First Order Fuzzy Transform for Images Compression

In this paper, we present a new image compression method based on the direct and inverse F-transform, a generalization of the concept of fuzzy transform. Under weak compression rates, this method improves the quality of the images with respect to the classical method based on the fuzzy transform.


Introduction
We present a new image compression method based on the discrete direct and inverse F 1 -transform which is a generalization of the classical fuzzy transform [1] [2] identified as F 0 -transform (for brevity, F-transform).
The F-transform compression technique [3] is a lossy compression method used in image and video analysis [4]- [18] and in data analysis [19]- [25] as well.
In [26], the concept of the F-transform was extended to the cases with various types of fuzzy partitions.In [1] [27], the F s -transform (s ≥ 1), a generalization of the F-transform, was presented: in other terms, the constant components of the F-transform were replaced by polynomials in order to capture more information of the original function.In particular, the F 1 -transform was used for the edge detection problem [1] [2].The aim of this paper is to improve the quality of the decoded images after their compression via the F 1 -transform-based method.
Strictly speaking, we divide images of sizes N × M into smaller images (called blocks) of sizes N(B) × M(B) and then we code each block into another one of sizes n(B) × m(B), where n(B) < N(B) and m(B) < M(B).The compression is performed by calculating the direct F 1 -transform components with first degree polynomials.Afterwards, we calculate the inverse F 1 -transform and obtain the corresponding decoded blocks, recomposed to obtain the final reconstructed image.In Figure 1, we describe this process in detail.
The compression rate is given by The quality of a decoded image is measured by the Peak Signal to Noise Ratio (PSNR) index.
In Section 2, we recall the definition of h-uniform generalized fuzzy partition and the concept of F 1 -transform.In Section 3, a F 1 -transform-based compression method is presented and it is applied to images considered as fuzzy relations: there every image is partitioned into smaller blocks and the direct and inverse F 1 -transforms are calculated for each block.Then the decoded blocks are recomposed and the PSNR index is calculated.In Section 4, tests are applied to grey image datasets and the results are compared with similar results obtained by using the classical F-transform compression method.Section 5 contains the conclusions.

Generalized Fuzzy Partition and F 1 -Transform
We recall the main concepts [2] that will be used in the sequel.We consider a set of points (called nodes) 0 . We say that the fuzzy sets and the following constraints hold:
and the following additional properties hold:  ( ) ( ) ( ) Likewise, we define the Hilbert space ( ) with weighted inner product: ( ) ( ) ( ) ( ) We consider an integer 0 s ≥ and all pairs of integers (i, j) such that 0 i j s ≤ + ≤ .We introduce a linear subspace ( ) having as orthogonal basis the following: where s is the maximum degree of polynomials ( ) ( ) l i j k P x Q y .For s = 1, the orthogonal basis of the linear space ( ) where the function . Then the following theorem holds: , where the coefficients ij kl c are given by Following [2], let { } , 1, , , 1, , , , 2 ( ) . For s = 1, the orthogonal basis of the linear subspace ( ) Let , where the three coefficients 00 10 01 , , Then the matrix [ ] ( ) . We define the inverse F 1 -transform of the function ( ) For sake of completeness, we point out the utility of the concept of inverse ( ) , , In other words, the Equality (13)

F 1 -Transform Image Compression Method
We are interested to the case discrete, i.e. we consider functions in two variables which assume a finite number of values in [ ] 0,1 like finite fuzzy relations.In- deed, let R be a grey image of sizes N M × , , respectively, where  9), ( 10), (11) in the following form, slightly modified): ( ) ( ) ( ) ( ) The Formula ( 14) is considered as a compressed image of the original image R.
1 nm R can be decoded by using the following inverse (discrete) F 1 -transform We divide the image R of sizes N M × in sub-matrices R B of sizes ( ) ( ) , via the discrete F 1 -transform, as Formula ( 14), of components 1B kl R given by We rewrite ( 15), ( 16), ( 17) as The basic functions generalized uniform fuzzy partition of ( ) ( )  , respectively.They are generated by the basic functions ( ) ( ) ( ) ( )  , respectively.Then we have that where ( )

l y y y y y B y h y y y y y B y h
where ( ) In Figure 2, we show the basic functions (23) for N = 16 and n = 4.

The compressed block
by using the inverse F 1 -transform defined for every which approximates the original block R B .Making the union of all the decoded blocks R 1B , we obtain a fuzzy relation (denoted with) R 1 of sizes N M × .Then we measure the RMSE (Root Mean Square Error) given by Figure 2. Cosine basic functions (N = 16, n = 4) ( ) which implies that PSNR is the following:

Test Results
We compare our method with the classical F-transform compression method, but here no comparison is made with the one inspired to the Canny method used in [2].
For our tests we have considered the CVG-UGR image database extracting grey images of sizes 256 × 256 (cfr., http://decsai.ugr.es/cvg/dbimagenes/).For brevity, we only give the results for three images as Lena, Einstein and Leopard whose sources are given in Figures 3(a)-(c), respectively.
In Table 1, we show the PSNR of the F-transform and F 1 -transform methods for some values of the compression rate in the image Lena.
We make the following remarks on Table 1: − for weak compression rates the quality of the decoded image under the F 1 -transform method is better than the one obtained with the F-transform method; − for strong compression rates the quality of the images decoded in the two methods is similar; − the difference between the two PSNR's in the two methods overcomes 0.1 for ρ > 0.25.
In Figure 4, we show the trend of the PSNR for the two methods.In Figures 5(a     In Table 2 and Figure 7, we show the PSNR obtained using the F-transform and F 1 -transform methods for some values of the compression rate in the image Einstein: this table confirms the same results obtained for the image Lena in Table 1.
In Figures 8(a In Table 3 we show the PSNR values obtained using the F-transform and F 1 -transform methods for some values of the compression rate in the image Leopard.Table 3 confirms the results obtained for the images Lena and Einstein: the quality of the decoded image obtained by using the F 1 -transform is better than the one obtained using the F-transform for weak compression rates.In Figure 10, we show the trend of the PSNR index obtained by using the two methods. In Figures 11(a)-(d) (resp., Figures 12(a)-(d)), we show the decoded images of Leopard obtained by using the F-transform (resp., F 1 -transform) method for ρ = 0.0.0625,0.16, 0.284444, 0.444444, respectively.
In Figure 13, we show the trend of the difference of PSNR by varying the compression rate for all the images in the dataset above considered.Summarizing, we can say that the presence of the coefficients of the F 1 -transform is negated by noise introduced during the strong compressions, while this effect increases considerably using weak compressions rates.

Conclusion
We give an image compression method based on the direct and inverse F 1 -transform.
The results show that the PSNR of the reconstructed images with the F 1 -transform-based compression method is better than the one obtained with the F-transform-based compression.In the tested dataset of images, we find that the difference between the two corresponding PSNR values is greater than 0.1 (resp., 0.25) for ρ = 0.25 (resp., ρ ≈ 0.5).In the next papers, we shall use the
x y A x B y x y c A x x B y y

F 1 -
transform which stands in the approximation of the function F 1 -transform of f with respect to this fuzzy partition.Moreover let f be four times continuously differentiable on [ ] [ ] , , a b c d × and A k (resp., B l ) be four times continuously differentiable on [ ] , a b (resp., [ ] , c d ).Then the following holds for every ( ) [ ] [ ] the normalized value of the pixel ( ), P i j , that is ( ) ( ) , , lev R i j P i j N = if Nlev is the length of the grey scale.Let { } 1 , , n A A  and { } 1 , , m B B  be two h-uniform generalized fuzzy partitions of [ ] [ ] as (by rewriting the Equations (

Figure 4 .
Figure 4. PSNR trend for the source image Lena.

Figure 7 .
Figure 7. PSNR trend for the source image Einstein.

Figure 10 .
Figure 10.PSNR trend for the source image Leopard.

Figure 13 .
Figure 13.PSNR trend for all the images in the dataset considered.

F 1 -
transform in data analysis problems.
then the ( ) says that we can approximate a function in two variables, four times continuously differentiable on [ ] [ ]

Table 1 .
PSNR of the F-transform and F 1 -transform methods for some values of the compression rate in the image Lena.

Table 2 .
PSNR results obtained for the source image Einstein.

Table 3 .
PSNR results obtained for the source image Leopard.