____________________________________________________

* This work is supported by FRGS/2007/FTMK(3)-F0050, Ministry of Higher Education, Malaysia.

IMAGE SUPER-RESOLUTION VIA DISCRETE TCHEBICHEF MOMENT*

NUR AZMAN ABU, WONG SIAW LANG AND SHAHRIN SAHIB

Faculty of Information and Communication Technology, Universiti Teknikal Malaysia Melaka (UTeM), Durian Tunggal, 76109 Melaka, Malaysia

Super-resolution is a set of methods of increasing or doubling image resolution. Super-resolution algorithms can be divided into frequency or space domain. In this paper, a simple technique shall be applied in both space and frequency domain of an image. This transform integrates a simplified mathematical framework technique using matrices, as well as a block-wise reconstruction technique. Tchebichef moment has been chosen here since it performs better than the popular Discrete Cosine Transform.

1. Introduction

Digital image has been an importance element in daily life. Users can easily capture images using small computing devices such as PDA, digital camera, mobile phone and so on. However, the displaying devices have been improved in a wide range from a small screen for mobile phone or PDA to a large LCD monitor with high definition. This leads to the importance of image resolution translation technique that can enlarge image size for suitable display on different devices.

Image enlargement is a method to convert from a lower resolution image into a higher resolution image. Some image enlargement applications are medical imaging[1], videoconferencing[2], and digital photographs[3]. Besides for displaying purposes, image enlargement can be used in zooming application as well. So far, various researches have been done in image enlargement. The most popular methods are interpolation by convolution, which includes nearest neighbor method[4], bi-linear interpolation[5], and bi-cubic interpolation[6].

This paper will present the Discrete Orthogonal Transform methods, which includes the Discrete Cosine Transform (DCT)[7] and Tchebichef Moment Transform (TMT), in image super-resolution. Although DCT for image enlargement has been proposed by Stephen A. Martucci[8], it has not been practically explored after a quarter century. Earlier, orthogonal moment functions are used in several computer vision and related image processing applications, such as pattern recognition, object identification, template matching, and pose estimation[9]. This paper will not

only inspect the image quality visually, but also analyze the reconstruction difference on re-enlargement images.

The organization of the paper is as follows. The next section will have brief description on the Tchebichef Moment Transform. The Image Algebra moment equation is reviewed in Section 3. Section 4 presents the experimental methods and results of discrete orthogonal transform on image super-resolution. Lastly Section 6 will conclude this paper.

2. Tchebichef Moment Transform

Let Tmn be TMT based on a discrete orthogonal polynomial set {tn(x)} defined directly on the image space [0, S–1], thus satisfying all the required analytical properties without any numerical approximation errors:

Tmn = ∑∑−

=

−

=

1

0

1

0

),()()(),(),(

1 S

i

S

jnm jifjtit

SnSm ρρ (1)

for m, n = 0, 1, 2, …, S −1

For a description of the properties of Tchebichef polynomials and the definitions of related terms such as the squared-norm ρ(), please refer to [9]. The Tchebichef orthogonal polynomials set {tn(x)} can be generated iteratively as follows,

,1)(0 =xt

,12

)(1 SSx

xt−+

=

( )

)(1

)1()()()12(

)(22

22

11

n

xtSnS

nxtxtn

xtnn

n

−− ⋅

−−−−⋅⋅−

= (2)

for n = 2, 3, …, S–1

2009 International Conference on Computer Technology and Development

978-0-7695-3892-1/09 $26.00 © 2009 Crown Copyright

DOI 10.1109/ICCTD.2009.61

315

2009 International Conference on Computer Technology and Development

978-0-7695-3892-1/09 $26.00 © 2009 Crown Copyright

DOI 10.1109/ICCTD.2009.61

315

Tchebichef Orthogonal Polynomials

-0.50

-0.25

0.00

0.25

0.50

0 1 2 3 4 5 6 7

x

t n (x )

t0 t1 t2 t3

Figure 1a. The discrete orthogonal Tchebichef Polynomial tn(x) for n =

0, 1, 2 and 3.

Discrete Orthogonal Cosine Functions

-0.50

-0.25

0.00

0.25

0.50

0 1 2 3 4 5 6 7

x

c n (x )

c0 c1 c2 c3

Figure 1b. The discrete cosine functions cn(x) for n = 0, 1, 2 and 3.

The first few discrete orthogonal Tchebichef

polynomials are shown in Figure 1a. The equivalent discrete orthogonal cosine functions are shown in Figure 1b. The above definition uses the following scale factor[10] for the polynomial of degree

β(n, S)= Sn (3)

The set {tn(x)}, has a squared-norm given by

{ }21

0

)(),( xtSn i

S

i∑−

=

=ρ

12

13

12

11

12

2

2

2

2

2

2

2

+

−⋅

−⋅

−⋅

−⋅

=n

S

n

SSSS L

(4)

Discrete orthogonal Tchebichef moment has its own advantages in image processing which has not been fully explored. Since computer image data operates on integers, discrete orthogonal Tchebichef moment is suitable for computer image processing. As shown in Figure 1a, the polynomial domain is discrete over natural numbers. Unlike the continuous orthogonal transform, discrete orthogonal Tchebichef moment is capable of performing image reconstruction exactly without any numerical errors[11].

3. Matrix Implementation of Moment Equations

This section provides a compact representation of the moment equations and the inverse moment transform. In the following discussion, from Eq. (1) the moment set consists of all orders of moments with the values for 0 < m, n < S of block size 0<S<N, and that the image size is N x N pixels. At first, the image matrix was subdivided into 4x4 pixels where the orthogonal moment on each block was computed independently. The block size S is taken to be S = N0 = 4 and then extendable to N1 = 8.

For simplicity, consider the discrete orthogonal moment definition (1) above, and define a kernel matrix K(4x4) as follows:

=

)3()3()3()3(

)2()2()2()2(

)1()1()1()1(

)0()0()0()0(

3210

3210

3210

3210

tttt

tttt

tttt

tttt

K (5)

Let the image block intensity matrix by F(4x4) with f()

denoting the intensity values:

=

)3,3()2,3()1,3()0,3(

)3,2()2,2()1,2()0,2(

)3,1()2,1()1,1()0,1(

)3,0()2,0()1,0()0,0(

ffff

ffff

ffff

ffff

F

(6)

The matrix T(4x4) of moments defined according to

Eq. (2) can now be formed as T(4x4) = KT

(4x4) F(4x4) K(4x4) (7)

The inverse moment relation used for

reconstructing the image block from the above moment set is now simply,

G(8x8) = K(8x4) T(4x4) KT(4x8)

(8)

316316

where G(8x8) denotes the matrix (image) of the reconstructed intensity values g(i, j). The visual representation of the matrix Eq. (8) is given in Figure 2.

Figure 2. The visual representation of the block matrices.

4. Experimental Results on Overlap DCT versus TMT

This section presents the experimental methods and results for image enlargement. There are 80 images chosen to be tested and went through the basic image super-resolution experimental validation. These images are classified into 40 real images and 40 graphical images respectively. Originally, all the images (24-bit RGB, 512x512 pixels) are down-sampled to ½ of their original size (24-bit RGB, 256x256 pixels) using bi-cubic interpolation method [6] whereby the output pixel value is a weighted average of pixels in the nearest 4x4 neighborhood. The down-sampled images are then enlarged using DCT and TMT methods to their original size.

In order to apply orthogonal transform, the image is divided into 4x4 blocks of pixels. The 4x4 blocks are processed from left-to-right and from top-to-bottom. The 4x4 block image shall be transformed into frequency domain under DCT or TMT according to Eq. (7). The 4x4 DCT frequency coefficients or TMT moment coefficients shall be used to reconstruct a larger 8x8 image according to Eq. (8). Out of this 8x8 image, only a portion shall be taken as an output depending on pixel shift s. For each layers of input image, the 4x4 input blocks may be shifted by one, two, three or four pixels at a time, thus the output image shall consists of

different pixel-blocks according to the pixel shift. Refer to Figure 3, when the 4x4 blocks of image pixel is shifted by 1-pixel, the output image taken would be 2x2 blocks of pixel. The same strategy apply to the 4x4 blocks of image shifted by 2, 3 and 4 pixels whereas the output image would be 4x4, 6x6 and 8x8 blocks of pixels respectively.

Figure 3. The visual representation of the strategy image super-resolution.

After the transformation and image reconstruction,

the image full reconstruction error is calculated for the difference between the original and re-enlargement images for each transform. The image full reconstruction error can be defined as

( ) ( ) ( )∑∑∑−

=

−

= =

−=1

0

1

0

2

0

,,,,3

1 M

i

N

j k

kjifkjigMN

SE (9)

where the third index in the intensity values refers to the three color RGB layers.

Table 1a. Average Error Score between DCT and TMT for 40 Real

Images.

s DCT TMT Diff.

1 6.0115 6.0362 0.0247

2 5.9222 5.8697 -0.0525

3 6.1260 6.6103 0.4843

4 6.0296 6.6419 0.6123

RGB image

Moment Function

Moment Coefficients

Moment Inverse

Input image 4x4 blocks with different pixel-shift,

s = 1, 2, 3, 4.

1

2

3

4

1 2 3 4

Output image with different block size,

2x2, 4x4, 6x6, 8x8

317317

Table 1b. Average Error Score between DCT and TMT for 40

Graphical Images. s DCT TMT Diff.

1 7.3425 7.3700 0.0275

2 7.2350 7.0800 -0.1550

3 7.5200 8.0100 0.4900

4 7.3050 7.9000 0.5950

Error Analysis for Real Images

5.6

5.8

6.0

6.2

6.4

6.6

6.8

1 2 3 4s

Err

or V

alue

DCT TMT

Figure 4a. The Illustration of Error Analysis for Real Images.

Error Analysis for Graphical Images

6.8

7.0

7.2

7.4

7.6

7.8

8.0

8.2

1 2 3 4s

Err

or V

alue

DCT TMT

Figure 4b. The Illustration of Error Analysis for Graphical Images.

The full reconstruction error performance of DCT

and TMT for real and graphical images are displayed in Figure 4a and 4b above.

On average, TMT does not perform better than DCT for real and graphical images as shown in Table 1a and Table 1b. However, TMT generates smaller reconstruction errors for s = 2. This indicates that the TMT enlarges better quality of super-sampling image when the input pixel image is overlapping by 2 pixels.

(a) Original

(b) DCT

(c) TMT

Figure 5. Comparison of Enlarged Images for Original, DCT and

TMT methods (s = 2, zoomed, enlarged 400%).

Apart from analyzing the reconstruction difference

on re-enlargement images, inspecting the image quality visually can be done as well. For fair evaluation purposes, a fixed parameter value is used in DCT and TMT methods. The best parameter value for pixel-shift, s is searched by looking at error value of re-enlargement image, and the parameters that appear to yield the smaller error value of TMT is selected. In this case, TMT performs best with s = 2. Figure 5 shows the 400% enlarged image of Lena, zoomed to the right eye and shoulder part, using DCT and TMT methods, along with the original image for s = 2.

318318

Based on Figure 5b, DCT gives smoother output. The output from DCT is ideally rounder whereas the output shoulder out of TMT is projected closer toward the original shoulder of Lena image in Figure 5c.

5. Conclusion

In this paper, a novel approach based on 4x4 discrete orthogonal Tchebichef Moment Transform for fast image super-resolution is proposed. This transform integrates a simplified mathematical framework technique using matrices, as well as a block-wise reconstruction technique to eliminate possible occurrences of numerical instabilities at higher moment orders. Under the condition of fixed parameter s, the comparison between Discrete Cosine Transform and Tchebichef Moment Transform has been done. The experimental results show that our proposed method performs better than Discrete Cosine Transform when image sub-block is shifted by s = 2 pixels. This can be confirmed by producing smaller reconstruction difference and inspecting image quality visually. The pixel shift s = 2 is a good compromise between efficiency and image quality.

References

1. T.M. Lehmann, C. Gonner and k. Spitzer, “Survey: Interpolation Methods in Medical Image Processing”, IEEE Trans. On Medical Imaging, Vol. 18, No. 11, pp. 1049-1075, Nov 1999.

2. Mei-Juan Chen, Chin-Hui Huang and Wen-Li Lee, “A Fast Edge-Oriented Algorithm for Image Interpolation”, Image and Vision Computing, Vol. 23, No.9 pp.791-798, Sep 2005.

3. M. Unser, “Splines: A Perfect Fit for Signal and Image Processing”, IEEE Signal Processing, No.11, pp. 22-38, Nov 1999.

4. P.Thevenaz, T. Blu and M. Unser, “Interpolation Revisited”, IEEE Transaction On Medical Imaging, Vol. 19, No.7, pp. 739-758, Jul 2000.

5. Y.T. Tsai, W.G. Hsu, S.Y. Tseng and S.L. Cheng, “Optimized Image Processing Algorithms for a Single Sensor Camera”, IEEE Pacific Conference on Communications, Computers and Signal Processing, Vol. 2, pp. 1010-1013, Aug 1997.

6. H.S. Hou and H.C. Andrews, “Cubic Spline for Image Interpolation and Digital Filtering, IEEE Transaction on Speech and Signal Processing, Vol. 26, No. 9, pp. 508-517, Dec 1978.

7. N. Ahmed, T. Neterajan and K. R. Rao, “Discrete cosine transform”, IEEE Trans. on Computers, Vol. 23, pp. 90-93, Jan 1974.

8. Stephen A. Martucci, “Image Resizing in the Discrete Cosine Transform Domain”, Proceedings International Conference Image Processing, Vol. 2, pp. 244-247, 1995.

9. R. Mukundan, “Some Computational Aspects of Discrete Orthonormal Moments”, IEEE Transactions on Image Processing, Vol. 13, No. 8, pp. 1055-1059, Aug 2004.

10. R. Mukundan, S. H. Ong and P. A. Lee, “Image Analysis by Tchebichef Moments”, IEEE Trans. on Image Processing, Vol. 10, No 9, pp. 1357–1364, Sep. 2001.

11. Nur Azman Abu, Nanna Suryana and R. Mukundan, “Perfect Image Reconstruction Using Discrete Orthogonal Moments”, Proceedings of The 4th IASTED International Conference on Visualization, Imaging, and Image Processing VIIP2004, Marbella, SPAIN, pp. 903-907, 6-8 Sep. 2004.

319319