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Instigation of Orthogonal Wavelet Transforms using Walsh, Cosine, Hartley, Kekre Transforms and their use in Image Compression

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Instigation of Orthogonal Wavelet Transforms using Walsh, Cosine, Hartley, Kekre Transforms and their use in Image Compression Powered By Docstoc
					                                                              (IJCSIS) International Journal of Computer Science and Information Security,
                                                                                                                        Vol. 9, No. 6, 2011

  Instigation of Orthogonal Wavelet Transforms using
  Walsh, Cosine, Hartley, Kekre Transforms and their
               use in Image Compression
    Dr. H. B.Kekre                   Dr. Tanuja K. Sarode                      Sudeep D. Thepade                      Ms. Sonal Shroff
     Sr. Professor,                     Asst. Professor                        Associate Professor,                      Lecturer,
  MPSTME, SVKM’s                    Thadomal Shahani Engg.                     MPSTME, SVKM’s                     Thadomal Shahani Engg.
NMIMS (Deemed-to-be                         College,                         NMIMS (Deemed-to-be                          College
University, Vileparle(W),           Bandra (W), Mumbai-50,                   University, Vileparle(W),            Bandra (W), Mumbai-50,
   Mumbai-56, India.                         India.                             Mumbai-56, India.                          India.


Abstract—In this paper a novel orthogonal wavelet transform                molecular dynamics, astrophysics, optics, quantum mechanics
generation method is proposed. To check the advantage of                   etc. This change has also occurred in image processing, blood-
wavelet transforms over the respective orthogonal transform in             pressure, heart-rate and ECG analyses, DNA analysis, protein
image compression, the generated wavelet transforms are applied            analysis, climatology, general signal processing, speech, face
to the color images of size 256x256x3 on each of the color planes
R, G, and B separately, and thus the transformed R, G, and B
                                                                           recognition, computer graphics and multifractal analysis.
planes are obtained. Form each of these transformed color                  Wavelet transforms are also starting to be used for
planes, the 70% to 95% of the data (in form of coefficients having         communication applications. One use of wavelet
lower energy values) is removed and image is reconstructed. The            approximation is in data compression. Like other transforms,
orthogonal transforms Discrete Cosine Transform (DCT), Walsh               wavelet transforms can be used to transform data then, encode
Transform, Hartley Transform and Kekre Transform are used                  the transformed data, resulting in effective compression [8].
for the generation of DCT Wavelets, Walsh Wavelets, Hartley                Wavelet compression can be either lossless or lossy. The
Wavelets, and Kekre Wavelets respectively. From the results it is          wavelet compression methods are adequate for representing
observed that the respective Wavelet transform outperforms the             high-frequency components in two-dimensional images.
original orthogonal transform.
                                                                               So far wavelets of only Haar transform have been studied.
                                                                           The paper presents the wavelet generation of transforms alias,
                       I.    INTRODUCTION                                  Walsh transform, DCT, Hartley transform and Kekre
                                                                           transform. Also the use of these transform wavelets is
The development of wavelets can be linked to several separate              proposed and strudied for image compression. The
trains of thought, starting with Haar's work in the early 20th             experimental results have shown better data compression can
century [16,17]. Wavelets are mathematical tools that can be               be achieved in transform wavelets than using image
used to extract information from many different kinds of data,             transforms themselves.
including images [21,22,24]. Sets of wavelets are generally
needed to analyze data fully. A set of "complementary"                                      II.   EXSISTING TRANSFORMS
wavelets will reconstruct data without gaps or overlap so that
the deconstruction process is mathematically reversible and is             This section discusses some of the existing transforms, Walsh,
with minimal loss. Generally, wavelets are purposefully                    DCT, Hartley and Kekre.
crafted to have specific properties that make them useful for              A. DCT
image processing. Wavelets can be combined, using a "shift,
                                                                               A discrete cosine transform (DCT) expresses a sequence of
multiply and sum" technique called convolution, with portions
                                                                           finitely many data points in terms of a sum of cosine functions
of an unknown signal(data) to extract information from the                 oscillating at different frequencies. In particular, a DCT is a
unknown signal. Wavelet transforms are now being adopted                   Fourier-related transform similar to the discrete Fourier
for a vast number of applications, often replacing the                     transform (DFT), but using only real numbers. DCTs are
conventional Fourier transform. They have advantages over                  equivalent to DFTs of roughly twice the length, operating on
traditional fourier methods in analyzing physical situations               real data with even symmetry. There are eight standard DCT
where the signal contains discontinuities and sharp spikes[1-              variants, of which four are common. The DCTs are important
4]. In fourier analysis the local properties of the signal are not         to numerous applications in science and engineering, from
detected easily. STFT(Short Time Fourier Transform)[5] was                 lossy compression of audio and images to spectral methods for
introduced to overcome this difficulty. However it gives local             the numerical solution of partial differential equations. For
properties at the cost of global properties. Wavelets overcome             compression, the cosine functions are much more efficient
this shortcoming of Fourier analysis [6,7] as well as STFT.                whereas for differential equations the cosines express a
Many areas of physics have seen this paradigm shift, including             particular choice of boundary conditions.




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                                                                                                      ISSN 1947-5500
                                                                           (IJCSIS) International Journal of Computer Science and Information Security,
                                                                                                                                     Vol. 9, No. 6, 2011
B. Walsh Transform
    The Walsh matrix was proposed by Joseph Leonard Walsh                                            ⎧     1            ,x ≤ y
in 1923 [18,19]. Each row of a Walsh matrix corresponds to a                                         ⎪
Walsh function. A Walsh matrix is a square matrix, with                                   K x, y   = ⎨− N + ( x + 1) , x = y + 1
dimensions a power of 2. The entries of the matrix are either +1                                     ⎪     0         ,x > y +1
or −1. It has the property that the dot product of any two
distinct rows (or columns) is zero [20,23,25]. The sequency
                                                                                                     ⎩
ordering of the rows of the Walsh matrix can be derived from
                                                                                                                                                           (1)
the ordering of the Hadamard matrix by first applying the bit-
reversal permutation and then the Gray code permutation[9].                              All diagonal elements and the upper diagonal elements are
The Walsh matrix (and Walsh functions) are used in computing                          one, while lower diagonal elements except the one exactly
the Walsh transform and have applications in the efficient                            below the diagonal are zero.
implementation of certain signal processing operations.
                                                                                          III.   GENERATING WAVELET FROM ANY ORTHOGONAL
C. Hartley Transform                                                                                         TRANSFORM
    Hartley transform was proposed by R. V. L. Hartley in                                  Wavelet transform matrix of size P2 x P2 can be generated
1942, as an alternative to the Fourier transform[10]. It is one of                    from any orthogonal transform M of size PxP. For example, if
many known Fourier-related transforms. Compared to the                                we have orthogonal transform matrix of size 9x9, then its
Fourier transform, the Hartley transform has the advantages of                        corresponding wavelet transform matrix will have size 81x81.
transforming real functions to real functions (as opposed to                          i.e. for orthogonal matrix of size P, wavelet transform matrix
requiring complex numbers) and of being its own inverse.                              size will be Q, such that Q = P2.
D. Kekre Transform                                                                       Consider orthogonal transform M of size pxp as shown
                                                                                      below.
    Kekre transform[11] matrix is the generic version of
Kekre’s LUV color space matrix[12-15]. Most of the other
transform matrices have to be in powers of 2. This condition is
not required in Kekre transform. Any term in the Kekre
transform is generated as
                                             M11        M12      ...       M1 (P-1)     M1P
                                             M21        M22      ...       M2 (P-1)     M2P
                                               .         .       ...          .          .
                                               .         .                    .          .
                                             MP1       MP2       ...       MP (P-1)     MPP
                                             Figure 1 : PxP orthogonal transform matrix
               1st column of M                                     2nd column of M                                 pth column of M
               Repeated P times                                    Repeated P times                                Repeated P times
             M11        M11        ...          M11        M12       M12         ...      M12      ...       M1P          M1P      ...         M1P

             M21         M21       ...         M21        M22        M22        ...        M22     ...       M2P        M2P        ...         M2P

              .           .         ...         .          .          .        ...        .         ...        .          .        ...          .
              .           .         ...         .          .          .        ...        .         ...        .          .        ...          .
             MP1         MP1        ...        MP1        MP2        MP2       ...       MP2        ...       MPP       MPP        ...         MPP
             M21         M22        ...        M2P         0       0         ...       0          ...       0         0          ...       0
         0           0            ...        0            M21        M22       ...       M2P      ...       0         0          ...       0
         .           .            .          .          .          .         .         .          .         .         .          .         .
         .           .            .          .          .          .         .         .          .         .         .          .         .
         0           0            ...        0          0          0         ...       0          ...         M21       M22        ...         M2P
                                  ...                                        ...                  ...
                                  ...                                        ...                  ...
             MP1         MP2        ...        MPP      0          0         ...       0          ...       0         0          ...       0
         0           0            ...        0            MP1        MP2       ...       MPP      ...       0         0          ...       0
         .           .            .          .          .          .         .         .          .         .         .          .         .
         .           .            .          .          .          .         .         .          .         .         .          .         .
         0           0            ...        0          0          0         ...       0          ...         MP1       MP2        ...         MPP
                                          Figure.2: QxQ wavelet transform generated from PxP orthogonal transform( (Q = P2 )




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                                                                                                                    ISSN 1947-5500
                                                                 (IJCSIS) International Journal of Computer Science and Information Security,
                                                                                                                           Vol. 9, No. 6, 2011
Figure 2 shows QxQ wavelet transform matrix generated from
PxP orthogonal transform matrix such that Q = P2. To                        Table 1,3,5 and 7 shows the comparison of MSE values
generate the wavelet matrix, the every column of the                        obtained from data compressed using DCT, Walsh, Hartley
orthogonal transform matrix is repeated P times. Then the                   and Kekre transforms applied on all the eleven test images
second row is translated P times to generate next P rows.                   respectively.
Similarly all rows are translated to generate P rows
corresponding to each row. Finally we get the wavelet matrix                Table 2,4,6 and 8 shows the comparison of MSE values
of the size QxQ, where Q = P2                                               obtained from data compressed using DCT wavelet, Walsh
                                                                            wavelet, Hartley wavelet and Kekre wavelet transforms
                                                                            applied on all the eleven test images respectively.
                    IV.   PROPOSED METHOD
In this section, the image compression using wavelet                        Figure 4: Comparison of average MSE with respect to 95% to
transform’s application is proposed.                                        70% of data compress using DCT wavelet, Walsh wavelet,
  Step 1.    Consider an image of size 256x256. The wavelet                 Hartley wavelet, Kekre wavelet, DCT, Walsh, Hartley and
             transform matrix of size 256x256 is generated                  Kekre transform.
             from orthogonal matrix of size 16x16.
                                                                            Figure 5,6,7,8 shows the results of Balls image obtained from
  Step 2.    The wavelet transform is applied on each of the
                                                                            DCT wavelet, Walsh wavelet, Hartley wavelet and Kekre
             image plane i.e. R-plane, G-plane, B-plane                     wavelet respectively for 70% to 95% of data compress.
             separately. Thus, transformed R-plane, G-plane,
             B-plane are obtained.
  Step 3.    From the transformed R-plane, G-plane, B-plane
             separately, the 70% to 95% coefficients having
             lowest energy values are removed. And then the
             image is reconstructed.
  Step 4.    Mean Square error between the reconstructed
             image and the original image is computed.


               V.     RESULTS AND DISCUSSION
In this section, the image compression using wavelet
transform’s application is proposed. The proposed method is
implemented using MatLab 7.0 on Core 2 Duo processor.
DCT, Walsh, Hartley and Kekre wavelets were generated by
the method discussed in the section 3. The eleven different                  Figure 3:Eleven original color test images namely Aishwariya, Balls, Bird,
color images belonging to different categories, of size                      Boat, Flower, Ganesh, Scenary, Strawberry, Tajmahal, Tiger and Viharlake
256x256 were compressed using the proposed method.                             (from left to right and top to bottom) belonging to different categories


Figure 3 shows the eleven color test images of size 256x256x3
belonging to different categories.
                  Table 1: Comparison of MSE values obtained for 95% to 70% data compressed using DCT applied on all eleven images.
                 %data compressed           95            90               85            80            75              70
                  %data retained             5            10               15            20            25              30
                    Aishwariya           16.0803        8.1392           4.542        2.7457         1.7756          1.2044
                       Balls             75.5739       62.2747         50.7583        40.078        30.6593         22.6352
                       Bird              23.4414       19.7856         17.1511        14.846        12.6658         10.6395
                       Boat              63.2849       56.6238         49.9363       43.0231         36.116         29.7334
                      Flower             23.2196       13.3896          8.2092         5.158         3.3155          2.1642
                     Ganesh              66.9069       60.1663         53.5089       47.1965        41.0909          34.991
                     Scenary             32.4582       26.3064         21.5738       17.7571        14.5049         11.6967
                   Strawberry            42.358        30.1477          21.656       15.8716        11.6467          8.5902
                    Tajmahal             49.7616        39.457          30.907       23.5085        17.5614         12.7884
                       Tiger             67.5201       53.9452          42.909       33.8272        26.4423         20.1247
                    Viharlake            42.4999       35.5583         29.7518       24.3079          19.42         15.3702
                     Average             45.7368      36.89035        30.08213      24.39269        19.56349        15.4489




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                                                                                                            ISSN 1947-5500
                                                        (IJCSIS) International Journal of Computer Science and Information Security,
                                                                                                                  Vol. 9, No. 6, 2011
 Table 2: Comparison of MSE values obtained for 95% to 70% data compressed using DCT wavelets applied on all eleven images.
    %data Compressed            95               90            85              80           75                70
      %data retained             5               10            15              20           25                30
         Aishwariya          13.9138          5.4578         2.5735         1.3751        0.7916           0.4853
            Balls            67.3254          51.4344       38.2372        27.1908       18.4033          11.9991
            Bird             15.1267          7.2307         3.5948          1.975        1.2121           0.7956
            Boat             54.5218          44.1746       35.3552        27.2747       19.6998          13.4656
           Flower            18.9017          7.4338         3.2424         1.5961        0.8935           0.5555
           Ganesh            65.2921          56.5384       48.5545        40.7238        33.151          25.9447
           Scenary           29.7195          20.5452       13.5051         8.2269        4.8059           2.7274
         Strawberry          40.4291           27.01        17.4447        10.6523        6.2435           3.5483
          Tajmahal           41.9902          29.0375       19.7188        12.8007        8.0717           5.0573
            Tiger            65.7406          49.9408       37.7845        27.6822       19.5049          13.2931
          Viharlake          38.3256          29.412        21.8512        15.5274       10.4845           6.8427
           Average           41.02605        29.83775       21.98745       15.91136      11.20562         7.701327

Table 3: Comparison of MSE values obtained for 95% to 70% data compressed using Walsh transform applied on all eleven images.
      %data Compressed            95               90            85             80              75              70
         %data retained           5                10            15             20              25              30
          Aishwariya           28.1012          18.5335       13.2878         9.6173         6.9535           4.9937
             Balls             81.3445          72.1406       63.6927        55.4022         47.672          39.9922
             Bird              29.0555          24.9949       21.8778        19.0824         16.3321          13.764
             Boat              66.874           60.8236       55.0104        49.0244         42.8836         36.6938
            Flower             36.1558          26.6554       20.6424        15.9753         12.2055          9.2632
            Ganesh             71.3259          65.8241       60.7583        55.4545         49.8489         43.9028
            Scenary            36.995           30.8505       26.1749        22.1124         18.3582         14.9854
          Strawberry           50.8574          42.5104        35.597        29.5755         23.963          18.9102
           Tajmahal            56.1151          46.753        39.0347        32.1825         26.1253           20.74
             Tiger             76.8846          67.4124       59.7075        52.2118         44.8674         37.3638
           Viharlake           46.1636          40.4746       35.3193        30.1336         25.1187          20.408
           Average            52.71569         45.17936      39.19116        33.70654       28.57529        23.72883

Table 4: Comparison of MSE values obtained for 95% to 70% data compressed using Walsh wavelets applied on all eleven images.
      %data compressed           95               90             85             80             75                70
        %data retained            5               10             15             20             25                30
          Aishwariya          24.2798          14.1835        8.4923         5.1467         3.2017            2.0188
             Balls            71.2873          59.9048        50.0267        40.7423       31.9771            24.201
             Bird              21.37           12.2772        6.7889         3.7529         2.1759            1.3296
             Boat             57.1472          47.5557        39.4294        31.806        24.5572           17.8547
            Flower            29.8175          18.8151        11.6577        6.9575         4.0902            2.3961
            Ganesh            68.7866          61.5798        54.7242        47.9489       41.1375           34.1747
            Scenary           33.2859          24.4968        17.4245        11.8686        7.8334            5.1252
          Strawberry          47.5166          37.5645        29.5413        22.3788       16.4636           11.6458
           Tajmahal           46.4039            34.44        25.3372        18.217        12.5737            8.4354
             Tiger            74.0328           62.677        52.7802        43.7662       35.2198            27.521
           Viharlake          41.6009          33.4509        26.2753        19.7348       14.0675             9.508
           Average            46.86623        36.99503       29.31615       22.93815      17.57251          13.11003

  Table 5: Comparison of MSE values obtained for 95% to 70% data compressed using Hartley transform applied on all eleven images.
       %data compressed           95               90             85             80              75             70
         %data retained            5               10             15             20               25            30
           Aishwariya          17.5702           9.2244        5.3507         3.3036            2.144         1.4455
              Balls            76.1777          62.8777       51.2288         40.6618          31.174        23.0542
              Bird             24.0468          19.8922       17.1642         14.8753         12.6649        10.6047
              Boat              63.743          56.9175       50.3089         43.4315          36.585        30.0655
             Flower             23.245          13.3901        8.1816          5.132           3.3002         2.1621
             Ganesh            67.4761          60.5399       54.0034         47.5902         41.4778        35.3603
             Scenary           33.3664          26.7334       22.0444         18.2306         14.8923        11.9873
           Strawberry          43.5429          31.7365        23.165         17.109          12.8251         9.5586
            Tajmahal            49.833          39.5094        30.807         23.4858         17.5716        12.8477
              Tiger            67.9142          54.7827       44.2022         35.0442         27.6414        21.3417
            Viharlake          43.0531          35.9083       30.1151         24.6871         19.7647        15.5462
            Average            46.36076        37.41019       30.59739       24.86828         20.00373       15.8158




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                                                                                                 ISSN 1947-5500
                                                        (IJCSIS) International Journal of Computer Science and Information Security,
                                                                                                                  Vol. 9, No. 6, 2011
Table 6: Comparison of MSE values obtained for 95% to 70% data compressed using Hartley wavelets applied on all eleven images.
       %data compressed           95               90            85             80              75               70
         %data retained           5                10            15             20              25               30
          Aishwariya           25.213           13.1629        7.0521         4.0416         2.4411            1.5258
             Balls             71.1273          57.2092       45.1996        34.8136         25.8726          18.5723
             Bird              23.6101          13.4562        7.3995         4.0684         2.3198            1.4062
             Boat              57.524           47.4436       38.7197        30.4678         22.8547          16.2068
            Flower             30.4708          17.2874        9.5996         5.2322         2.9442            1.6909
            Ganesh             68.6207          59.8072       51.8952        44.4754         36.9332          29.7399
            Scenary            33.7757          24.0033       16.4779        10.8463         6.9661            4.4146
          Strawberry           47.5595          35.0583        25.241          17.6          12.0036           8.0276
           Tajmahal            45.9356          33.1194       23.4711        16.1064         10.6663           6.9495
             Tiger             71.4263          57.522        46.3082        36.2968         27.6191          20.1696
           Viharlake           40.7793          31.5152       23.8878        17.1808          11.82            7.7949
           Average            46.91294         35.41679      26.84106        20.10266       14.76734         10.59074

Table 7: Comparison of MSE values obtained for 95% to 70% data compressed using Kekre transform applied on all eleven images.
      %data compressed            95              90             85             80              75               70
        %data retained            5               10             15             20              25               30
          Aishwariya          104.7137         98.3782        89.9261        79.8792         71.1881         63.4727
             Balls             95.4663         94.6395        91.7457        85.9287         78.2736         70.4866
             Bird              71.0262         66.9461        61.3682         53.14          44.6532         36.7741
             Boat              96.431           91.469        85.2311        77.9438         70.1285          62.199
            Flower             75.3232          73.098        70.8086        66.3523         61.4276         54.4408
            Ganesh             89.5643         85.6352         79.994        73.6673         66.8255         59.6006
            Scenary            69.4835         66.4225        60.9814        54.9584         48.8438         42.7133
          Strawberry           91.5023         86.9626        82.2165        76.2722         69.2823         61.9791
           Tajmahal            87.7596          81.797        74.3466        66.3947         58.3154         50.4829
             Tiger            103.1722          96.723         90.382        84.1157         77.8424         71.2272
           Viharlake           65.3572         60.0596        54.1362        48.2416         42.1389         36.1324
           Average            86.34541         82.01188      76.46695        69.71763       62.62903        55.40988

Table 8: Comparison of MSE values obtained for 95% to 70% data compressed using Kekre wavelets applied on all eleven images.
       %data compressed           95              90            85              80             75              70
         %data retained            5              10            15              20             25              30
           Aishwariya          41.1364         30.8323        22.567         15.7857       10.5192           6.8614
              Balls            78.2216         69.5023       60.7559         51.6477       42.7972           34.186
              Bird             30.9089         18.9243       11.0784         6.3362         3.7573           2.3177
              Boat             61.2107         51.5525       43.1938         35.063        27.1555          20.0594
             Flower            43.864          33.5713       24.2075         16.0388        9.7355           5.4288
             Ganesh            73.9496         67.8945       61.3198         54.3184       46.8956          39.3169
             Scenary           40.3419         30.1718       22.2297         15.6323       10.7463           7.3131
           Strawberry           58.15          49.6929       41.7717         34.4112       27.2761          20.7789
            Tajmahal           53.9875         41.6384       32.1871         23.958        16.8148          11.1398
              Tiger            86.4872         76.9422       68.3485         60.3057       51.9975          43.7026
            Viharlake          46.5487         39.6689       32.3033         25.1071       18.4207           12.622
            Average            55.8915        46.39922       38.17843       30.78219       24.19234         18.5206

Table 9: Comparison of average MSE values obtained for 95% to 70% data compressed using DCT, Walsh, Hartley, Kekre transforms and their
                                          corresponding wavelets applied on all eleven images.


       %data compressed                95              90                85           80              75             70
        %data retained                  5              10                15           20              25             30
        DCT Wavelets             41.02605        29.83775          21.98745     15.91136        11.20562       7.701327
        Walsh Wavelets           46.86623        36.99503          29.31615     22.93815        17.57251       13.11003
       Hartley Wavelets          46.91294        35.41679          26.84106     20.10266        14.76734       10.59074
        Kekre Wavelets            55.8915        46.39922          38.17843     30.78219        24.19234        18.5206
             DCT                  45.7368        36.89035          30.08213     24.39269        19.56349        15.4489
            Walsh                52.71569        45.17936          39.19116     33.70654        28.57529       23.72883
           Hartley               46.36076        37.41019          30.59739     24.86828        20.00373        15.8158
            Kekre                86.34541        82.01188          76.46695     69.71763        62.62903       55.40988




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Figure 4: Comparison of average MSE with respect to 95% to 70% of data compressed using DCT wavelet, Walsh wavelet, Hartley wavelet, Kekre wavelet,
                                                     DCT, Walsh, Hartley and Kekre transform.




                        Figure 5: Results of Balls image obtained from DCT wavelet for 70% to 95% of data compressed.




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Figure 6: Results of Balls image obtained from Walsh wavelet for 70% to 95% of data compressed.




Figure 7: Results of Balls image obtained from Hartley wavelet for 70% to 95% of data compressed.




                                              131                                  http://sites.google.com/site/ijcsis/
                                                                                   ISSN 1947-5500
                                                                         (IJCSIS) International Journal of Computer Science and Information Security,
                                                                                                                                   Vol. 9, No. 6, 2011




                               Figure 8: Results of Balls image obtained from Kekre wavelet for 70% to 95% of data compressed.

      From Table 9, it is observed that performance of all                                   Proceedings of the International Computer Music Conference (ICMC-
                                                                                             87, Tokyo), Computer Music Association, 1987.
      wavelet transforms is better than that of their respective
                                                                                      [6]    S. Mallat, "A Theory of Multiresolution Signal Decomposition: The
      orthogonal transforms as indicated by lower MSE values.                                Wavelet Representation," IEEE Trans. Pattern Analysis and Machine
      Figure 4 compares the average MSE with respect to 95%                                  Intelligence, vol. 11, pp. 674-693, 1989.
      to 70% od data compress using. DCT wavelet, Walsh                               [7]    Strang G. "Wavelet Transforms Versus Fourier Transforms." Bull.
      wavelet, Hartley wavelet, Kekre wavelet, DCT, Walsh,                                   Amer. Math. Soc. 28, 288-305, 1993.
      Hartley and Kekre transform.                                                    [8]    P. P. Kanjilal, “Adaptive Prediction and Predictive Control”, IET, p 210,
                                                                                             1995
                           VI.     CONCLUSION                                         [9]    Yuen, C. “Remarks on the Ordering of Walsh Functions”, IEEE
                                                                                             Transactions on Computers, C-21: 1452, 1972.
      In this paper, novel orthogonal wavelet transform
                                                                                      [10]   Hartley, R. V. L., “A more symmetrical Fourier analysis applied to
      generation method is proposed. The proposed method can                                 transmission problems”, Proc. IRE 30, 144–150, 1942.
      be used to generate the wavelet transform from any                              [11]   H.B.Kekre, Sudeep D. Thepade, “Image Retrieval using Non-
      orthogonal transform. To test the efficiency of wavelet                                Involutional Orthogonal Kekre’s Transform”, International Journal of
      transform, they are applied on the eleven different color                              Multidisciplinary Research and Advances in Engineering (IJMRAE),
                                                                                             Ascent Publication House, 2009, Volume 1, No.I, 2009. Abstract
      images for the purpose of data compression. The                                        available online at www.ascent-journals.com
      orthogonal transforms used in this paper are DCT, Walsh,                        [12]   H. B.Kekre, Sudeep D. Thepade, “Image Blending in Vista Creation
      Hartley and Kekre. From the results, it can be concluded                               using Kekre's LUV Color Space”, SPIT-IEEE Colloquium and Int.
      that wavelet transforms outperforms their respective                                   Conference, SPIT, Andheri, Mumbai, 04-05 Feb 2008.
      orthogonal transform as indicated by lower MSE values                           [13]   H.B.Kekre, Sudeep D. Thepade, “Boosting Block Truncation Coding
                                                                                             using Kekre’s LUV Color Space for Image Retrieval”, WASET Int.
                              REFERENCES                                                     Journal of Electrical, Computer and System Engineering (IJECSE),
                                                                                             Vol.2, Num.3, Summer 2008. Available online at
                                                                                             www.waset.org/ijecse/v2/v2-3-23.pdf
[1]   K. P. Soman and K.I. Ramachandran. ”Insight into WAVELETS From                  [14]   H.B.Kekre, Sudeep D. Thepade, “Color Traits Transfer to Grayscale
      Theory to Practice”, Printice -Hall India, pp 3-7, 2005.                               Images”, In Proc.of IEEE First International Conference on Emerging
[2]   Raghuveer M. Rao and Ajit S. Bopardika. “Wavelet Transforms –                          Trends in Engg. & Technology, (ICETET-08), G.H.Raisoni COE,
      Introduction to Theory and Applications”, Addison Wesley Longman,                      Nagpur, INDIA. Available on IEEE Xplore.
      pp 1-20, 1998.                                                                  [15]   H.B.Kekre, Sudeep D. Thepade, “Creating the Color Panoramic
[3]   C.S. Burrus, R.A. Gopinath, and H. Guo. “Introduction to Wavelets and                  Viewusing Medley of Grayscale and Color Partial Images”, WASET Int.
      Wavelet Transform” Prentice-hall International, Inc., New Jersey, 1998.                Journal of Electrical, Computer and System Engg. (IJECSE), Volume 2,
[4]   Amara Graps, ”An Introduction to Wavelets”, IEEE Computational                         No. 3, Summer 2008. Available online at www.waset.org/ijecse/v2/v2-3-
      Science and Engineering, vol. 2, num. 2, Summer 1995, USA.                             26.pdf
[5]   Julius O. Smith III and Xavier SerraP“, An Analysis/Synthesis Program           [16]   Dr. H.B.kekre, Sudeep D. Thepade, Adib Parkar, “A Comparison of
      for Non-Harmonic Sounds Based on a Sinusoidal Representation'',                        Haar Wavelets and Kekre’s Wavelets for Storing Colour Information in
                                                                                             a Greyscale Image”, International Journal of Computer Applications




                                                                                132                                      http://sites.google.com/site/ijcsis/
                                                                                                                         ISSN 1947-5500
                                                                          (IJCSIS) International Journal of Computer Science and Information Security,
                                                                                                                                    Vol. 9, No. 6, 2011
       (IJCA), Volume 1, Number 11, December 2010, pp 32-38. Available at              Professor at MPSTME, SVKM’s NMIMS. He has guided 17 Ph.Ds, more than
       www.ijcaonline.org/archives/volume11/number11/1625-2186                         100 M.E./M.Tech and several B.E./ B.Tech projects. His areas of interest are
[17]   Dr. H.B.kekre, Sudeep D. Thepade, Adib Parkar “Storage of Colour                Digital Signal processing, Image Processing and Computer Networking. He
       Information in a Greyscale Image using Haar Wavelets and Various                has more than 270 papers in National / International Conferences and Journals
       Colour Spaces”, International Journal of Computer Applications (IJCA),          to his credit. He was Senior Member of IEEE. Presently He is Fellow of IETE
       Volume 6, Number 7, pp.18-24, September 2010. Available online at               and Life Member of ISTE Recently 11 students working under his guidance
       http://www.ijcaonline.org/volume6/number7/pxc3871421.pdf                        have received best paper awards. Two of his students have been awarded Ph.
[18]   Dr.H.B.Kekre, Sudeep D. Thepade, Juhi Jain, Naman Agrawal, “IRIS                D. from NMIMS University. Currently he is guiding ten Ph.D. students.
       Recognition using Texture Features Extracted from Walshlet Pyramid”,
       ACM-International Conference and Workshop on Emerging Trends in                 Dr. Tanuja K. Sarode has Received Bsc.(Mathematics) from Mumbai
       Technology (ICWET 2011),Thakur College of Engg. And Tech.,                                                  University     in    1996,    Bsc.Tech.(Computer
       Mumbai, 26-27 Feb 2011. Also will be uploaded on online ACM Portal.                                         Technology) from Mumbai University in 1999,
                                                                                                                   M.E. (Computer Engineering) degree from
[19]   Dr.H.B.Kekre, Sudeep D. Thepade, Akshay Maloo, “Face Recognition
                                                                                                                   Mumbai University in 2004, Ph.D. from Mukesh
       using Texture Features Extracted form Walshlet Pyramid”, ACEEE
                                                                                                                   Patel School of Technology, Management and
       International Journal on Recent Trends in Engineering and Technology
                                                                                                                   Engineering, SVKM’s NMIMS University, Vile-
       (IJRTET), Volume 5, Issue 1, www.searchdl.org/journal/IJRTET2010
                                                                                                                   Parle (W), Mumbai, INDIA. She has more than 12
[20]   Dr.H.B.Kekre, Sudeep D. Thepade, Juhi Jain, Naman Agrawal,                                                  years of experience in teaching. Currently working
       “Performance Comparison of IRIS Recognition Techniques using                                                as Assistant Professor in Dept. of Computer
       Wavelet Pyramids of Walsh, Haar and Kekre Wavelet Transforms”,                                              Engineering at Thadomal Shahani Engineering
       International Journal of Computer Applications (IJCA), Number 2,                College, Mumbai. She is life member of IETE, member of International
       Article                  4,                March                  2011,         Association of Engineers (IAENG) and International Association of Computer
       http://www.ijcaonline.org/proceedings/icwet/number2/2070-aca386                 Science and Information Technology (IACSIT), Singapore. Her areas of
[21]   Dr.H.B.Kekre, Sudeep D. Thepade, Akshay Maloo, “Face Recognition                interest are Image Processing, Signal Processing and Computer Graphics. She
       using Texture Features Extracted from Haarlet Pyramid”, International           has 90 papers in National /International Conferences/journal to her credit.
       Journal of Computer Applications (IJCA), Volume 12, Number 5,
       December          2010,        pp       41-45.        Available      at         Sudeep D. Thepade has Received B.E.(Computer) degree from North
       www.ijcaonline.org/archives/volume12/number5/1672-2256                                                          Maharashtra University with Distinction in
[22]   Dr.H.B.Kekre, Sudeep D. Thepade, Juhi Jain, Naman Agrawal, “IRIS                                                2003. M.E. in Computer Engineering from
       Recognition using Texture Features Extracted from Haarlet Pyramid”,                                             University of Mumbai in 2008 with
       International Journal of Computer Applications (IJCA), Volume 11,                                               Distinction, currently submitted thesis for
       Number       12,   December       2010,    pp    1-5,   Available    at                                         Ph.D. at SVKM’s NMIMS, Mumbai. He has
       www.ijcaonline.org/archives/volume11/number12/1638-2202.                                                        more than 08 years of experience in teaching
[23]   Dr.H.B.Kekre, Sudeep D. Thepade, Akshay Maloo, “Performance                                                     and industry. He was Lecturer in Dept. of
       Comparison of Image Retrieval Techniques using Wavelet Pyramids of                                              Information Technology at Thadomal Shahani
       Walsh, Haar and Kekre Transforms”, International Journal of Computer                                            Engineering College, Bandra(w), Mumbai for
       Applications (IJCA) Volume 4, Number 10, August 2010 Edition, pp 1-                                             nearly 04 years. Currently working as
       8, http://www.ijcaonline.org/archives/volume4/number10/866-1216                                                 Associate Professor in Computer Engineering
                                                                                                                       at Mukesh Patel School of Technology
[24]   Dr.H.B.Kekre, Sudeep D. Thepade, Akshay Maloo, “Query by image                  Management and Engineering, SVKM’s NMIMS, Vile Parle(w), Mumbai,
       content using color texture features extracted from Haar wavelet                INDIA. He is member of International Association of Engineers (IAENG) and
       pyramid”, International Journal of Computer Applications (IJCA) for the         International Association of Computer Science and Information Technology
       special edition on “Computer Aided Soft Computing Techniques for                (IACSIT), Singapore. He is member of International Advisory Committee for
       Imaging and Biomedical Applications”, Number 2, Article 2, August               many International Conferences. He is reviewer for various International
       2010. http://www.ijcaonline.org/specialissues/casct/number2/1006-41             Journals. His areas of interest are Image Processing Applications, Biometric
[25]   Dr.H.B.Kekre, Sudeep D. Thepade, “Image Retrieval using Color-                  Identification. He has about 110 papers in National/International
       Texture Features Extracted from Walshlet Pyramid”, ICGST                        Conferences/Journals to his credit with a Best Paper Award at International
       International Journal on Graphics, Vision and Image Processing (GVIP),          Conference SSPCCIN-2008, Second Best Paper Award at ThinkQuest-2009
       Volume 10, Issue I, Feb.2010, pp.9-18, Available online                         National Level paper presentation competition for faculty, Best paper award at
       www.icgst.com/gvip/Volume10/Issue1/P1150938876.html                             Springer international conference ICCCT-2010 and second best research
                                                                                       project award at ‘Manshodhan-2010’.
                          AUTHORS PROFILE
Dr. H. B. Kekre has received B.E. (Hons.) in Telecomm. Engineering. from               Ms. Sonal Shroff has Received B.Sc.(Physics) from University of Mumbai in
                             Jabalpur University in 1958, M.Tech                                                1996, B.Sc.Tech.(Computer Technology) from
                             (Industrial Electronics) from IIT Bombay in                                        University of Mumbai in 1999. She has more than
                             1960, M.S.Engg. (Electrical Engg.) from                                            10 years of experience in teaching. Currently
                             University of Ottawa in 1965 and Ph.D.                                             working as Lecturer in Dept. of Computer
                             (System Identification) from IIT Bombay                                            Engineering at Thadomal Shahani Engineering
                             in 1970 He has worked as Faculty of                                                College. She is life member of ISTE. Her areas of
                             Electrical Engg. and then HOD Computer                                             interest are Image Processing, Signal Processing
                             Science and Engg. at IIT Bombay. For 13                                            and Computer Graphics.
                             years he was working as a professor and head
                             in the Department of Computer Engg. at
Thadomal Shahani Engineering. College, Mumbai. Now he is Senior




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                                                                                                                        ISSN 1947-5500