Fast Wavelet Histogram Techniques for Image Indexing

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1 Fast Wavelet Histogram Techniques for Indexing M K Mandal, and T Aboulnasr School of Information Technology and Engineering University of Ottawa, Ottawa, Canada - K mandalm@margegenieuottawaca S anchanathan epartment of Computer Science and Engineering Arizona State University, Tempe, Arizona, USA panch@asuedu Abstract Content based image indexing is emerging as an important research area with application to digital libraries and multimedia databases A majority of indexing techniques is based on pixel domain feature such as histogram, color, texture, and shape However, with recent advances in image compression, compressed domain indexing techniques are gaining popularity due to their low complexity Recently, a wavelet histogram technique which exploits the directional properties of wavelet transform has been proposed in the literature Although, this technique provides a good retrieval performance for texture images, its complexity is very high In this paper, we propose three techniques to reduce the complexity of the wavelet histogram method These techniques together provide a superior performance at a substantially reduced complexity, and hence can be considered as a potential candidate for developing a joint wavelet based image storage and retrieval system 1

2 1 Introduction and video indexing techniques have become important with recent advances in image and video compression standards such as JEG- [1], and MEG- [] These techniques store and retrieve images and video based on their contents [3], and have wide applications in several areas such as multimedia information systems, digital libraries, movie industry, and video on demand The block schematic of a typical image storage and retrieval system is shown in Fig 1 Traditional databases use keywords as labels to quickly access large quantities of text data However, representation of visual data with text labels needs a large amount of manual processing In addition, the retrieval results might not be satisfactory when the query is based on features not abstracted by the associated keyword indexing techniques are generally based on features such as color [], texture [5], sketch and shape [6] These features represent different properties of an image and a particular feature is employed depending on the application at hand The color (or gray level) feature of an image is chosen when it is desired to retrieve an image with specific color (or gray level) content The gray level or color content of an image is generally represented using histograms Texture feature is employed to distinguish images based on textural pattern Sketch or shape is employed to retrieve images containing objects of a specific shape We note that the features mentioned above have different orders of complexity It has been seen that the color/histogram feature provides a good indexing and retrieval performance while being computationally inexpensive [7] Here, the histogram of an image is employed as an index Retrieval is done by comparing the histograms of the query image with histograms of all candidate images in a database A sub-set of images with least difference of image histograms (OIH) is then retrieved as similar images Histogram is invariant to translation and rotation of images As a result, the histogram-based methods are robust to small camera operations These techniques generally perform well for natural images with large objects However, it fails to provide a good indexing performance for texture images, since different texture images can have very similar histograms The image features such as histogram, and texture are extracted directly from the image pixels Recently, image and video compression standards such as JEG, and MEG, have been proposed to reduce the bandwidth and storage requirements Hence, images and videos are expected to be stored in compressed form This has led to the proliferation of a number of compressed domain indexing techniques [8] in the literature Here, indexing is performed directly on the compressed data and hence the need to decompress the image data is eliminated, resulting in a reduction in complexity The compressed domain techniques are generally based on discrete cosine transform (CT) which provides excellent compression performance Feig and Li [9] have recently proposed low complexity histogram techniques based on compressed data, such as CT and FT (iscrete Fourier Transform) data Recently, discrete wavelet transform () based techniques have become popular for indexing applications for several reasons First, indexing can be done hierarchically exploiting the multiresolution property of Secondly, edge and shape of objects can be estimated easily in domain Finally, directional information from various directional subbands can be employed to enhance indexing performance Mandal et al [1] have proposed an indexing technique based on histogram of wavelet coefficients of different bands Liang et al [11] have proposed a joint image coding and indexing in wavelet domain where wavelet packet tree structure and subband significance are employed as features for indexing Wang et al [1] have proposed a technique where the magnitude of coefficients are compared for image indexing The comparison starts at the lowest resolution, and the results are progressively refined using higher resolution coefficients This technique fails to retrieve images even with small camera operations or object motions Smith and Chang [13] have recently proposed a technique for automatic extraction of texture regions in the wavelet domain Here, the coefficients are first thresholded based on their energy and a bi-level image is generated corresponding to each subimage A morphological operator is then applied on the bi-level images to eliminate the spurious components Each channel is then upsampled to the original size of the image and texture feature sets can be generated from these subimages which can be employed for image segmentation The objective in many applications is to retrieve an overall similar image and segmentation of images is not desired In these cases, a multi-dimensional wavelet histogram (WH) can be calculated from these subimages, which is then employed as an index of the overall image However, the computational complexity of WH technique (WHT) is very high In addition, it does not provide a good indexing performance for natural images Hence, it is necessary to develop a computationally inexpensive WH technique that provides good indexing performance

3 In this paper, we propose three techniques to reduce the computational complexity of WH techniques The first technique is based on multiresolution property of the The second technique is based on employing a smaller feature space, resulting in a reduced size feature vector Finally, a coarse resolution WH technique is proposed to reduce the complexity The organization of the paper is as follows A detailed review of WHT is provided in section The proposed techniques and associated complexity analysis are presented in section 3 and, respectively erformance of the proposed techniques is presented in section 5, which is followed by conclusions Wavelet Histogram Technique In this section, we provide a brief review of the wavelet histogram technique proposed by Smith and Chang [13] Here, the image is first decomposed to M stages using wavelets The wavelet bands form a pyramid of M levels, where the level-k bands are the highpass bands after the k -th stage decomposition A three-stage decomposition is shown in Fig, where level-1 bands consists of { A 7, A8, A }, level- bands consists of 9 { A, A5, A6 }, and level-3 bands consists of { A, A1, A, A3 } Intensity wavelet bands (IW) are then generated using the magnitudes of coefficients from highpass bands of all levels Each of these IW's is upsampled to the full-size image by inserting zeros and subsequently passed through appropriate filters to obtain a texture channel In order to reduce computational complexity, upsampling and filtering can be substituted [13] by a simple pixelreplication routine The entire process for a three-stage wavelet histogram generation is illustrated in Fig 3 Here, nine texture channels are generated from nine highpass bands A texture point is then defined as a 9- vector by considering texture channel values from the same location of all nine bands Thus for a M image, there will be M 9- vectors Each element of the 9- vector is thresholded to two levels high (1) and low () ue to the binary nature of the vector elements, 51 (= 9 ) different vectors span the entire 9- feature space The histogram (with 51 bins) of these vectors is employed as the index of the image Since, the histogram is created in the wavelet domain, we term it as wavelet histogram (WH) A face and a texture image along with their WH's are shown in Fig and 5, respectively It is observed that the WH's have peaks at a regular interval The relative strength of the frequencies at a given histogram bin depends on several factors, including correlation among wavelet bands, and the directional information present in an image The image indexing technique based on WH is compute intensive The complexity of WH generation is shown in Table 1 It is observed that the total complexity is approximately ( 3 + 1) * where is the number of channels employed for WH generation, and is the total number of pixels in the image On the other hand, the complexity of feature generation (ie, the computation of histogram) for OIH (ifference Of Histogram) technique is only operations Hence, the complexity of WHT technique is significantly higher compared to OIH technique The higher complexity is mainly due to the fact that all highpass bands are to be upsampled, and thresholded This multi-channel nature of the entire operation increases the complexity by a significant amount However, the retrieval complexity of the OIH/WHT techniques are proportional to the number of histogram bins 3 roposed Fast Wavelet Histogram Techniques The WHT technique [13] upsamples and filters all highpass wavelet bands such that each upsampled band contains the same number of coefficients as present in the original image (which can be considered as level- k k wavelet band) In other words, the level-k bands are upsampled by a factor of It is known that there is significant redundancy present in the upsampled and subsequently filtered images Hence, there is a potential to reduce this redundancy without degrading the performance These techniques are based on three principles First, we propose a technique based on WHT at a selected level Once a particular decomposition level is chosen, the IW s at various levels are appropriately upsampled or downsampled to generate identical number of coefficients A WHT is then generated from these IW s In the second technique, a smaller feature space is employed Instead of employing all (3M+1) bands for M-level decomposition, fewer numbers of bands are employed to generate the WHT In the third technique, the WHT is generated as in [13] However, a coarse resolution histogram is employed for image retrieval We now present the details of these techniques 3

4 31 WHT at a Selected Level Here, we propose a fast WHT technique (FWHT) exploiting the multiresolution nature of wavelet representation Let us assume that the WHT technique employs M -stage wavelet decomposition The proposed FWHT technique can be of M different types with varying complexities In scheme-k FWHT technique, various highpass wavelet bands are appropriately upsampled or downsampled to have the same number of coefficients as in a level-k band, k { 1,, M} In addition, the zeroth band (ie, lowpass subband at level- M ) is also included in WH generation Fig 6 shows histogram generation for scheme-1 FWHT technique with a 3-level decomposition Here, the coefficients from level-1 bands are not altered, while the coefficients from level- and level-3 bands are upsampled by a factor of (x), and 16 (x), respectively A texture point is then defined as a 1- (since 1 bands have been employed) vector consisting of values from the same location of all ten wavelet channels Each element of the 1- vector is thresholded to two levels high (1) and low () Thus for an image with pixels, there will be / 1- vectors A wavelet histogram with 1 bins is created from these vectors WH generation for scheme- FWHT is shown in Fig 7 Here, the coefficients from level- bands are not altered while the coefficients from level-1 bands are downsampled by a factor of x and the coefficients from level-3 are upsampled by a factor of x Following a procedure similar to the scheme-1 FWHT, / vectors are created A wavelet histogram with 1 bins is then created from these vectors Finally, the wavelet histogram for scheme-3 FWHT (shown in Fig 8) can be generated by downsampling the coefficients from level-1, and level- bands by factors of 16 (x) and (x), respectively and following the above procedure of the modification we have proposed above is to include the lowest frequency band ( A ) However, if we include A as it is, it will make the feature vector variant to image brightness, which is not desirable Hence, we make this band zero-mean, before applying the thresholding This enables us to use the lowpass information without being affected by the brightness variation which may be caused by a camera flash 3 Smaller Feature Space Histograms corresponding to selected FWHT techniques are shown in Fig 9 It is observed that wavelet histograms generally have high peaks at a regular interval The histograms are sparse at higher levels because of lower number of feature points In addition, when the zeroth band is included, the number of bins increases from 51 (because of 9-bands) to 1 (because 1-bands), resulting in a sparser histogram Hence, it may be possible to employ a smaller feature space without degrading the performance We now propose a modified version of FWHT techniques where only the first few of the low-resolution wavelet bands are employed for WH generation The histograms of the modified FWHT are more smooth and dense, since there are fewer histogram bins Fig 9 (f) shows the wavelet histogram, having 18 histogram bins, for modified scheme- FWHT using seven wavelet bands ( A - A 6 ) In addition to the increased smoothness of histogram, the complexity of this modified FWHT is lower than FWHT The information loss resulting from ignoring some subbands may lead to a degradation in the indexing performance However, out of the ten bands, three bands provide horizontal information, three bands provide vertical information, and three bands provide diagonal information Since information along a particular direction is provided by at least three subbands (although at different scales), some of the higher resolution bands may be ignored without performance degradation We note that the above two techniques perform well when the low frequency bands contain most of the image information These techniques may not provide good results for images with sparse structure [1] 33 Coarser Resolution Wavelet Histogram The complexity of comparing wavelet histograms for retrieval purposes is directly proportional to the number of histogram bins In addition, a large disk space is required to store the feature vector if repeated computation of WH is to be avoided In order to reduce the complexity as well as storage space requirement, a histogram with fewer bins is desirable method of doing this is to employ a smaller feature space (ie, employing fewer number of

5 bands) which has been discussed in section 3 An alternative method is to employ a coarser resolution wavelet histogram Coarser resolution histograms can be obtained by passing the full resolution histogram through a lowpass filter Here, we employ a lowpass wavelet filter for this purpose as it has several useful properties Let the full resolution WH be denoted by h [] where h The WH at 1/ resolution (ie, at scale 1) can be computed as: 1[ k m k] = h [ m] g[ m ] (1) h p [q] = Lowpass coefficient of p th scale at q th location g [] = Wavelet filter coefficients h has one-half the number of bins compared to h [] We note that 1[] obtain coarser resolution WH's with fewer numbers of histogram bins Computational Complexity of FWHT Eq (1) can be employed recursively to In this section, we provide an estimate of the computational complexity of the FWHT techniques There are two types of complexities involved in indexing- i) the complexity of feature vector generation ( ξ G ), and ii) the complexity of feature vector comparison ( ξ C ) The feature vector may be generated only once, and stored along with each image (it requires extra storage space) On the other hand, the complexity of feature comparison is involved each time a retrieval is performed The complexity of WH generation for scheme-k FWHT technique is provided in Table We note that WHT can be considered as scheme- FWHT Further, for scheme- and scheme-1 only upsampling is required, while for scheme-, both upsampling and downsampling are required In all schemes, upsampling and filtering has been implemented by simple pixel replication It is observed from Table that the complexity of scheme-1 (column 3) is proportional to the number of bands and output coefficients For scheme- FWHT, downsampling has been implemented as a weighted average of x or x coefficients Thus, the complexity of generating an output pixel by downsampling is greater than the corresponding complexity of generation by upsampling Hence, the former has been given a larger weight (=) in row 3/column 5 of Table We observe that the complexity of scheme-1, scheme-, and scheme-3 FWHT techniques is 3, 8, and times the complexity of WHT technique, respectively This reduction in complexity of WH computation is mainly due to two factors: i) the reduction in the number of effective coefficients for histogram calculation, ii) reduction in the number of bands (only 7 bands out of 1 bands) employed for upsampling and downsampling Table 3 compares the complexity of WH generation for scheme- FWHT using 7 and 1 bands It is observed the complexity for the case of 7 bands is approximately half the complexity of the 1-band case The run-time complexity of FWHT techniques is similar to any OIH/WHT technique, and is proportional to the number of histogram bins 5 erformance of the roposed Techniques In this section, we evaluate the performance of the proposed techniques We employ the retrieval efficiency as the performance criterion [7] This is defined as follows: for each image i, in a database of size K, we manually list the similar images found in the database Let, i, 1 i K, be the number of such images We then apply an indexing technique for a query image- q, and retrieve the first ( + τ) images Here, τ is a positive integer, and is used as a tolerance for retrieval If n q is the number of successfully retrieved images, the efficiency of retrieval can then be defined as: q 5

6 η K q= R = K q= n q q Here, we have employed two databases of images The first database, I1, contains images of (size 8 56 ) wide varieties, including faces, natural scenes, animals, birds Each image has five derivatives corresponding to i) normal, ii) translated rightward, iii) translated leftward, iv) rotated clockwise, v) rotated anticlockwise Thus, we have a total of images in I1 The second database, I, contains 5 rodatz textures of size Each texture has four similar images, ie, the total number of images is A tolerance τ of 5 has been used in all cases aubechies 8 tap minimum phase wavelet has been used for WH generation The 1 histograms are compared in L metric The performance of OIH, WHT, and FWHT techniques on I1 and I databases is shown in Table It is observed that WHT technique provides a performance of 85% on I database The retrieval efficiency of WHT on I database is superior to that of OIH technique This is because the texture images in I have a strong directional property that is captured by the WHT technique On the other hand, the OIH technique captures only the global description of an image in spatial domain and is hence not as efficient For the general database I1, the OIH and WHT techniques provide a performance of 93% and 711%, respectively Since the I1 database contains images with a few large objects, the image histogram is generally influenced by these objects Hence, his tograms of images containing similar objects are very similar For this reason, OIH technique provides a good performance However, WHT technique provides a lower retrieval efficiency since the directional information is less prominent in these images The performance of FWHT techniques on both I1 and I databases is shown in Fig 11 For the I database, we observe the following: i) best performance is achieved with scheme-, ii) WH generated with all wavelet bands provide the best performance, and iii) the inclusion of zeroth band does not seem to influence the retrieval efficiency On the other hand, for the I1 database, we observe the following: i) best performance is achieved at level-1, ii) best performance is achieved around 7 bands, and iii) the inclusion of zeroth band significantly improves the retrieval efficiency The above observations confirm the expectations based on the properties of the images in the two databases s in I database have strong directional features and hence employing all highpass subbands provides the best result Inclusion of the zeroth band is not important for this database However, for the I1 database, the zeroth subband provides crucial information about the low frequency texture information Hence, inclusion of this band is very important In addition, the subbands at the highest resolution are not crucial for the images in I1 The directional information achieved by a few subbands of lower resolution is generally sufficient The above experimental results suggest that the complexity of the wavelet histogram generation can be reduced significantly while retaining (or even improving) the performance level Table shows the complexity and performance WHT and FWHT techniques for I1 and I database, respectively It is observed that for I, a complexity reduction by a factor of 1 is possible without degrading the performance On the other hand, for I1 database, a complexity reduction factor of 3 is possible with improvement in performance The complexity of WH comparison for retrieval is directly proportional to the number of histogram bins We proposed two techniques in section 3 to reduce the number of histogram bins The first technique (ie, truncated feature space) is effective for I1 database where the best performance (see Fig 11) is obtained for FWHT with 7 bands (ie, with 18 histogram bins) However, for I database, the best performance is obtained with 1 bands (with 1 histogram bins) Hence, the complexity is significantly higher for I database We now evaluate the performance of coarse resolution WH for image retrieval Table 5 compares the performance of FWHT(,,1) with its coarser resolution counterparts for I database These WH's are obtained using aubechies 8 tap wavelet filter It is observed that the computational complexity can be reduced substantially using these coarse resolution WH's with a marginal degradation in retrieval performance 6

7 6 Conclusions Fast wavelet histogram techniques have been proposed in this paper to improve performance of image indexing systems For natural images, the FWHT techniques provide a superior performance compared to WHT, at a reduced complexity, due to the inclusion of the zeroth wavelet band On the other hand, for texture images, FWHT techniques provide a performance comparable to that of the WHT technique at a reduced complexity These techniques generally perform well for most images where coarse resolution coefficients provide substantial information about the full resolution image Although, the techniques have been proposed in the wavelet domain, they can be extended easily to other domains, such as CT References 1 ISO/IEC JTC1/SC9/WG1, ocument 39R, ew Work Item: JEG image coding system, March 1, 1997 IEEE Trans on Circuits and Systems for Video Technology, Special issue on MEG-, Feb Furht, S W Smoliar and H Zhang, Video and rocessing in Multimedia Systems, Kluwer Academic ublishers, 1995 M J Swain and H allard, Color indexing, International Journal of Computer Vision, Vol 7, o 1, R W icard and F Liu, A new Wold ordering for image similarity, roc of the International Conference on Acoustics, Speech and Signal rocessing, Vol V, pp 19-13, April K Hirata and T Kato, Rough sketch-based image information retrieval, EC Research and evelopment, Vol 3, o, pp 63-73, April M Mehtre, M S Kankanhalli, A arasimhalu and G C Man, Color matching for image retrieval, attern Recognition Letters, Vol 16, pp , March M K Mandal, F Idris, and S anchanathan, " and video indexing in the compressed domain: a critical review," to appear in and Vision Computing Journal 9 E Feig and C S Li, Computing image histogram from compressed data, roc of SIE, Vol 898, pp 118-1, M K Mandal, T Aboulnasr and S anchanathan, indexing using moments and wavelets, IEEE Trans on Consumer Electronics, Vol, o 3, Aug K C Liang and C -C J Kuo, "Retrieval and progressive transmission of wavelet compressed images," roc of ISCAS, pp , Hong Kong, J Z Wang, et al, Wavelet-based image indexing techniques with partial sketch retrieval capability, roc of the Forum on Res and Tech Adv in ig Lib, May J R Smith and S F Chang, "Automated binary texture feature sets for image retrieval," roc of ICASS, Vol, pp 39-, Atlanta, May J Field, What is the goal of sensory coding, eural Computation, Vol 6, pp , 199 7

8 Table 1 Complexity ( ξ G, in operations/image) of computing wavelet histogram in WHT technique [13] : Total number of channels (bands) to be upsampled and filtered, : umber of pixels in an image Main Modules Sub-Modules Complexity Typical Value 1 Thresholding Absolute Value Calculation * 59e+5 Thresholding * 59e+5 Histogram Feature oint Generation * 59e+5 Calculation Histogram Calculation 65e+ Total Thresholding + Histogram ( 3 + 1) * 18e+6 Calculation 1 for =65536 (56x56), =9 Feature point generation refers to computing of histogram bins from the 9- feature vector Table Complexity ( ξ G, in operations) of computing wavelet histogram in scheme-{1,,3} U : Total number of channels to be upsampled and filtered, : Total number of channels to be downsampled, : Total number of channels, : umber of image pixels Main Modules Sub-Modules Approximate umber of Operations Scheme-1 Scheme- Scheme-3 Upsampling & Absolute Value Thresholding Calculation U * / U * / 16 * U * / 6 Thresholding U * / U * / 16 U * / 6 ownsampling Absolute Value - * * / * * / 6 & Thresholding Calculation Thresholding - * / 16 * / 6 Histogram Feature oint * / Calculation Generation * / 16 * / 6 Histogram Calculation / 16 / 6 Total ( U + + 1) ( U ) ( 5U ) * * * 16 6 Total (typ value) 1e+5 a 1e+5 b 79e+ c =1, =65536, a for U =7, b for U =, =3, c for U =3, =3 8

9 Table 3 Complexity of FWHT technique at level- o of ands ( ) includes the zeroth band U : Total number of channels to be upsampled, : umber of image pixels of channels to be downsampled, o of ands 1 for U =, : Total number Complexity Typical Value 1 (O/image) + 1) * / 78e+ ( U ) * / 16 1e+5 =3, ( U =1, =65536 Table Effieicncy ( η R ) versus complexity of OIH, WHT and FWHT (scheme-) techniques ξ : complexity of feature generation in operations/image, G ξ : complexity of feature C comparison FWHT employs 7 bands for I1 and 1 bands for I (zeroth band is included in both cases) I1 (General image) I (Texture image) Technique ξ η (in %) ξ η (in %) ξ G C R OIH 65e e WHT 16e e FWHT 63e e ξ G C R Table 5 Retrieval effieicncy ( η R ) of FWHT(,,1) at different resolutions for I database Resolution o of Histogram ins Efficiency ( η ) in % R Original / / / / / /

10 ew Input igitization Analysis & Coding atabase Query igitization Analysis Matching Retrieved s Figure 1 Schematic of an image storage and retrieval system a A A 1 A A3 A A 7 b Original S 1 A 5 A 6 A8 A9 (a) (b) Figure and Structure of three-stage wavelet transform a) original image, b) Various directional bands A 9, S 9 - A 8 A 7,, S 8 S 7 A 6 A 5 A A 3 A A 1 A, S6 ot Used 8 8 8,, 8,8 8,8 8,8 S 5 S S 3 S S 1 Figure 3 Schematic of wavelet histogram generation [13] 1

11 1 Frequenc WHT Histogram ins (a) (b) Figure A face image and its wavelet histogram 3 Frequenc WHT Histogram ins (a) (b) Figure 5 A texture image and its wavelet histogram 11

12 - A 9 A 8 A 7 S 9 S 8 S 7 A 6 A 5 A A 3 A S6 A 1 A,,,,, S 5 S S 3 S, S1, S Figure 6 Schematic of fast wavelet histogram generation (FWHT) at level-1 - A 9 A 8 A 7 S 9 S 8 S 7 A 6 A 5 A A 3 A A 1 A S6,, S 5 S S 3 S, S 1, S Figure 7 Schematic of fast wavelet histogram generation (FWHT) at level- 1

13 - A 9 A 8 A 7 S 9 S 8 S 7 A 6 A 5 A A 3 A A 1 S6 S 5 S S 3 S S1 A S Figure 8 Schematic of fast wavelet histogram generation at level-3 13

14 FWHT(1,1,9) 1 5 FWHT(,1,9) Histogram ins Histogram ins (a) (b) 6 5 FWHT(3,1,9) 3 1 FWHT(1,,1) Histogram ins Histogram ins (c) (d) FWHT(,,1) 3 1 FWHT(,,7) Histogram ins Histogram ins (e) (f) Figure 9 Fast wavelet histograms of the image shown in Fig 3(a), at various levels FWHT ( p, q, r) refers to WH calculated at level- p using subbands A, } { q A q+ r 1 1

15 15 Original -1 1 Original Histogram ins (a) Histogram ins (b) Histogram ins Histogram ins (c) (d) Figure 1 A typical wavelet histogram, FWHT(,,1) at different resolution (a) original, (b) 1/ resolution, (c) 1/ resolution, (d) 1/8 resolution aubechies 8 tap wavelet has been employed to obtain the lower resolution histograms 85 I1 76 I1 Retrieval Efficiency Scheme-1 Scheme- Scheme o of ands (a) Retrieval Efficiency Scheme-1 Scheme- Scheme o of ands (b) 88 I 88 I Retrieval Efficiency o of ands Scheme-1 Scheme- Scheme-3 Retrieval Efficiency o of ands Scheme-1 Scheme- Scheme-3 (c) (d) Figure 11 Comparison of indexing performance of various wavelet-histogram techniques Scheme-α : coefficients of all bands have been appropriately upsampled or downsampled to have same number of coefficients as in scale α { A β 1 A } (for Fig a and Fig c) or A } wavelet histogram generation { 1 β "o of bands=β" means that bands A (for Fig b and Fig d) have been employed for 15

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