Texture Discrimination Using an Adaptive Multiresolution Network Filter

sizes. It is also known that the neighbourhood size affects This paper proposes a new discrimination method for the discrimination ability in other methods [8]. textured images using an adaptive multiresolution network This paper proposes a new discrimination method filter. First, the local transformation function is determined using GMDH, which is applied to actual textured images by a modified GMDH (Group Method of Data Handling) so with unknown element sizes. The first part of this paper as to have a high discrimination ability. Next, the method is describes a new process to obtain the local transformation extended to obtain a function in terms of multiresolution function, which is expressed by the polynomial of the pixel densities for the discrimination of an image with an selected pixels in a bounded neighbourhood so as to obtain unknown texture element size. Finally, the grouping of areas the discrimination ability as high as possible. The next part having the same texture characteristics is achieved by describes the modification of the process to the applying the same procedure to the output of the function. discrimination at several resolutions. Finally, the proposed The practicability of this filter is experimentally confirmed method is applied to several kinds of textured images. The using several kinds of images. practicability of the method is confirmed through these experiments. INTRODUCTION BASIC CONCEPT The purpose of texture discrimination is to detect an area having a specific uniform texture from the entire image. When a human recognizes each textured pattern in an Texture discrimination is important for image segmentation image, it seems that smaller areas having same texture especially in remote sensing and shape-from-texture areas. characteristics are combined with each other to become a Up to date, some useful methods for texture discrimination larger area. This is after he has observed each textured have been developed. These are based on cooccurence Pattern at every resolution and preattentively selected several matrices [I], Fourier analysis [2], locally-defined filters representative characteristics at a certain resolution to [3],[4], a Markov random field model [5], and an discriminate the object textured pattern among others. Three autoregressive model 161. In the discrimination method 141, a important points are learned from this human recognition non-linear local transformation function which is expressed process. by the polynomial of some neighbouring pixels is obtained, (1) Some characteristics which are useful to discriminate the and the texture class of each pixel is determined by object textured pattern from only neighbouring textured estimating the output of the function at the pixel. To obtain patterns and not from all other patterns in the image, are the transformation function, GMDH (Group Method of Data selected even if the entire image includes many kinds of Handling) [7] developed by Ivakhnenko is used. The texturedpanerns. advantages of this discrimination method are; (2) The characteristics are selected from textured patterns at (i) The way to select neighbouring pixels and combine them a certain resolution where the ability to discriminate the so as to have the best discrimination ability is automatically object textured pattern from others is the greatest. determined in the process of making the transformation (3) Smaller areas having the same characteristics are function. Other methods need two steps which are combined with each other to form the whole object textured characteristics determination and discriminant function pattern to bedetected. determination in feature space. T h ~ s paper proposes a new discrimination method (ii) The method can be easily translated into hardware for using an adaptive multiresolution network filter based on high speed processing once the transformation function is GMDH to achieve this human recognition process. oGained, because the-function is composed of a number of multiply and add operations between neighbouring pixels ADAPTIVE MULTIRESOLUTION NETWORK and coefficients. FILTER On the other hand, problems associated with this Let an image be composed of two textured patterns method are; Ta and Tb, and let I(x,y) denote the density of the pixel (i) The discrimination ability depends on the neighbourhood located at the coordinate (x, y) in the image. The original size to be considered. density I(x,y) can be changed into the density Ik(x,y) at (ii) The computational cost to obtain the transformation resolution k by the following transformation with a twofunction becomes expensive because the number of dimensional Gaussian filter; neighbouring pixel combinations significantly increase when the large neighbourhood is required. 2 2 Though the discrimination ability is high only when 1 the neighbourhood size equals to approximately that of the Ik(x.y) = ------/+OOJ+m~(~,() exp((x-S) +(y-i) I texture element size, as described in [4], it suggests that %hidk -0° -0° 2 many trials are needed to decide the suitable neighbourhood 20k size when the method is applied to an actual image which is composed of several textured patterns with unknown element dkd5 , (1) where ak is a standard deviation and k is used to express the difference of resolution. Next, as shown in Fig.1, Ik(x,y) at several resolutions are calculated from I(x,y) and several neighbouring pixels at (x+Axi, y+Ayi) are selected at each resolution. Approximate ranges of the filter operation are denoted by several squares and the selected neighbouring pixels are denoted by the + symbol in the figure. Consider a local transformation function F, hereinafter called LTF, which is expressed by a high-order polynomial in terms of pixels Iki (=Ik(x+Axi, y+Ayi)) and satisfies the following condition; 1 forall(x,y)~Ta, F(Ioo, I,,, ... , Ik i . ... ) = -1 for all (X,Y)E Tb , (2) where i is used to express the difference of the selected neighbouring pixel. This LTF can realize three points in the human recognition process described in the previous section because the first and third points correspond to the LTF transforming every pixel density of Ta and Tb into 1 and -1, respectively. Moreover, the determination of LTF means not only the determination of the order and coefficients of the polynomial but also the selection of some useful neighbouring pixels at a certain resolution as variables. These determinations and selections correspond to the second point in the human recognition process. Fig.2 shows the outline of an LTF, which is the multiresolution network filter this paper proposes. This filter is composed of a series of neigbourhood transformation I functions, NTF's. Each NTF has the same structure and is a non-linear function in terms of neighbouring pixel densities at several resolutions of each input image. Partial areas having same characteristics on the original textured image are combined with each other eraduallv throueh each NTF. and finally the classified ima& whic6 has prxel density areas approximately 1 and -1 corresponding to the oringinal Ta and Tb is obtained. Fig.3 briefly explains the structure of an NTF, and gives a case of detecting four neighbouring pixels each at four resolutions (k=O, 1, 2, 3). The NTF is composed of several 0th basic transformation functions, 0th BTF's, in terms of some useful neighbouring pixel densities on the input image at several resolutions. The NTF is also composed of several 1st through nth BTF's in terms of several outputs of their previous BTF's. The 0th BTF's contributes the multiresolution of the filter, and the 1st through nth BTF's make the discrimination ability of the filter as high as possible by combining some neighbouring pixel densities in the form of a high order polynomial. The following sections describe how to determine each function corresponding to the portion 3.1 and 3.2 in Fig.3, and the portion 3.3 in Fig.2. Determination of the neighbourhood transformation function: This section describes how to select and combine the pixels in a bounded neighbourhood of Io(x,y) at a resolution 00 so as to obtain the best discrimination ability. In addition, it describes how to make the neighbourhood function NTF by using those pixels. In short, the learning area ta and tb are first set within the textured pattern area Ta and Tb, respectively, and then the NTF, which can transform every pixel density on the former and the latter area into 1 and -1, is determined by the pixel density information from the learning areas. [Step 11 Set the learning area ta and tb in the textured areas Ta and Tb, respectively. [Step 21 Obtain some pixel densities 100,101, ... , IOi, ... in a bounded neighbourhood of Io(x,y), and select two pixels IOi and I0j ( i + j ) from them. Calculate the coefficients aOO, a01, a02 and a03 of the 0th basic transformation function, Fie.1 Selection of neiehbourine nixels at evew resolution, NlF : Ns#phbourhood Transformat~on Function Fie.2 Outline of an adnolive multircsolution network f i l t c~ Fie.3 Outline of a neiehbourhd transformation functi~n. 0th BTF, foij(x,y) expressed in the form of Equ.(3) by Lagrange's method of indeterminate coefficient so as to satisfy Equ.(4) and minimize the summation sOij of dispersions ~ao i j and sboij in Equ.(5) as shown in Fig.4. a b So;, = S oij + s O i j 9 where A and B are the total pixel numbers of the learning area ta and tb. The function f o i j has x and y variables because IOi and I0j are pixel densities of which positional relationships against the processed pixel position (x, y) are preserved. This step corresponds to part I of the portion 3.1 in Fig.3, and is executed just for the number of combinations of two different pixels from the number of pixel densities 100, 101, ... , IOi, ... . Therefore, the number of 0th BTF obtained in this step is the same as the combination number. The purpose of this step is to select neighbouring pixels and determine the coefficients with high discrimination ability under the condition that the average output of the obtained 0th BTF becomes 1 in the area ta and 1 in the area tb. [Step 31 Select h of the 0th BTF's which have a high discrimination ability, in other words, a small so i j Regard the outputs of the selected 0th BTF's as the inputs to the following 1st BTF's. Let the indeces ij of the selected BTFs be iojo, i l j l , ... , ih-ljh-1, and hereinafter f n p (n=O in this step) is used instead of f o i j , where l s p l h , p=l corresponds to iojo, p=2 corresponds to i l j 1, ... , p=h corresponds to ihljh1. [Step 41 Calculate the coefficients amo, aml, am2 and am3 of the mth BTF fmpq(x,y) (m=n+l, I l p l h , I l q l h , p'a) expressed in the form of Equ.(6) by Lagrange's method of indeterminate coefficients so as to satisfy Equ.(7) and monomize the summation smpq of dispersions sampq and sbmpq in Equ.(8) as shown in Fig.5. The purpose of this step is to obtain the next mth BTF by combining two nth BTF's. The mth BTF become a higher order polynomial than the nth BTF since Equ.(8) has a non-linear term, and is expressed only in terms of original neighbouring pixels 100,101, ... , IOi, ... . The number of mth BTF obtained in this step is the same as the number of combinations of two differnt nth BTFs from h. [Step 51 Check if the smallest smpq is less than the predetermined small value or larger than the smallest value of the last s n p q In either of these cases the final step is executed. Otherwise the next step is executed. [Step 61 Select h' number of mth BTF's of which smpq are small and less than the smallest value of the last sn q, and regard their outputs as the inputs to Step 4. Steps ?and 4, and the iterations of Step 4 from Step 6 correspond to part 11 of portion 3.1 in Fig.3. By these iterations of Step 4 the BTF gradually becomes a higher order polynomial with a smaller smp [Final s t e p h i n d the mth BTF with the smallest sm construct a neighbourhood transformation function ascending order continuously from the BTF to its parent BTF's. The construction of the NTF means the fixing of the network route in Fig.3 or the acquisition of the formulas in the equation form. One of characteristics of this discrimination method is based on the iterations of Step 4. By these iterations the BTF becomes a higher order polynomial with the pixels in a bounded neighbourhood. In the case of h=3, for example, Fie4 Dctcrmination o f 0th transformation function, Fin.5 Dcrcrmination o f mth transformation function, three 0th BTF's, f01 , f 0 2 and f03, of which variables are pixel densities in the combinations of (io, jo), ( i l , j l ) and (i2, j2), respectively, are obtained in Step 3. In Step 4 three 1st BTF's, f 112, f 113, and f 123, are obtained by the combinations of (f 01, f02) , (f 01, f 03) and (f 02, f 03). respectively. These BTF's are 4th order polynomials with pixel density variables in the combination of (io, jo. i i , j i) , (io, jo, i2, j2) and ( i i , j l , i2, j2), respectively. By each iteration the order of the BTF polynomial is doubled and the polynomial is expected to have a higher discrimination ability than the last iteration. This is because the polynimial is nonlinearly composed only of two less-ordered polynomials which have high discrimination abilities in the last iteration. Multiresolution network: To deal with an image with an unknown texture element size, it is necessary to use a somewhat larger estimate of the neighbourhood size, and to use manv uixel densities within the neighbourhood for discriminatibn'. In this case the combination i f pixel densities for obtaining characteristics of the texture is significantly increased and clearly the computational cost is expensive. To avoid this problem, the network described in the previous section is modified to the multiresolution network, which is able to adaptively use low or high resolutional pixel densities for the large or small texture element image. To achieve this, the function which can use several neighbouring pixel densities Iki for discrimination is added to the network as shown in the portion 3.2 of Fig.3. These pixel densities Iki can be obtained by applying a Gaussian filter with a standard deviation 4 to the input image at the coordinates (x+Axi, y+Ayi), where the distance AX^, Ayi) from (x, y) is relative to its deviation value. By this modification, the density information of a larger neighbourhood is expressed by less number of pixel densities, and that prevents the explosion of combinations. The following is for the modification of the previous procedure. [Step 21 Obtain some pixel densities IkO, Ikl, ... , Iki, ... in a neighbourhood of Ik(x,y), and select two pixels Iki and Iki ( i + j ) from them at every resolution ok. Determine the 0th BTF fokij(x,y) at every resulution 0k so as to minimize the summation SOkij of dispersions. The conditions for this 0th BTF are the same as in Equ.(3), (4) and (5) except for thier notations. [Step 31 Select h of the 0th BTF's which have a small SOkij and regard the outputs of the selected 0th BTF's as the inputs to the following 1st BTF's. Let the indeces kij of the selected BTF's be koiojo, k l i l j 1, ... , kh1 ihljh1, and hereinafter notate f n p (n=O in this step) as fokij, where l l p l h , p=l corresponds to koiojo, p=2 corresponds to k l i l j l , ... . p=h corresponds to kh-lib-ljh-1. This modification makes it possible to select the optimum combination of pixel densities for discrimination automatically even if the object image has a different texture element size. Synthesis of textured area: It is necessary for the final discrimination to combine the partial areas having similar texture charactaristic to form one area. For this purpose the NTF is operated repeatedly on the output image of the last NTF as shown in the portion 3.3 of Fig.2. Hereinafter, one operation of the NTF is called a "generation". Since the operation with pixel densities within a certain neighbourhood is executed in Step 2 of every generation, the successive generation results in the combination of a series of pixel densities within a larger neighbourhood in the useful form for discrimination. The stop condition for generation is for the smallest Snkij in the present generation to become less than the predetermined value. The local transformation function F, is constructed by using all NTF's of the first to the present generation. ( a ) S a n ~ l e I ( b ) S a m ~ l c : Fie.6 Texture w e r n sam~lesj