Variational image decomposition for estimation of fringe orientation and density from electronic speckle pattern interferometry fringe patterns with greatly variable density

Highlights

• We propose a new method for estimating fringe orientation and density.

• We propose a new image decomposition model the $\mathit{BL}–\mathit{Hilbert}$ model.

• It is easy to select parameters for our decomposed images with uniform density.

Abstract

Fringe orientation and density are important properties of fringes. The estimation of fringe orientation and density from electronic speckle pattern interferometry (ESPI) fringe patterns with greatly variable density is still a challenging problem faced in this area. We propose an effective method based on variational image decomposition to estimate fringe orientation and density simultaneously. The $\mathit{BL}–\mathit{Hilbert}$ model is proposed to successfully decompose an ESPI fringe pattern with greatly variable density into two images: one only includes low density fringes and the other high density fringes. The density of the two decomposed images are uniform. We estimate the orientation and density of the two decomposed images by existing methods. The whole fringe orientation and density can be obtained by combining the corresponding results of the two decomposed images. We evaluate the performance of our method via application to the computer-simulated and experimentally obtained ESPI fringe patterns with greatly variable density and comparison with the widely used three methods.

Introduction

Fringe orientation and density are important properties of fringes for directing electronic speckle pattern interferometry (ESPI) fringes processing such as image filtering [1], [2], [3] and skeleton extraction [4]. Obviously, errors in estimation of fringe orientation or density will affect noise reduction and as a consequence the accuracy of the fringe analysis. Therefore, the computations of fringe orientation and density form very critical steps and are of fundamental importance.

In the past few years, there have been various methods proposed for the estimation of fringe orientation and density. For fringe orientation, these mainly include the gradient-based method (GM) [5], the plane-fit method (PM) [6], and the fast Fourier transform method (FFTM) [7]. For fringe density, these mainly include the accumulate-differences method (ADM) [8], the continuous wavelet transform (CWT) [9], and the isotropic adaptive bandpass filter (IDBF) [10]. It is worth mentioning that the estimation of fringe density is usually more difficult than that of fringe orientation. All these methods have significant advantages and disadvantages. The GM [5] is popularly used for its efficiency, but its accuracy can easily be lowered by speckle noise. The PM [6] can get accurate orientations in spite of high noise, but it is strict with the size of the calculating window. The FFTM [7] can give desired results of fringe orientation even in the regions with low or high fringe density and in noisy and poor contrast conditions, but it requires long computation times. The ADM [8] performs well in the presence of noise. The main drawback of this method is that, it cannot give satisfactory results if the quality of an image is poor [7]. The CWT [9] is suitable for fringe density estimation. The main drawbacks of this method are the required long computation times, computation complexity and low accuracy results [10]. The IDBF [10] works remarkably well for obtaining local fringe density map, but it depends on many parameters.

Although there have been various methods for the estimation of fringe orientation and density, according to our own experience in this area, it is still difficult to obtain desired estimation results for ESPI fringe patterns with greatly variable density as shown in the following. Fig. 1(a) is a computer-simulated ESPI fringe pattern with the size of 512×512 pixels, and Fig. 1(b) is an experimentally obtained one with the size of 400×400 pixels. Obviously, the fringe density of the two ESPI fringe patterns varies greatly. It is well known that the estimation results obtained by the above-mentioned methods greatly depend on the parameters in each method. When applying the above-mentioned methods for Fig. 1, we need to laboriously find a good balance between low density fringes and high density fringes through the fine tuning of input parameters, which is difficult and sometimes impossible as shown later. Therefore, the estimation of orientation and density for this case is still a challenging problem faced in this area. For the estimation of this type of orientation and density, the input parameters applied in high density regions should be different from that in low density regions. It is thus natural and reasonable to highly expect that low density fringes and high density fringes can be described separately and processed individually.

Recently the variational image decomposition (VID) technique is becoming an active area of research in image processing and attracts more and more attention. The basic idea of the methods is that they decompose an image into different components representing different information, and each component is modeled by a different function space. Then the functional is established by combining corresponding norms over the irrespective spaces. Each component can be obtained by minimizing the functional. In the past few years, there have been various image decomposition models proposed. In [11], Rudin et al. have proposed the earliest Rudin–Osher–Fatemi image decomposition model, which split an image f into two components $f=u+v$, such that u represents a cartoon or geometric component of f, while v represents the oscillatory component of f. In [12], Meyer has proposed the $\mathit{TV}–G$ model, which use the total variation (TV) of an image to model u and the norm G to model v. Also, other decomposition models such as $\mathit{TV}–H^{-1}$ model [13], $\mathit{TV}–\mathit{Hilbert}$ model [14] and so on were proposed. Particularly, in [15], [16], [17], Zhu et al. have proposed the variational image decomposition for optical fringe filtering and automatic background removing.

Most of the above-mentioned image decomposition models have been studied extensively as a useful tool for noise removal and background removing. However, the purpose of our paper is greatly different from the previous ones. In this paper, we focus on the estimation of orientation and density with greatly variable density, which is still a challenging problem faced in this area. As far as we know, this is the first attempt to introduce the variational image decomposition (VID) to propose an effective method for obtaining the orientation and density for this kind of fringe. The basic idea of our method is that we decompose an ESPI fringe pattern f into two images: one image u only includes low density fringes and the other image v high density fringes. Through the VID, the original ESPI fringe pattern with greatly variable density is transformed into two fringe images with uniform density. For the two decomposed fringe images u and v, it is easy to estimate the orientation and density separately. We further get the orientation and density of u and v by the existing effective methods. The whole fringe orientation and density of the original ESPI fringe pattern can be obtained by the simple combination of the corresponding results of u and v. For clarity, the rest of the paper is organized as follows. In Section 2, we describe our proposed method in detail for estimation of fringe orientation and density from electronic speckle pattern interferometry fringe patterns with greatly variable density. We present the experimental results and discussions in Section 3 and summarize the paper in Section 4.