The F-transform in Terms of Image Processing Tools

In the proposed contribution, we have discussed the usefulness of the higher degree Fp-transforms, p = 0,1, for various tasks of image processing. We show that an image can be efficiently represented as a matrix of its F-transform components. We analyze the details and discuss advantages of this type of representation. We show that in a particular case, components of the F0-transform can be obtained with the help of the operation of convolution, and components of the F1-transform can be obtained with the help of convolution with three different kernels. We give image illustrations of all made assertions.


Introduction
The images are commonly represented by the matrices of their pixel values.These values represent levels of the color or light intensities.Depending on desired processing, corresponding ways of image representation should be chosen.We recall the most frequently used processing tasks and corresponding to them representations: compression and the JPEG standard or wavelet representation (JPEG2000), computer vision and representation with Haar-like features [1], shape detection and Hough transform [2], etc.In the proposed contribution, we aim at showing that an image can be efficiently represented as a matrix of its F-transform components.We analyze the details and discuss advantages of this type of representation.Among various soft computing methods, the F-transform [3] is the most similar to conventional transforms that are used in image processing [4].This technique can be considered as a universal one due to its higher order variants [5] and already existed applications [6,7,8,9].In this contribution, we show that the theory of F-transforms, including its higher degree versions, has good potential in many image processing tasks.To prove this claim, we analyze F 0 and F 1 versions of the F-transform including their reductions to the discrete case and their representations with the help of typical tools used in image processing.By this we mean convolution with various kernels and their matrix representations in the integer scale.We aim at showing that the F 0 -transform with a uniform fuzzy partition can be represented with the help of a convolution.Moreover, in the integer representation, the particular case of the F 0 -transform with uniform fuzzy partition, generated by a triangular shaped function, is a sparse version of a convolution with the 3 × 3 Gaussian mask (up to the scaling factor).In the case of F 1 version of the F-transform, it is necessary to apply three convolutions: the first one is similar to that of the F 0 -transform, and the second-third have kernels that are similar to the Sobel kernels.These results explain why these two particular F-transforms and the whole theory in general are effective tools in image processing.The structure of the paper is a follows.Section 2 describes preliminaries and the used notation.The next two sections describe the F 0 and F 1 transforms and section 5 contains description of the parallel between the F-transform and the convolution operation.Section 6 contains some examples and the conclusion is given in section 7.

Preliminaries and notations
Let us fix the following notation and use it throughout the paper.The one channel image I is a 2D function such as where M − 1 stands for the image width, N − 1 stands for the image height and [0, 255] stands for the pixel intensity.We denote [0, M] Z = {0, 1, 2, . . ., M} and [0, N] Z = {0, 1, 2, . . ., N}.The F-transform is a result of a linear transformation (weighted orthogonal projection) on the set of supports of a fuzzy partition [3].It was shown that this transform is useful in various tasks of image processing such as inpainting [6], compression, upsampling, edge detection [7], etc.The direct F-transform (where it is applied to digital images) changes the image representation from the set of pixels to the set of F-transform components.The inverse Ftransform (applied to the set of components) returns the pixel representation of an image.This returned representation approximates the original one.The following sections describe the details.

F 0 -transform
We recall that 2D images are identified with intensity functions of two arguments.The F 0 -transform of an image is given by a corresponding matrix of components.We remind the definition of the F-transform [3], and at the beginning, recall the notion of a fuzzy partition (For the sake of simplicity, we consider this notion for a one dimensional universe).Fuzzy sets (basic functions) A 0 , . . ., A m , 1 < m < M, identified with their membership functions A 0 , . . ., and strictly decreases on where k = 1, . . ., m; We say that the fuzzy partition given by A 0 , . . ., A m , is an h-uniform fuzzy partition if nodes x k = hk, k = 0, . . ., m, are equidistant, h = M/m and two additional properties are met: Parameter h will be referred to as a radius.
Let fuzzy partition of [0, M] be given by basic functions A 0 , . . ., A m : [0, M] → [0, 1] and similarly, fuzzy partition of . We remark that the set of pixels coordinates is sufficiently dense with respect to the chosen partitions.This means that (∀k and (∀l)(∃ j ∈ [0, N] Z ) B l ( j) > 0, and this follows from the condition 5) above.
We say that the m × n matrix of real numbers F 0 mn [I] = (F 0 kl ) is called the (discrete) F 0 -transform of the image I with respect to {A 0 , . . ., A m } and {B 0 , . . ., B n }, if for all k = 0, . . ., m; l = 0, . . ., n, The elements F 0 kl are called components of the F 0 -transform.The inverse F 0 -transform with respect to the same basic functions {A 0 , . . ., A m } and {B 0 , . . ., B n } is identified with the reconstructed image O 0 where The function O 0 approximates the original function (image) I on its domain P.

F 1 -transform
In this section, we recall the (direct) F 1 -transform as it has been presented in [7].
where the weight function is equal to A k ).be a linear span of the set consisting of two orthogonal polynomials where 1 is a denotation of the respective constant function.
) be a linear span of the set consisting of three orthogonal polynomials ), and F 1 kl be the orthogonal projection of We say that matrix F 1 mn [I] = (F 1 kl ), k = 0, . . ., m, l = 0, . . ., n, is the F 1 -transform of I with respect to {A k × B l | k = 0, . . ., m, l = 0, . . ., n}, and F 1 kl is the corresponding F 1 -transform component.The F 1 -transform components of I are linear polynomials in the form where the coefficients are given by is the inverse F 1 -transform of I.

F-transform Components as Convolutions
Because of the nature of the F p -transform p = 0, 1, we can process an input image "piece-by-piece", i.e. each of its restriction to a certain subdomain covered by a combination of basic functions independently of each other.The subdomain covered by A 1 × B 1 is shown in Fig. 1.In the case of a h-uniform fuzzy partition, every basic function A k (B l ) is a translation of the so called generating function A (B). (5.7) The illustration of the triangular shaped generating function A and the corresponding h-uniform fuzzy partition is given in Fig. 2. Assume that the radius h of both partitions A 0 , . . ., A m and B 0 , . . ., B n is the same natural number.In this case, we can represent the combination of generating functions g(x, y) = A(x)B(y), x, y ∈ [−h, h] Z (this combination is considered as a kernel function) by the corresponding matrix g = A T B of its values.Matrix g is a digital mask of the kernel g.In the terminology of kernels or masks, expression (3.1) is a convolution with sliding windows.This fact unifies the image processing on the basis of the F 0 -transform with the conventional one.Another consequence of this relationship is parallelization in the computation of the F 0 -transform components.This justifies the effectiveness of the F-transform technique.It is not difficult to see that the similar correspondence can be established between the F 1 -transform coefficients c 00 kl , c 10 kl , c 01 kl in (4.5).The expression for c 00 kl coincides with that in (3.1) for the F 0 -transform component F kl and therefore, the same kernel can be used for the convolution.If we analyze the expressions for c 10 kl and c 01 kl , then we extract the kernels g x (x, y) = xA(x)B(y) and g y (x, y) = yA(x)B(y), where x, y ∈ [−h, h] Z .Below, we give examples of all considered above kernels g, g x , g y for the two types of basic functions with the radius h = 2: triangular shaped and rectangular shaped.We will see that the corresponding to these kernels masks are very well known in image processing.In particular, they are used in filtering and edge detection.We draw a parallel with the following three frequently used kernels in image processing [4].At first, we recall the expression for the Gaussian kernel G (up to a normalizing coefficient) with the 3 × 3 mask and see from (5.8) that G = 4 × g.
At second, we recall the expressions for the 3 × 3 and vertical Sobel masks (both are used for edge detection [10]), respectively: It is easy to see from (5.8) that S x = 4 × g x and S y = 4 × g y .
It has been shown in [7], that the coefficients c 01 kl and c 10 kl are approximations of partial derivatives.On the basis of this fact, the F-transform edge detector has been proposed.The use of the Sobel masks in the computation of c 01 kl and c 10 kl (in the case of triangular shaped basic functions) justifies the effectiveness of a conventional and the F-transform based approaches in the problem of edge detection.

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The 3 × 3 masks of the respective kernels g, g x , g y are as follows (we ignored the zero values): (5.9) We recall the expression for the averaging kernel A (it is widely used in image filtering) with the 3 × 3 mask and easily see that A = 1 9 g where g is given in (5.9).Another example with the horizontal and vertical 3 × 3 Prewitt masks P x and P y (both are used in the problem of edge detection [11]) is considered below.
It is easy to see from (5.9) that P x = g x and P y = g y .
Both types of masks: the Sobel and Prewitt are widely used in the problem of edge detection.Similar to the Sobel edge detector, the correspondence between the Prewitt masks and the F-transform digital kernels used in the computation of c 01 kl and c 10 kl (the case of rectangular shaped basic functions) justifies the effectiveness of a conventional and the F-transform based approaches in the problem of edge detection.
As a summarization of this section, we can say that the theory of F 0 and F 1 -transform serves as a universal platform for the various image processing techniques.A particularization of a fuzzy partition (the main parameter of the F-transform) leads to a particular conventional technique, which proved itself in this branch of science.

Examples and demonstration
Let us illustrate the F 0 and F 1 -transform actions on an input image I and compare approximation qualities of both mentioned transforms.For better illustration, the images made of pure components are also visualized (the components are used as intensity levels).The F 0 -transform is shown in Fig. 3 and F 1 -transform in Fig. 4. Visual comparison of the reconstructed by inverse F-transform images can be made from Fig. 5.     Based on the visual perception of the Fig. 5, we can say that the image O 1 reconstructed by the inverse F 1 -transform International Scientific Publications and Consulting Services is less blurry, than O 0 .It is not a surprise that the computation of O 1 is more time consuming.The theoretical justification of this claim is in [7].We illustrate this assertion by the two below given RMSE distances between the original image I and its two inverse F 0 -and F 1 -transform reconstructions denoted by O 0 and O 1 , respectively: RMSE(I, O 0 ) = 8.74 RMSE(I, O 1 ) = 5.12.
In the below shown figures, we illustrate that the coefficients c 01 kl and c 10 kl of the F 1 -transform components approximate corresponding partial derivatives approximation (the theoretical result is in [7]).The partial derivatives approximations related to the x and y axis are illustrated in Fig. 6. kl .The images has been adjusted using intensity and contrast for better visibility.

Conclusion
In this paper, we have demonstrated the usefulness of the higher degree F p -transforms, p = 0, 1 for various tasks of image processing.We have made use of the fact that the F 1 -transform is an extension of the F 0 -transform on the space with two basis polynomials.On the basis of this fact, we made visual perception of the two reconstructed images by the inverse F 0 -and F 1 -transforms.We showed that the second image is less blurry, than the first one.The components obtained using F 0 nicely represent the texture part of an image, whereas F 1 components nicely represent edges.We have drawn a parallel with convolution operations where the F 0 kernel on the basis of triangular shaped basic functions matches with the Gaussian kernel, and the F 1 kernels match with the Sobel kernels.In general, we confirmed the great potential of the higher degree F p -transforms in a wide spectrum of image processing tasks.

. 5 )
We define the inverse F 1 -transform of the function I similar to the case of the inverse F 0 -transform as a weighted combination of basic functions A k × B l with weights given by linear polynomials F 1 kl .Let F 1 nm [I] = (F 1 kl ), k = 0, . . ., m, l = 0, . . ., n be the F 1 -transform of function I ∈ L 2 ([a, b] × [c, d]) with respect to International Scientific Publications and Consulting Services generalized fuzzy partition {A k × B l | k = 0, . . ., m, l = 0, . . ., n}.We say that the function O 1 : [a, b] × [c, d] → R represented by

Figure 1 :
Figure 1: The subdomain covered by A 1 × B 1 and corresponding to it fuzzy component F 11 for the h = 2-uniform partition.

Figure 2 :
Figure 2: Generating function A of an h-uniform fuzzy partition.

Figure 3 :
Figure 3: a) Original image I; b) image composed by the F 0 -transform components; c) reconstructed image O 0 given by the inverse F 0 -transform.

Figure 4 :
Figure 4: a) Original image I; b) image composed by the F 1 -transform components; c) reconstructed image O 1 given by the inverse F 1 -transform.

10 klFigure 6 :
Figure 6: Reconstruction of the partial derivative with respect to (left image) y on the basis of c 01 kl and (right image) x on the basis of c 10 kl .The images has been adjusted using intensity and contrast for better visibility.