Generating Suitable Basic Functions Used in Image Reconstruction by F-Transform

Image reconstruction technique based on F-transform uses clearly defined basic functions. These functions have strong impact on the quality of reconstruction. We can use some predefined shape and radius, but also we can create a new one from the scratch. The aim of this paper is to analyze the creating process and based on that find best basic function for input set of damaged testing images. 1. Introduction Image reconstruction aims at recovering damaged parts. In real situation this damage can be caused by various sources on various surfaces. Input information has to be digitalized as a first step and after that divided on damaged and nondamaged parts. Typical situation lays on photography. We can distinguish among many types of damage like stains, scratches, text, or noise. We can say that every kind of damage covers typical way of damaging process. Unwanted time or date stamp is covered by text. Scratches, cracks, and folds are covered by inpaint. These are shown in Figure 1 which includes also noise. Figure 1: (a) Inpaint damage; (b) text damage; (c) noise damage. Target of reconstruction is removing damaged parts from input image and replacing them by the parts with recomputed values. These values are computed from the neighborhood ones. The technique mentioned in this paper is based on F-transform which brings particular way of valuation neighborhood pixels and their usage in the computation [1, 2]. Other methods [3, 4] differ by the choice at the technique. In this paper, we will focus on the valuation part of the computation. Quality of reconstruction will be measured by RMSE value (RMSE stands for the root mean square error). In Figure 2 you can see damaged input images and reconstructed ones with usage of the ideal basic function described later. We will analyse influence of changing parameter on target quality. Figure 2: (a) Original image; (b) damaged image; (c) reconstructed image. 2. F-Transform In image reconstruction, a discrete version of the F-transform is used. Details can be seen in [5, 6]. In this section two-dimensional (2D) variant and also conditions for proper functioning will be briefly introduced. 2.1. Fuzzy Partition with Ruspini Condition Let be fixed nodes within such that , and . We say that the fuzzy sets , identified with their membership functions defined on , establish a fuzzy partition with Ruspini condition of if they fulfill the following conditions for :(1) ,？？ ;(2) if , where for uniformity of notation, we set and ;(3) is continuous;(4) , for , increases on , and , for , strictly decreases on ;(5)for all , The