 DCT block, B

d

.

In Mukherjee and Mitra’s IHAC algorithm, four 2 ×2

adjacent blocks (

ˆ

B

k

) are derived from the correspon-

ding 4 ×4 DCT blocks (B

k

) using subband approxi-

mation and then the 2 ×2 blocks

ˆ

B

k

are recomposed

to form a single 4 ×4 block B

d

using the conversion

matrix (Mukherjee and Mitra, 2005).

To double an image, Mukherjee and Mitra em-

ployed DCT block decomposition (Mukherjee and

Mitra, 2005). In the IDDA algorithm, as shown in

Fig. 2, a 4 ×4 DCT block denoted as B is ﬁrst de-

composed using the conversion matrix to four 2 ×2

DCT blocks,

ˆ

B

k

. Then each of these blocks is trans-

formed into a 4×4 DCT block, B

k

, by using subband

approximation and zero-padding.

To halve this image the IHAC resizing algorithm

would contain the following composition step:

B

d

= A

"

ˆ

B

(2×2)

1

ˆ

B

(2×2)

2

ˆ

B

(2×2)

3

ˆ

B

(2×2)

4

#

A

T

= A ·

ˆ

B ·A

T

, (1)

Where · denotes matrix multiplication and the conver-

sion matrix, A, is as:

A =

1/

√

2 0 1/

√

2 0

0.6533 0.2706 −0.6533 0.2706

0 1/

√

2 0 −1/

√

2

−0.2706 0.6533 0.2706 0.6533

.

(2)

Note that the rows of A are orthogonal and have unit

norms, which is a necessary condition for an ortho-

gonal block transformation. All the entries in A re-

quire processors to approximate irrational numbers.

A ﬁxed-point approximation is equivalent to scaling

each row of the conversion matrix, A, and rounding

to the nearest integer. To this end, the conversion ma-

trix is multiplied by 2.5 and then rounded. Therefore,

we have C deﬁned as:

C = round(2.5 ·A)

=

2 0 2 0

2 1 −2 1

0 2 0 −2

−1 2 1 2

. (3)

We selected the scaling constant of 2.5 because it was

same one that the H.264/AVC designers used to de-

velop their ﬁxed-point approximation of 4-point DCT

(Hallapuro et al., 2002; Malvar et al., 2003). To re-

store the orthonormal property of the original matrix

of A, all the values of c

i j

in row r are multiplied by

1

r

∑

j

c

2

r j

:

A = C •R, (4)

VISAPP 2018 - International Conference on Computer Vision Theory and Applications

144

B

1

B

2

B

3

B

4

ˆ

B

1

ˆ

B

2

ˆ

B

3

ˆ

B

4

B

d

ˆ

B

3

ˆ

B

4

ˆ

B

2

ˆ

B

1

- -

Subband approximation Block composition

Block

ˆ

B

Figure 1: IHAC algorithm-Four 2 ×2 approximated DCT coefﬁcients of adjacent blocks are composed into one 4 ×4 DCT

block (Mukherjee and Mitra, 2005).

B

ˆ

B

3

ˆ

B

4

ˆ

B

2

ˆ

B

1

ZerosZeros

ZerosZeros

ˆ

B

1

ˆ

B

2

ˆ

B

3

ˆ

B

4

- -

Block decomposition

Subband approximation

with zero-padding

Figure 2: IDDA algorithm-An 4 ×4 DCT block is decomposed into four

4

2

×

4

2

blocks where each is approximated to an 4×4

DCT block with zero-padding (Mukherjee and Mitra, 2005).

where R is deﬁned as:

R =

1/

√

8 1/

√

8 1/

√

8 1/

√

8

1/

√

10 1/

√

10 1/

√

10 1/

√

10

1/

√

8 1/

√

8 1/

√

8 1/

√

8

1/

√

10 1/

√

10 1/

√

10 1/

√

10

,

(5)

and the operator • denotes an element-by-element

multiplication. The two-dimensional transformation

in Equation (1) can be rewritten as:

B

d

= A ·

ˆ

B ·A

T

= [C •R] ·

ˆ

B ·[C

T

•R

T

]. (6)

Rearranging to extract the scaling arrays R:

B

d

= [C ·

ˆ

B ·C

T

] •[R •R

T

]

= [C ·

ˆ

B ·C

T

] •S, (7)

Where

S = R •R

T

=

1/8 1/

√

80 1/8 1/

√

80

1/

√

80 1/10 1/

√

80 1/10

1/8 1/

√

80 1/8 1/

√

80

1/

√

80 1/10 1/

√

80 1/10

. (8)

Using the new ﬁxed-point approximation of the con-

version matrix C with its scaling matrix S, we mo-

diﬁed the IHAC algorithm. Fig. 3 outlines our new

proposed image halving algorithm that takes advan-

tage of these matrices in the block composition step.

Similarly, we modiﬁed the IDDA algorithm in

the decomposition step to leverage the ﬁx-point ap-

proximation of the conversion matrix with its sca-

ling matrix. Fig 4 displays the outline of our propo-

sed ﬁx-point approximation of the IDDA algorithm

(IDDA

f pa

).

Input: 4 ×4 block based DCT encoded image.

Output: 4 ×4 block based DCT encoded downs-

ampled image.

for every four adjacent 4×4, B

1

,B

2

,B

3

,B

4

of the

input image do

1. Subband approximation: Get correspon-

ding 2 × 2 two-point DCT blocks {

ˆ

B

(2×2)

k

},

where k ∈ {0,1, 2,3} using low-pass truncated

approximation, as follows.

ˆ

B

(i, j)

k

=

1

2

[B

(i, j)

k

], f or 0 ≤i, j ≤ 1.

2. Block composition: Convert four 2 × 2

DCT blocks to a 4 ×4 DCT block B

d

as fol-

lows.

B

d

=

C

"

ˆ

B

(2×2)

1

ˆ

B

(2×2)

2

ˆ

B

(2×2)

3

ˆ

B

(2×2)

4

#

C

T

!

•S

end for

Figure 3: Our proposed resizing algorithm, IHAC

f pa

, for

halving an image based on a 4 ×4 DCT block framework.

3 QUALITY ASSESSMENT

We developed three experiments to evaluate image

quality performance of our proposed ﬁxed-point ap-

proximation of the IHAC and IDDA algorithms.

In the ﬁrst experiment, for each image in the sam-

ple set I

orig

of size N ×N is ﬁrst spatially downsam-

pled using MATLAB bicubic interpolation with anti-

aliasing to an image I

d

of size N/2 ×N/2. These

halved images provided a reference so PSNR can be

computed when comparing images produced from va-

Low Complex Image Resizing Algorithm using Fixed-point Integer Transformation

145

Input: 4 ×4 block based DCT encoded image.

Output: Upsampled image in the compressed

domain.

for each 4 ×4 blocks B do the following: do

1. Block decomposition: Convert the block to

four 2 ×2 DCT blocks as follows.

"

ˆ

B

(2×2)

1

ˆ

B

(2×2)

2

ˆ

B

(2×2)

3

ˆ

B

(2×2)

4

#

=

C

T

BC

•S

2. Subband approximation and zero pad-

ding: Compute the approximate 4 ×4-point

DCT coefﬁcients from each of

ˆ

B

(2×2)

k

, where

k ∈{0,1, 2,3}, low-pass truncated approxima-

tion.

ˆ

B

(i, j)

k

= 2

h

ˆ

B

(i, j)

k

i

, f or 0 ≤i, j ≤ 1.

Form four 4 ×4 DCT blocks by zero padding

each of them (the high frequency components

are assigned to zero).

end for

Figure 4: Our resizing algorithm IDDA

f pa

for doubling

images based on a 4 ×4 DCT block framework.

rious halving algorithms. The original images I

orig

in

the sample set are raw grayscale (8 bits/pixel) images

with a resolution of 512 ×512 pixels. In all the expe-

riments, we applied MATLAB dct2 function to trans-

form all the images I

orig

from the spatial to the DCT

domain to represent compressed images formatted as

blocks of 4×4 DCT coefﬁcients. We applied the DCT

resizing algorithms on the compressed I

orig

and out-

putted a compressed halved image. Using MATLAB

idct2, we transformed these newly compressed hal-

ved images back into the spatial domain where we

computed an PSNR with the reference halved images.

We also compared our proposed halving algorithm

with halved images that were resized in the spatial

domain by bilinear interpolation, which will be refer-

red as IHS. Tables 1 and 2 show the PSNR and SSIM

values computed from comparing the halved images

generated from our proposed halving algorithm with

IHAC and IHS. As can be seen in these tables, our

proposed halving algorithms produces images with

slightly lower PSNR or higher SSIM values compared

to its ﬂoating-point implementation (IHAC), which

was expected because of rounding errors associated

with integer transforms with scaling. In average, our

IHAC

f pa

algorithm generates images that are 0.13 dB

lower PSNR value or 0.0003 lower SSIM value than

ones generated from IHAC algorithm; however, ima-

ges generated from our halving algorithm have 2.27

dB higher quality than the images generated by IHS

algorithm.

Table 1: Experiment 1. PSNR values from image halving

experiment.

Image IHS IHAC IHAC

f pa

Fishing boat 39.94 42.00 41.67

Cameraman 40.47 46.09 45.47

Elaine 45.13 45.87 45.98

Goldhill 42.39 43.63 43.37

House 46.49 50.48 50.92

Jetplane 40.22 44.27 43.99

Lake 39.11 42.80 42.54

Lena 42.00 45.62 45.45

Livingroom 40.06 42.01 41.70

Mandril 36.54 36.49 36.32

Peppers 42.54 44.94 45.00

Pirate 40.83 43.25 43.07

Walkbridge 37.89 39.74 39.43

Watch 36.69 41.04 40.76

Woman blonde 41.61 42.72 42.60

Woman darkhair 48.73 51.19 51.65

mean(PSNR) 41.29 43.88 43.75

Table 2: Experiment 1. SSIM values from image halving

experiment.

Image IHS IHAC IHAC

f pa

Fishing boat 0.9920 0.9929 0.9925

Cameraman 0.9950 0.9985 0.9982

Elaine 0.9956 0.9930 0.9929

Goldhill 0.9922 0.9934 0.9931

House 0.9971 0.9992 0.9991

Jetplane 0.9951 0.9976 0.9974

Lake 0.9934 0.9958 0.9956

Lena 0.9947 0.9965 0.9963

Livingroom 0.9913 0.9935 0.9932

Mandril 0.9823 0.9845 0.9839

Peppers 0.9963 0.9960 0.9959

Pirate 0.9919 0.9949 0.9946

Walkbridge 0.9874 0.9922 0.9917

Watch 0.9952 0.9981 0.9980

Woman blonde 0.9936 0.9937 0.9935

Woman darkhair 0.9979 0.9984 0.9984

mean(SSIM) 0.9932 0.9949 0.9946

Figure 5 show the image ”Lena” generated by our

IHAC

f pa

algorithm as well as the ones obtained by

IHAC and IHS algorithms. Figure 5a is an image hal-

ved through bicubic interpolation with anti-aliasing,

which is the reference image so PSNR can be compu-

ted. Figure 5b is an image halved by our proposed al-

gorithm, IHAC

f pa

, with PSNR value of 45.45 dB. Fi-

gure 5c was downsampled through ﬂoating-point im-

plementation of the IHAC algorithm, which provide

a PSNR value of 45.62 dB. Figure 5d was a spatial

downsampled image using bilinear interpolation that

produced a PSNR value of 42.00 dB.

For the second experiment to evaluate image qua-

lity performance of our IDDA

f pa

algorithm, we ﬁrst

VISAPP 2018 - International Conference on Computer Vision Theory and Applications

146

(a) I

d

(b) IHAC

f pa

(c) IHAC (d) IHS

Figure 5: Resultant images downsampled in experiment 1.

spatially downsampled image I

orig

to create an image

I

d

that are compressed then upsampled in the DCT

domain. We used MATLAB bicubic interpolation

with anti-aliasing to halve images in the spatial dom-

ain to produce image I

d

. Using MATLAB dct2

function, we transformed the spatially halved image

I

d

to the DCT domain to represent a compressed

image formatted as blocks of 4 ×4 DCT coefﬁcients.

Then we applied the doubling algorithms to resize

and output compressed image to the original resolu-

tion. We transformed the compressed doubled image

back into the spatial domain using MATLAB idct2

function so a PSNR value can be computed between

the upsampled image with the original image I

orig

. Si-

milar in the ﬁrst experiment, we doubled the halved

image I

d

in the spatial domain using bilinear interpo-

lation for comparison, which will be referred as IDS

in the paper. Tables 3 and 3 display the results of

this experiment. As can be seen by these results, on

average our doubling approach created images only

0.07 dB lower than images that were spatially resi-

zed; however, in point of SSIM assessment there is

about 0.01 improvement. Again, our IDDA

f pa

algo-

rithm generates images with negligibly lower PSNR

and SSIM values compared to the ones generated

from IHAC algorithm; although, our IDDA

f pa

algo-

rithm is considerably less complex as shown in the

next section.

In the third experiment, an image is ﬁrst halved

and then doubled. The resulting upsampled image is

compared with the original image I

orig

. We compa-

red the PSNR value computed from images genera-

ted from resizing algorithms used in tandem: spatial

resizing using IHS-IDS, IHAC-IDDA and our pro-

posed halving and doubling algorithm. The results

of this experiment are shown in Tables 5 and 6. As

can be seen from the results of this experiment, using

our proposed algorithms in tandem provides PSNR

values about 0.48 dB lower PSNR value and about

0.0048 lower SSIM value than the IHAC-IDDA pai-

ring but we observed an 1.27 dB PSNR gain or about

0.0476 SSIM gain over the spatial approach imple-

Table 3: Experiment 2. PSNR values from image doubling

experiment.

Image IDS IDDA IDDA

f pa

Fishing boat 28.94 29.40 29.11

Cameraman 33.21 33.68 32.91

Elaine 32.53 32.75 32.50

Goldhill 30.62 31.03 30.78

House 41.72 41.18 39.66

Jetplane 30.17 30.68 30.28

Lake 29.35 29.77 29.37

Lena 32.69 33.17 32.72

Livingroom 28.59 28.94 28.69

Mandril 23.05 23.49 23.38

Peppers 31.23 31.62 31.31

Pirate 29.99 30.43 30.14

Walkbridge 26.23 26.70 26.47

Watch 27.11 27.55 27.14

Woman blonde 28.99 29.38 29.19

Woman darkhair 39.84 40.04 39.44

mean(PSNR) 30.89 31.24 30.82

mented using IHS-IDS pairing.

These limited experiments demonstrated to us that

our algorithms could produced images equal or better

than resizing images in the spatial domain using bi-

linear interpolation while beneﬁting from the compu-

tational savings derived from our approach. The next

section will discuss how computationally efﬁcient our

algorithms are.

4 COMPUTATIONAL COST

This section outlines the computational cost of our

proposed resizing algorithms. By using a ﬁxed-point

approximation of the conversion matrix in our propo-

sed algorithms, we hope to decrease the computati-

onal cost. Leveraging ﬁxed-point arithmetic, multi-

plying or dividing values that are a power of two can

be accomplished by binary shift operations. As can be

seen in Equation (3), the conversion matrix C entries

are either 0, 1’s or 2’s. Therefore, matrix multiplica-

Low Complex Image Resizing Algorithm using Fixed-point Integer Transformation

147

Table 4: Experiment 2. SSIM values from image doubling

experiment.

Image IDS IDDA IDDA

f pa

Fishing boat 0.8274 0.8478 0.8414

Cameraman 0.9551 0.9624 0.9539

Elaine 0.7930 0.8074 0.8038

Goldhill 0.8304 0.8527 0.8475

House 0.9838 0.9837 0.9777

Jetplane 0.9288 0.9373 0.9300

Lake 0.8610 0.8776 0.8705

Lena 0.9021 0.9140 0.9084

Livingroom 0.8194 0.8419 0.8357

Mandril 0.6576 0.7178 0.7143

Peppers 0.8829 0.8911 0.8832

Pirate 0.8565 0.8774 0.8708

Walkbridge 0.7648 0.8071 0.8009

Watch 0.9383 0.9462 0.9369

Woman blonde 0.8328 0.8526 0.8481

Woman darkhair 0.9590 0.9619 0.9587

mean(SSIM) 0.8621 0.8799 0.8739

Table 5: Experiment 3. PSNR values from image halving

followed by image doubling.

Image IHS IHAC IHAC

f pa

IDS IDDA IDDA

f pa

Fishing boat 27.99 29.64 29.37

Cameraman 31.24 33.94 33.16

Elaine 31.89 32.96 32.64

Goldhill 29.81 31.27 31.02

House 38.65 41.73 39.68

Jetplane 29.02 30.88 30.47

Lake 28.15 30.00 29.57

Lena 31.41 33.43 32.92

Livingroom 27.71 29.15 28.93

Mandril 22.52 23.71 23.62

Peppers 30.35 31.83 31.44

Pirate 29.04 30.66 30.36

Walkbridge 25.43 26.92 26.71

Watch 25.88 27.75 27.33

Woman blonde 28.34 29.59 29.40

Woman darkhair 38.43 40.39 39.48

mean(PSNR) 29.74 31.49 31.01

tions can be carried out multiplier-free.

Let assume that n

m

and n

a

are the total number of

multiplications and additions required for the image

resizing algorithm, respectively. The IHAC

f pa

algo-

rithm ﬁrst performs subband approximation and mul-

tiplies each element in the input 4 ×4 DCT coefﬁ-

cients by half, which can be implemented as a right

shift operation; thus, there is no cost. The composi-

tion step contains two matrix multiplications by ap-

plying the conversion matrix C on input DCT coefﬁ-

Table 6: Experiment 3. PSNR values from image halving

followed by image doubling.

Image IHS IHAC IHAC

f pa

IDS IDDA IDDA

f pa

Fishing boat 0.7972 0.8591 0.8546

Cameraman 0.9345 0.9648 0.9586

Elaine 0.7775 0.8192 0.8158

Goldhill 0.8003 0.8644 0.8605

House 0.9728 0.9845 0.9797

Jetplane 0.9089 0.9408 0.9359

Lake 0.8335 0.8856 0.8803

Lena 0.8823 0.9200 0.9153

Livingroom 0.7854 0.8537 0.8495

Mandril 0.6038 0.7455 0.7430

Peppers 0.8678 0.8974 0.8906

Pirate 0.8260 0.8867 0.8817

Walkbridge 0.7156 0.8240 0.8197

Watch 0.9191 0.9497 0.9420

Woman blonde 0.8097 0.8635 0.8596

Woman darkhair 0.9507 0.9644 0.9609

mean(SSIM) 0.8366 0.8890 0.8842

cients twice. Since matrix multiplication with C can

be carried out multiplier-free, only additions count,

which there are n

a

= 8. An element-by-element mul-

tiplication is applied with the scaling matrix, S. As

shown in Equation (8), the scaling matrix S contains

four elements equal to 1/8, which can be implemen-

ted as a right shift operation; and twelve elements

that need to perform multiplication. thus n

m

= 12 and

n

a

= 0. Finally, the four 4 ×4 input DCT blocks re-

present 64 pixels of the input image. Therefore, on

average our method will consume n

m

= 0.1875 and

n

a

= 0.25 per pixel of the original image.

Similarly, our IDDA

f pa

algorithm contains two

matrix multiplication using the conversion matrix

C. Also, scaling matrix and subband approximation

are similar. Therefore, the proposed doubling algo-

rithm requires n

m

= 0.1875 and n

a

= 0.25 per pixel

of the upsampled image.

Table 7 and 8 compare the computational com-

plexity for the IHAC

f pa

and IDDA

f pa

algorithms

with other image halving and doubling algorithms, re-

spectively. As shown by these tables, both proposed

image halving and doubling algorithms are more efﬁ-

cient than their ﬂoating-point implementations. When

comparing our proposed halving algorithm, IHAC

f pa

,

with its ﬂoating-point implementation, it is 63% more

efﬁcient in n

m

and 75% in n

a

. Regarding doubling

algorithm, IDDA

f pa

, it is 91% more efﬁcient in n

m

and 83% in n

a

when comparing with its ﬂoating-point

version, IDDA. When comparing both of our propo-

sed algorithms with spatial resizing, the computatio-

nal saving is about 85% in n

m

and 97% in n

a

.

VISAPP 2018 - International Conference on Computer Vision Theory and Applications

148

Table 7: Computational complexity of halving algorithms.

ops IHS IHAC IHAC

f pa

n

m

1.25 0.5 0.1875

n

a

8.25 1 0.25

Table 8: Computational complexity of doubling algorithms.

ops IDS IDDA IDDA

f pa

n

m

1.25 2 0.1875

n

a

8.25 1.5 0.25

5 CONCLUSION

In this paper, we developed a ﬁxed-point approxima-

tion of a conversion matrix that is included in IHAC

and IDDA resizing algorithms. Matrix multiplications

with ﬁxed-point approximation of the conversion ma-

trices are multiplier-free, which reduce the computati-

onal cost in the proposed resizing algorithms. Images

generated by the proposed ﬁxed-point resizing algo-

rithms have PSNR values negligibly lower than PSNR

of those images generated by the ﬂoating-point imple-

mentations; however, the total number of multiplica-

tions and additions are substantially decreased. The

proposed ﬁxed-point approximation also can be ex-

tended so one can resize images by integral or arbi-

trary factors. Also, there are several resizing algo-

rithms that use conversion matrices for composition

and decomposition such as IHCA, IDAD, LMDS and

LMUS, which would be good candidates for this ap-

proach.

ACKNOWLEDGEMENTS

The authors would like to acknowledge that this re-

search was supported by NSERC Strategic Project

Grant: Hi-Fit: High Fidelity Telepresence over Best-

Effort Networks.

REFERENCES

(2012). Advanced video coding for generic audiovisual ser-

vices.

(2012). Information technology – coding of audio-visual

objects – part 10: Advanced video coding.

Aggarwal, A. and Singh, C. (2015). Hybrid dct-zernike

moments-based approach for image up-sampling. In

2015 Annual IEEE India Conference (INDICON), pa-

ges 1–6.

Chang, S.-F. and Messerchmitt, D. G. (1995). Manipula-

tion and compositing of MC-DCT compressed video.

IEEE Journal on Selected Areas in Communications,

13(1):1–11.

Dugad, R. and Ahuja, N. (2001). A fast scheme for image

size change in the compressed domain. IEEE Tran-

saction on Circuits and Systems for Video Technology,

11(4):461–474.

Hallapuro, A., Karczewicz, M., and Malvar, H. S. (2002).

Low-complexity transform and quantization. In Joint

Video Team (JVT) of ISO/IEC MPEG and ITU-VCEG.

Hung, K. W. and Siu, W. C. (2014). Novel dct-based

image up-sampling using learning-based adaptive k

-nn mmse estimation. IEEE Transactions on Cir-

cuits and Systems for Video Technology, 24(12):2018–

2033.

Jiang, J. and Feng, G. (2002). The spatial relationship of

DCT coefﬁcients between a block and its sub-blocks.

IEEE Transaction on Signal Processing, 50(5):1160–

1169.

Malvar, H. S., Hallapuro, A., Karczewicz, M., and Kerof-

sky, L. (2003). Low-complexity transform and quanti-

zation in H.264/AVC. IEEE Transaction Circuits and

Systems Video Technology, 13:598–603.

Meher, P. K., Park, S. Y., Mohanty, B. K., Lim, K. S., and

Yeo, C. (2014). Efﬁcient integer dct architectures for

hevc. IEEE Transactions on Circuits and Systems for

Video Technology, 24(1):168–178.

Merhav, N. and Bhaskaran, V. (1997). Fast algorithms for

DCT-domain image down-sampling and for inverse

motion compensation. IEEE Transaction on Circuits

and Systems for Video Technology, 7(3):486–476.

Mukherjee, J. and Mitra, S. K. (2002). Image resizing in the

compressed domain using subband DCT. IEEE Tran-

saction on Circuits and Systems for Video Technology,

12(7):620–627.

Mukherjee, J. and Mitra, S. K. (2005). Arbitrary resizing

of images in the DCT space. In IEE Proceeding Vi-

sion, Image and Signal Processing, volume 152, pa-

ges 155–164.

Mukhopadhyay, J. and Mitra, S. (2004). Resizing of ima-

ges in the DCT space by arbitrary factors. In Image

Processing, 2004. ICIP ’04. 2004 International Con-

ference on, volume 4, pages 2801–2804.

Nam, H. M., Jeong, J. Y., Byun, K. Y., Kim, J. O., and

Ko, S. J. (2010). A complexity scalable h.264 decoder

with downsizing capability for mobile devices. IEEE

Transactions on Consumer Electronics, 56(2):1025–

1033.

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149