I. Introuction With the evolution of imaging technology, an increasing number of image moalities becomes available. In remote sensing, sensors are use

Transcription

1 A multivalue image wavelet representation base on multiscale funamental forms P. Scheuners Vision Lab, Department ofphysics, University ofantwerp, Groenenborgerlaan 7, 00 Antwerpen, Belgium Tel.: +3/3/ Fax: +3/3/ scheun@ruca.ua.ac.be Abstract In this paper, a new wavelet representation for multivalue images is presente. The iea for this representation is base on the first funamental form that provies a local measure for the contrast of a multivalue image. In this paper, this concept is extene towars multiscale funamental forms using the yaic wavelet transform of Mallat. The multiscale funamental forms provie a local measure for the contrast of a multivalue image at ifferent scales. The representation allows foramultiscale ege escription of multivalue images. A variety of applications is presente, incluing multispectral image fusion, colour image enhancement an multivalue image noise filtering. In an experimental section, the presente techniques are compare to single value an/or single scale algorithms that were previously escribe in the literature. The techniques, base on the new representation are emonstrate to outperform the others. EDICS: -WAVP, -COLO

2 I. Introuction With the evolution of imaging technology, an increasing number of image moalities becomes available. In remote sensing, sensors are use that generate a number of multispectral bans. In meical imagery, istinct image moalities reveal ifferent features of the internal boy. Examples are MRI images, sensitive to ifferent imaging parameters (T, T, proton ensity, iffusion,...), CT an nuclear meicine moalities. A particular type of multivalue images are colour images. A lot of attention has been evote to the classification an/or segmentation of multimoal ata. In remote sensing, classification of texture regions is performe for ientification purposes [] []. In meical imaging, segmentation of ifferent tissue regions is aime at [3] [4]. However, usually only a fraction of the ata provies unique or useful information. Moreover, reucing the ata set reuces the complexity of a classification proceure. For this reason, a fusion of the bans into one or a few greylevel images is common practice [5] [6]. Another proceure merges ifferent bans (e.g. a high resolution greylevel image with a low resolution colour image) for enhancement [7]. Other image processing proceures to facilitate the classification an segmentation of multivalue images inclue noise filtering an image enhancement. It is obvious that all these image processing an analysis techniques woul benefit from the combine use of the ifferent bans. Nevertheless, in most cases single-value processing an analysis techniques are applie to each of the bans separately. The results for each component are then combine in a usually heuristic manner. A large part of image processing an analysis techniques makes use of the image ege information, that is containe in the image graient. Aniceway of escribing multivalue eges is given in [8]. Here, the images "first funamental form", a quaratic form, is efine for each image point. This is a local measure of irectional contrast base upon the graients of the image components. This measure is maximal, at each image point, in a particular irection, that in the greylevel case is the irection of the graient. Base on this efinition, in [9], a colour ege etection algorithm was escribe an a colour image anisotropic iffusion algorithm was escribe in [0]. In single-value images, multiresolution techniques are use to escribe eges. The

3 3 wavelet transform e.g. is successfully applie to compression, noise-reuction, enhancement, classification an segmentation of greylevel images. However, when applie to multivalue images, it is applie to each ban separately. As an example, I woul like to mention multispectral image fusion an merging [7], [] an multimoal meical image segmentation [4]. In this paper, a new multivalue image wavelet representation is presente. This representation allows for a multiscale ege escription of multivalue images. The iea for the representation is base on the first funamental form of [8] an the yaic wavelet representation of Mallat, presente in []. The latter ecomposes an image into etail images that are convolutions of the image with erivatives of a smoothing function. These etail images can be written as the erivatives of the image, smoothe at ifferent scales. This observation allows for a efinition of multiscale funamental forms. The eigenvectors an eigenvalues of these quaratic forms escribe the irections an rates of change of the multivalue image at that particular scale. To emonstrate the use of the presente representation, I will escribe several applications of multivalue image processing an analysis. First, a technique for multispectral image fusion will be elaborate. Secon, by merging the new representation with the original bans, a colour image enhancement proceure will be evelope. Finally, base on the work of [0], a multivalue anisotropic iffusion filter will be constructe. The evelope techniques will be compare to the single-value an/or the single scale versions of the algorithms. The paper is organize as follows: in the next section, the first funamental form an in section 3, the reunant wavelet transform is reviewe. In section 4, the multiscale funamental form is presente. In section 5, the three ifferent applications will be explaine an emonstrate by some experiments. II. Multivalue ege representation using the first funamental form For the erivation of the first funamental form, we will follow [8]. Let I(x; y) be a multivalue image with omain R an real-value components I n (x; y);n=; :::N. The value of I at a given point isan-imensional vector. To escribe the graient information of I, let us look at the ifferential of I, which is assume to exist. In the Eucliean space

4 4 R N : x + an its square norm is given by (sums are over all bans of the image): kik = = = 0 x y 0 x y 0 x y T 0 C B A T 0 C B T 0 C B k @y P @ Gxx G xy 0 C B x C A y 0 C B P x C n n 0 C B k k G xy G yy C A () This expression is calle the first funamental form. The matrix 0 B G Gxx G xy G xy G yy C A (3) is a symmetric, nonnegative efinite matrix, which ensures that its eigenvalues are real an nonnegative. It reflects the change in a multivalue image. The irection of maximal an minimal change are given by the eigenvectors of G. The corresponing eigenvalues enote the rates of change. For a greylevel image (N = ), it is easily calculate that the largest eigenvalue is given by + = krik, i.e. the square graient magnitue. The corresponing eigenvector lies in the irection of the graient. The other eigenvalue equals zero. For a multivalue image, the eigenvectors an eigenvalues escribe an ellipse in the image plane. When + fl, the graients of all bans are more or less in the same irection. When ' +, there is no preferential irection, i.e. the eigenvectors can be chosen arbitrarily. The conjecture is that the multivalue ege information is reflecte by the eigenvectors an eigenvalues of the first funamental form. A particular problem that occurs is that the iagonalization oes not uniquely specify the sign of the eigenvectors. This has been extensively stuie in [9]. There, it was proven that the eigenvectors can be uniquely oriente in simply connecte regions where 6= +. Base on this, an algorithm was propose to orient the eigenvectors, keeping the angle-function continuous in local regions.

5 5 III. The yaic wavelet transform The first funamental form of the previous section reflects only ege information at a single scale. Many greylevel image processing applications make use of higher scale ege information using multiresolution techniques. The aim of this paper is to exten the multivalue ege representation towars multiscale eges. The wavelet transform employe in this work is base on non-orthogonal (reunant) iscrete wavelet frames introuce by Mallat [], [3]. Define a -D smoothing function (x; y), i.e.: lim (x; y) =0 x;y! Z (x; y)xy = (4) Supposing is ifferentiable, efine ψ (x; y) (x; an ψ (x; y) (x; (5) The fact that satisfies (4) guarantees that ψ (x; y) an ψ (x; y) are wavelets, i.e. have zero mean. The wavelets that were chosen in [], an that we also will apply, are quaratic spline wavelets of compact support, an is a cubic spline smoothing function. The wavelet transform of an image I(x; y) is then efine by: Ds(x; y) =I Λ ψs(x; y) ands(x; y) =I Λ ψs(x; y) (6) where Λ enotes the convolution operator an ψs(x; y) = s ψ ( x s ; y s )anψ s(x; y) = s ψ ( x s ; y s ) (7) enote the ilations of the functions ψ i. s is the scale parameter which commonly is set equal to j with j =;:::;. This yiels the so calle yaic wavelet transform of epth. D an D are referre to as the etail images, since they contain horizontal an vertical j j etails of I at scale j. In practice,for igital images etail values are only require at integer positions (m; n) Z. For finite image sizes, borer problems are solve by imposing perioic bounary conitions. In [3], a fast algorithm has been esigne for this transform by iterative

6 6 filtering with a set of -imensional low an high pass filters H, G, K an L associate with the wavelets ψ an ψ (for this, it is necessary that the wavelets can be written as separable proucts of -D functions). These filters have finite impulse responses, because ψ an ψ have compact support, which makes the transform fast an easy to implement. We enote A p the iscrete filter obtaine by putting p zeros between each of the coefficients of the filter A. We enote by (A; B) Λ C the separable convolution of the rows an columns, respectively, of the iscrete image C with the iscrete -D filters A an B. L = I an D is the unit-impulse filter whose impulse response equals at 0 an 0 otherwise. The proceure at scale j is given by: L j+(m; n) = (H j ;H j ) Λ L j(m; n) D j+(m; n) = (D; G j) Λ L j(m; n) D j+(m; n) = (G j;d) Λ L j(m; n) (8) For a proof, we refer to []. Thus the wavelet representation of epth of the image I consists of the low resolution image L an etail images fd i j g i=; j=;:::;. Similarly, a reconstruction algorithm was esigne, to reconstruct the original image from its wavelet representation. At scale j: L j (m; n) = (K j ;L j ) Λ D j(m; n) + (L j ;K j ) Λ D j(m; n) + ( ~ Hj ; ~ Hj ) Λ L j(m; n) (9) with ~ Hp the filter whose fourier transform is the complex conjugate of the fourier transform of H p. Substitution of (5) an (7) in (6) yiels the following interesting property: 0 D (x; y) j D (x; y) j C A = j (I j)(x; (I j)(x; y) C A = j r(i Λ j)(x; y) (0) This stipulates that the wavelet transform of a greylevel image consists of the components of the graient ofthe image, smoothe by the ilate smoothing function j.

7 7 IV. The multiscale funamental form Base on (0), for multivalue images a funamental form can be constructe at each scale. Similar to (), an applying (0), the square norm of the ifferential of (IΛ j)(x; y) is given by: k(i Λ j)k 0 T 0 = j x C B y 0 T 0 = j x C B A y P P n D n;j n D D n; j n; j P Pn D D n; j n; j n D Gxx G xy j j G xy G yy j j 0 C B x y 0 C B x C A y C A () where D n; j an D n; j are the j-th scale etail coefficients of the n-th ban image. The matrices 0 B G j Gxx G xy j j G xy G yy j j C A () are symmetric, nonnegative efinite matrices, with real an nonnegative eigenvalues. Expression () will be referre to as the j-th scale funamental form. It reflects the change in the j-th scale smoothe image an therefore the ege information at the j-th scale. The irection of maximal an minimal change are given by the eigenvectors v + j an v j of G. The corresponing eigenvalues + j an j enote the rates of change. The eigenvectors an eigenvalues escribe an ellipse in the image plane, where the longest axis enotes the irection of the graient at scale j an the shortest axis the variance of graient at scale j aroun that irection. For a greylevel image, one obtains + j (x; y) = j h (D j) (x; y)+(d j) (x; y) i = kr(i Λ j)k v + j (x; y) = r(i Λ j ) kr(i Λ j )k i.e. the first eigenvector enotes the irection of the graient of the j-th scale smoothe image, while its corresponing eigenvalue enotes its length. For N =, the matrices G j have rank, i.e. j (x; y) =0. Remark that: D q j(x; y) = + v + (x; y) j j ;x (3) D q j(x; y) = + v + (x; y) (4) j j ;y

8 8 i.e. the original representation is obtaine by projecting the first eigenvector, multiplie by the square root of the corresponing eigenvalue onto the x an y-axes. In multivalue images the ege information is containe in both eigenvalues. The eigenvectors an eigenvalues of the multiscale funamental forms escribe the ege information of a multivalue image in a multiresolution way. The multivalue image can be represente at each scale by 4 etail images: D ;+ j (x; y) = D ;+ j (x; y) = D ; j (x; y) = D ; j (x; y) = q + v + (x; y) j j ;x q + v + (x; y) j j ;y q v (x; y) j j ;x q v (x; y) (5) j j ;y In figure, a schematic overview of the Multivalue Image Wavelet Representation (MIWR) is given. It is important to remark that this representation is by no means complete, i.e. the original bans cannot be reconstructe from this representation. It was not the goal of this work to erive a complete representation but only to escribe the ege information of multivalue images. The same problem as in the single scale case occurs with calculating the MIWR: the matrix iagonalization oes not uniquely specify the signs of the eigenvectors. This phenomenon translates in the multivalue image problem as arbitrariness of the graients orientation. From (5), this orientation reflects on the sign of the etail coefficients that can flip incoherently from one pixel to another. Therefore the orientation must be etermine before a reconstruction can be calculate. Instea of following the proposal of [9], we propose the following (more simple) solution to this problem. The orientation of the graient is approximate by the orientation of the graient of the average of all bans. The average of the bans is calculate an wavelet transforme. The scalar prouct of the obtaine etail coefficients D an j D j with the first eigenvectors then etermines the signs: if D j v+ + j ;x D j v+ 0 then the sign of the eigenvectors is not change, if the j ;y scalar prouct is negative, then the signs of v + j ;x an v+ j ;y are flippe. The sign of v j chosen so that the angle of its irection is ß more than the angle of v+ j. is

9 9 V. Applications an experiments In this section, several applications of the MIWR representation are escribe. In principle, all greylevel processing techniques that are base on graient or multiscale contrast can be extene towars multivalue images using the MIWR representation. In this paper, three applications are iscusse, because they are obvious choices or they were previously escribe in the literature. To compare, single value techniques, i.e. where the processing is performe on each ban separately, or single scale techniques, i.e. where the first funamental form is use, are applie. A. Multispectral image fusion Image fusion is the process of combining several, perfectly registere images into one greylevel image. This technique is applie to multispectral satellite imagery [], [] as well as to biomeical multimoal imagery [3], with the purpose of visualisation an of reucing the complexity for classification an segmentation tasks. In a multiresolution approach, the wavelet representations of all bans are to be combine into one greylevel image wavelet representation. In [] e.g. the etail coefficients between ifferent bans are compare an for each pixel the largest one is chosen to represent the fuse image. Using the propose MIWR representation, a single greylevel wavelet representation can be constructe in the following way. Starting from (5), one ignores the secon eigenvalues. A low resolution image is obtaine by averaging the low resolution images of the original bans: L μ P = N N n= L n;. The obtaine representation is then given by: μ L an fd i;+ j g i=; j=;:::;. By applying the iscrete wavelet reconstruction of (9) to this representation, one obtains a greylevel image that contains the fuse ege information of the ifferent bans. In figure a schematic overview of this fusion algorithm is given. To emonstrate the propose fusion technique, the following experiment is conucte. As a test image remote sensing ata is use: a Thematic Mapper image, containing 7 bans of 5x5 images from the U.S. Lansat series of satellites [4]. The first four images are fuse into one greylevel image. In figure 3, the result is shown. On top, the propose technique is applie. On the bottom, the wavelet fusion technique of [] is applie. The same reunant wavelet representation as in the first image is applie on every ban. In

10 0 both applications, the images were wavelet ecompose into 4 levels of resolution. For each pixel position an at each scale, the largest etail coefficient istaken to be the etail coefficient of the fuse image: ~ D i j (x; y) = max n D i n; j (x; y). One can observe an improve overall contrast using the propose technique. This effect can be attribute to the superior escription of the ege information in the MIWR representation. At first sight, it might seem surprising that the fuse images are reconstructe without visual artifacts. After all, the etail images fd i;+ j g are constructe pixelwise, irrespective of etail values at neighbouring positions an scales. This seems not to affect the reconstruction. We are not able to give rigorous mathematical evience for this, but several arguments are liste below. ffl For most positions, j are small compare to + j. This is the case because most ege positions appear in more than one ban an are equally oriente. If ege positions of ifferent bans are equally strong, they are fuse accoringly. If an ege position of one of the bans is stronger than the others, it will ominate the fusion process. In those cases where j ' + j, the graients are mostly weak. These positions correspon to noisy an locally texture areas. ffl Real eges have continuous behaviour both in space an scale. This means that a strong graient at one position at a particular scale will also be foun at the same or a neighbouring position at the other scales, an this will be also the case for all positions along the ege. This is probably the main reason for the fuse reconstruction to work properly. In [], a thorough stuy of the behaviour of eges accross scales is performe, revealing e.g. that an image can almost perfectly be reconstructe from the local maxima of the etail images only. ffl Most of the other wavelet-base fusion techniques that have been escribe in the literature, all are base on fusion rules that o not take into account neighbouring etail information in space or scale. No visual artifacts of the reconstructions were reporte in these cases. The continuity arguments are also vali in those cases. The fact that the propose technique combines graient information of the ifferent bans is an extra argument for a more continuous behaviour of the values of fd i;+ j g accross space an scale.

11 B. Colour image enhancement A problem, relate to image fusion is that of image merging. Here, two images are combine, forming an enhance final image. This technique is applie in multispectral satellite imagery, where e.g. a high-resolution greylevel image is merge into a lower resolution multispectral image to enhance its resolution. In [7], a wavelet-base approach is propose, by replacing etail images of the multispectral bans by the etail images of the higher resolution greylevel image. This concept of image merging can, together with the propose multivalue wavelet representation, be use for colour image enhancement in the following way. First, the three bans of a colour image (RGB or another colour space) are wavelet transforme an the MIWR representation calculate using (5). As in the case of image fusion, only the first eigenvalue is retaine, leaing to a single greylevel representation. Since the first eigenvector represents the irection where the multiscale graientofthemultivalue image is largest, an since the secon eigenvalue is neglecte, a representation is obtaine with enhance ege information. This representation is then merge with the representations of the original bans in the following way. The low resolution image of each ban is retaine. For each ban an at each scale, the etail images of the original bans are replace by the corresponing etail image from the MIWR representation. Then, the three bans are reconstructe. As a result a colour image is obtaine with enhance ege information. In figure 4, a schematic overview of the multi-scale colour image enhancement proceure is given. To emonstrate the color image enhancement proceure, the RGB image LENA is processe. In figure 5, the results are shown (a small part of the image is zoome in). The first image is the original, the secon is the enhance image, using the propose technique. To compare, the same merging proceure was applie for image 3, but this time the etail images from the bans were replace by the wavelet representation obtaine when fusing the three bans using []. In both cases, the images were ecompose into 4 levels of resolution. One can observe an improve enhancement when applying the propose technique. Again, one can be surprise that the reconstruction works fine, without isturbing visual

12 artifacts. The same arguments as for fusion hols. If strong graients are present in one color ban, the enhancement proceure merely confirms them. If a strong graient appears in only one of the bans, it will also be superimpose in the other bans. Another point ofconcern is the spectral preservation of the reconstruction. One might expect that the proceure mixes the colors. Apparently, this is not the case. The main reason for this is that the low resolution images of the ifferent bans are preserve, guaranteeing that most chrominance etail is preserve. Again, this is confirme in other wavelet-base merging techniques (see e.g. [7]). In orer to give a more quantitative measure for this behavior, a spectral correlation between the original an enhance images can be calculate. For the correlation between images A an B: Cor(A; B) = < (A <A>)(B <B>) > q < (A <A>) >< (B <B>) > (6) where < > enotes the average over all pixels. This is one for each ban separately. For the LENA image, we obtaine (0.95, 0.98, 0.93) for the correlations between original an enhance R, G an B bans respectively. For the comparing technique, we obtaine (0.94, 0.97, 0.99). C. Multivalue image noise filtering In this section we will restrict ourselves to aaptive filtering techniques, base on anisotropic iffusion [5]. In [6] an [0], anisotropic iffusion technique were escribe for colour images, base on the first funamental form. In this section, a multiscale version of multivalue anisotropic iffusion is constructe, base on the multiscale funamental forms. Let us first escribe the single scale version. From a multivalue image, the first funamental form is calculate using (). The eigenvectors an corresponing eigenvalues of the first funamental form escribe an ellipse in the image plane. Since the first eigenvector is irecte along the graient, it will be irecte across an ege. The secon eigenvector will then be irecte along the ege. The goal of anisotropic iffusion is to smooth an image preferably along eges while preserving high-frequency information across eges. Using the first funamental form, a locally aapte Gaussian smoothing kernel is con-

13 3 structe (see [7] for more information): ψ G(r) = exp ψ (r:v+ ) ff!! + (r:v ) ff (7) with r the two-imensional position vector an v + funamental form. The stanar eviations are given by: an v the eigenvectors of the first ff = ff = ff ( a) +C ff +C (8) where a = is a measure for the anisotropy anff is the stanar eviation of the image noise. C is the corner strength. During smoothing, corners shoul be preserve. A corner is ientifie as an anisotropic situation with a large local graient strength. We can calculate a measure for the corner strength by: C = ( a)kik, (I is the image that is convolve with the kernel) an ivie the stanar eviations ff an ff by+c. The avantage of this smoothing kernel is that it is more extene along eges an less extene across eges. The algorithm was originally esigne for greylevel images, where instea of the first funamental form (), a quaratic form was use, in which the sum was taken over a local winow aroun each pixel [8]. The obtaine gaussian kernel was then convolve with the image. In the case of multivalue images, a gaussian kernel can be calculate for each ban separately. In [0], it is argue that it is better to apply the same anisotropic iffusion process to all three bans, by using the first funamental form. The single scale multivalue anisotropic iffusion algorithm then looks as follows. From a colour image, the first funamental form is calculate using (). From this, a gaussian kernel is calculate using (7) an (8). Each of the three bans is then convolve with this kernel. How shoul one exten this algorithm to a multiscale one? Using the multiscale funamental forms of (), the obtaine eigenvectors an eigenvalues escribe the irections an rates of the graient of the smoothe image (I Λ j )(x; y). One can now construct a gaussian kernel for each scale, that is aapte to the ege information at that particular scale. To apply the algorithm as propose in the previous paragraph, the smoothe images shoul be convolve with their corresponing kernel. One has however no irect

14 4 access to the smoothe images, but only to their erivatives (i.e. the etail images). Since taking the erivative an convolving with a gaussian are interchangeable, one can as well convolve the etail images with the corresponing kernels. After reconstruction, a noise filtere image is obtaine. The algorithm then looks as follows: ffl For each ban, calculate its yaic wavelet representation, using (8). ffl For all scales, calculate the multiscale funamental forms, using (). ffl For each scale, calculate the gaussian kernels, using (7) an (8). ffl At each scale, convolve the obtaine gaussian kernel with the etail images from every ban separately. ffl Reconstruct each noise-filtere ban separately. To emonstrate the technique the following experiment is conucte. The original RGB image PEPPERS is corrupte with gaussian white noise, with a variance of 50 (PSNR=.4 b). The propose algorithm is applie (up to 4 scales), leaing to a noise filtere image with apsnrof 5.5 b. To compare, the single scale algorithm, using the first funamental form, is applie, leaing to a PSNR of 4. b. In figure 6, the results (a small part is zoome in) are isplaye. One can clearly observe an improve noise reuction, while keeping the ege information, using the propose technique. For this application, less problems with reconstruction or color mixing are expecte. The etail images at each ban are merely low-pass filtere, an each ban is reconstructe separately. No fusion or merging of etail images is performe. VI. Conclusions In this paper, a new multivalue image wavelet representation is presente. For this, the concept of multiscale funamental forms is introuce. This concept is base on the first funamental form that provies a local measure of contrast for multivalue images. Using the yaic wavelet transform of Mallat, the extension towars multiscale funamental forms is obtaine. They provie a local measure of contrast for multivalue images in a multiresolution framework. The new representation allows for a multiscale ege escription of multivalue images. In principle, all greylevel image processing techniques, base on graient or multiscale contrast can be extene towars multivalue images using the new representation. In this

15 5 paper, three moel applications were escribe: multispectral image fusion, colour image enhancement an anisotropic iffusion filtering of multivalue images. The algorithms, base on the new representation are emonstrate to outperform single value an/or single scale versions that were previously escribe in the literature. In the near future, the propose applications will be stuie in more etail, an larger scale experiments are planne in the application fiels of multispectral satellite imagery an biomeical imaging. Also, a stuy on other possible image processing techniques is planne, incluing multivalue image texture characterization an multivalue image segmentation.

16 6 ACKNOWLEDGEMENT The author woul like to thank Dr. Jan Sijbers for proviing him with helpfull assistance on the anisotropic iffusion algorithm.

17 7 References [] I.L. Thomas, V.M. Benning, an N.P.Ching,Classification of remotely sense images, Aam Hilger, Bristol, 987. [] C Lee an D.A. Langrebe, Analyzing high-imensional multispectral ata," IEEE Trans. Geosci. Remote Sensing, vol. 3, no. 4, pp , 993. [3] T. Taxt an A. Lunervol, Multispectral analysis of the brain using magnetic resonance imaging," IEEE Trans. Me. Imaging, vol. 3, no. 3, pp , 994. [4] C. Busch, Wavelet base texture segmentation of multi-moal tomographic images," Comput. & Graphics, vol., no. 3, pp , 997. [5] G. Harikumar an Y. Bresler, Multi-spectral image visualization by projection pursuit," Visual. Biome. Comput '9, vol. SPIE-808, pp , 99. [6] A. Manuca, Multispectral image visualization with nonlinear projections," IEEE Transactions on Image Processing, vol. 5, no. 0, pp , 996. [7] D.A. Yocky, Image merging an ata fusion by means of the iscrete two-imensional wavelet transform," J. Opt. Soc. Am. A, vol., no. 9, pp , 995. [8] S. Di Zenzo, A note on the graient of a multi-image," Comput. Vision Graphics Image Processing, vol. 33, pp. 6 5, 986. [9] A. Cumani, Ege etection in multispectral images," CVGIP: Graphical Moels an Image Processing, vol. 53, pp. 40 5, 99. [0] G. Sapiro an D.L. Ringach, Anisotropic iffusion of multivalue images with applications to color filtering," IEEE Trans. Image Processing, vol. 5,, pp , 996. [] H. Li, B.S. Manjunath, an S.K. Mitra, Multisensor image fusion using the wavelet transform," Graphical Moels an Image Processing, vol. 57, no. 3, pp , 995. [] S. Mallat an S. Zhong, Characterization of signals from multiscale eges," IEEE Trans. Pattern Anal. Machine Intell., vol. 4, pp , 99. [3] S. Mallat, E., A wavelet tour of signal processing, vol. -n eition, Acaemic Press, 999. [4] The images were provie by r. C.C. Hung from the Center for Hyrology, Soil Climatology, an Remote Sensing from the Alabama A&M University,".

18 8 [5] P. Perona an J. Malik, Scale-space an ege etection using anisotropic iffusion," IEEE Trans. PAMI, vol.,7, pp , 990. [6] A. Chambolle, Partial ifferential equations an image processing," Proceeings of the IEEE International Conference on Image Processing, vol., pp. 6 0, 994. [7] G.Z. Yang, P. Burger, D.N. Firmin, an S.R. Unerwoo, Structure aaptive anisotropic filtering for magnetic resonance image enhancement," Proceeing of CAIP, pp , 995. [8] J. Sijbers, P. Scheuners, M. Verhoye, A. Van er Linen, D. Van Dyck, an E. Raman, Watershe-base segmentation of 3 MR ata for volume quantization," Magnetic Resonance Imaging, vol. 5,6, pp , 997.

19 9 FIGURE CAPTIONS Fig. : Schematic overview of the Multivalue Image Wavelet Representation (MIWR). Fig. : Schematic overview of the multiscale fusion algorithm. Fig. 3: Fusion of the first 4bansof a Lansat image; a: using the propose technique; b: using wavelet maxima fusion. Fig. 4: Schematic overview of the multiscale colour image enhancement algorithm. Fig. 5: Colour image enhancement; a: original; b: using the propose technique; c: using the wavelet maxima proceure. Fig. 6: Anisotropic iffusion; a: original PEPPERS image; b: gaussian noise ae (ff = 0); c: filtering using the funamental form; : filtering using the multiscale funamental forms.

20 0 I I N D, D, D N, D N,,+ D,+ D, D, D D, D, D N, D N,,+ D,+ D, D, D L, LN, Fig.. Schematic overview of the Multivalue Image Wavelet Representation (MIWR) I I N I F D, D, D N, D N,,+ D,+ D D, D, D N, D N,,+ D,+ D L, LN, _ L Fig.. Schematic overview of the multiscale fusion algorithm

21 Fig. 3. Fusion of the first 4 bans of a Lansat image; a: using the propose technique; b: using wavelet maxima fusion

22 I R I G I B I ~ R I ~ G I ~ B D R, D R, D G, D G, D B, D B,,+ D,+ D D R, D R, D G, D G, D B, D B,,+ D,+ D LR, LG, LB, Fig. 4. Schematic overview of the multiscale colour image enhancement algorithm

23 3 Fig. 5. Colour image enhancement; a: original; b: using the propose technique; c: using the wavelet maxima proceure

24 4 Fig. 6. Anisotropic iffusion; a: original PEPPERS image; b: gaussian noise ae (ff = 0); c: filtering using the funamental form; : filtering using the multiscale funamental forms.