Collage Error Research Articles

Solving inverse problems for steady-state equations using a multiple criteria model with collage distance, entropy, and sparsity

In this paper, we extend the previous method for solving inverse problems for steady-state equations using the Generalized Collage Theorem by searching for an approximation that not only minimizes the collage error but also maximizes the entropy and minimizes the sparsity. In this extended formulation, the parameter estimation minimization problem can be understood as a multiple criteria problem, with three different and conflicting criteria: The generalized collage error, the entropy associated with the unknown parameters, and the sparsity of the set of unknown parameters. We implement a scalarization technique to reduce the multiple criteria program to a single criterion one, by combining all objective functions with different trade-off weights. Numerical examples confirm that the collage method produces good, but sub-optimal, results. A relatively low-weighted entropy term allows for better approximations while the sparsity term decreases the complexity of the solution in terms of the number of elements in the basis.

Prediction of the PSNR Quality of Decoded Images in Fractal Image Coding

With many observations, we find that there exists a logarithmic relationship between the average collage error (ACER) and the PSNR quality of decoded images. By making use of ACER in the encoding process, the curve fitting result can help us to predict the PSNR quality of decoded images. Then, in order to reduce the computational complexity further, an accelerated version of the prediction method is proposed. Firstly, a low limit of percentage of accumulated collage error (LLPACE) is proposed to evaluate the actual percentage of accumulated collage error (APACE). If LLPACE reaches a large value, such as 90%, the corresponding APACE can be proved to be limited in a small range (90%–100%) and the APACE can be estimated approximately. Thus, the remaining range blocks can be neglected and the corresponding computations can be saved. With the approximated APACE and the logarithmic relationship, the quality of decoded images can be predicted directly. Experiments show that, for four fractal coding methods, the quality of decoded images can be predicted accurately. Furthermore, the accelerated prediction method can provide competitive performance and reduce about one-third of total computations in the encoding process. Finally, the application of the proposed method is also discussed and analyzed.

COLLAGE-BASED INVERSE PROBLEMS FOR IFSM WITH ENTROPY MAXIMIZATION AND SPARSITY CONSTRAINTS

We consider the inverse problem associated with IFSM: Given a target function f, find an IFSM, such that its invariant fixed point f is sufficiently close to f in the Lp distance. In this paper, we extend the collage-based method developed by Forte and Vrscay (1995) along two different directions. We first search for a set of mappings that not only minimizes the collage error but also maximizes the entropy of the dynamical system. We then include an extra term in the minimization process which takes into account the sparsity of the set of mappings. In this new formulation, the minimization of collage error is treated as multi-criteria problem: we consider three different and conflicting criteria i.e., collage error, entropy and sparsity. To solve this multi-criteria program we proceed by scalarization and we reduce the model to a single-criterion program by combining all objective functions with different trade-off weights. The results of some numerical computations are presented. Numerical studies indicate that a maximum entropy principle exists for this approximation problem, i.e., that the suboptimal solutions produced by collage coding can be improved at least slightly by adding a maximum entropy criterion.

An effective image retrieval method based on Kernel Density Estimation of collage error and moment invariants

In this paper, we propose a new method that combines collage error in fractal domain and Hu moment invariants for image retrieval with a statistical method — variable bandwidth Kernel Density Estimation (KDE). The proposed method is called CHK (KDE of Collage error and Hu moment) and it is tested on the Vistex texture database with 640 natural images. Experimental results show that the Average Retrieval Rate (ARR) can reach into 78.18%, which demonstrates that the proposed method performs better than the one with parameters respectively as well as the commonly used histogram method both on retrieval rate and retrieval time.

Solving inverse problems for DEs using the Collage Theorem and entropy maximization

Abstract In this work, we focus on the inverse problem associated with a DE: Given a target function x , find a DE such that its solution x is sufficiently close to x in the sup norm distance. We extend the previous method for solving inverse problems for DEs using the Collage Theorem along a new direction. We search for a set of coefficients that not only minimizes the collage error but also maximizes the entropy. This approach is motivated by some promising results for IFS and probabilities. In our new formulation, the minimization of the collage error can be understood as a multi-criteria problem: two different and conflicting criteria are considered, i.e., collage error and the entropy. In order to deal with this kind of scenario we propose to scalarize the model, which reduces the multi-criteria program to a single-criterion program by combining all objective functions with different trade-off weights. Numerical examples confirm the sub-optimality of the Collage Theorem and we show that, by adding the entropy term, we obtain a better approximation of the solution.

Prediction on decoded image quality for fractal image coding

A new decoded image quality prediction algorithm for fractal image coding was proposed.According to the results of many experiments,there exists an approximately exponential discipline between the average collage error and the PSNR of decoded image quality.The collage error of each range block was computed in the fractal encoding process and the PSNR of decoded image quality was predicted according to the average collage error.The experimental results show that for different sizes of range blocks and fast fractal image encoding,the proposed algorithm can obtain satisfying prediction results of decoded image quality.

Fractal indexing with the joint statistical properties and its application in texture image retrieval

Fractal image coding is a block-based scheme that exploits the self-similarity hiding within an image. Fractal parameters generated by the block-based scheme are quantitative measurements of self-similarity, and therefore they can be used to construct image signatures. By combining fractal parameters and collage error, a set of new statistical fractal signatures, such as histogram of collage error (HE), joint histogram of contrast scaling and collage error (JHSE), and joint histogram of range block mean and contrast scaling and collage error (JHMSE) is proposed. These fractal signatures effectively extract and reflect the statistical properties intrinsic in texture images. Hence, they provide new statistical features for use in texture image retrieval and identification. Furthermore, in order to reduce computational complexity of the JHMSE signature, the JHMSE signature is simplified to HM (histogram of range block mean) +JHSE and HM+HS (histogram of contrast scaling) +HE, based on the independence and distance equivalence. Mathematical analysis of the simplification scheme is also carried out. The proposed fractal signatures are compared with the existing fractal signatures. Experimental results show that the proposed signatures, HM+JHSE and HM+HS+HE, achieve a higher retrieval rate with a lower computational complexity.