STEP-EDGE RECONSTRUCTION USING 2D FINITE RATE OF INNOVATION PRINCIPLE Changsheng Chen, Pina Marziliano, and Alex C. Kot School of Electrical and Electronic Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798 {chen0535,epina,eackot}@ntu.edu.sg ABSTRACT Parametric signals that have a finite number of degrees of freedom per unit of time are defined as signals with Finite Rate of Innovation (FRI). Sampling and reconstruction schemes have been developed based on the 1D FRI principle and applied to reconstructing step edge images on a row by row basis. In this paper, we derive the 2D FRI principle by exploiting the separability of the B-spline sampling kernel. The proposed 2D FRI principle regards the sampling and reconstruction as block by block operations. The step-edge parame- ters can be retrieved in high accuracy with no post-processing. The performance on synthetic images shows that our proposed technique is more precise than the row by row approaches on Signal-to-Noise Ratio (SNR) levels larger than 4 dB. Experimental results on real images demonstrate that the proposed method can reconstruct the step-edge precisely under noisy and practical sampling conditions. Index Terms— Two dimensional Finite Rate of Innovation, Step-edge reconstruction, B-spline kernel. 1. INTRODUCTION The signals with finite rate of innovation (FRI) are defined such that they can be expressed in a parametric form and have a finite number of degrees of freedom per unit of time [1]. New sampling schemes have been developed for perfect reconstruction of the FRI signals by using Sinc or Gaussian sampling kernels. Dragotti et al. generalized the sampling scheme to three types of kernels with compact support [2], i.e., polynomial reproducing kernels, exponential reproducing kernels, and rational kernels. Following [2], Shukla et al. extended the sampling schemes to multi-dimension with the polynomial reproducing kernels and proposed three reconstruction schemes [3]. Nevertheless, these reconstruction schemes were developed under the noise free as- sumption. Baboulaz et al. developed a local reconstruction scheme for the step-edge with a polynomial reproducing kernel [4], i.e., the B-spline kernel. The extracted local features can be applied to the registration step in a super-resolution task, and had better performances compared to the traditional approach. Hirabayashi et al. proposed reconstruction schemes [5, 6] with the trigono- metric and hyperbolic E-spline kernels to achieve better accuracy. Baboulaz’s and Hirabayashi’s approaches were still based on the 1D FRI principle which treats the images row by row. However, in the image acquisition process the samples are always affected by both horizontal and vertical neighborhoods. In this paper, the image is regarded on a block by block ba- sis so as to exploit the vertical correlations between different rows. We derive the 2D FRI principle by exploiting the separability of the B-spline kernel. The 2D FRI principle regards the sampling and re- construction as block by block operations. Reconstruction results on both synthetic and real step-edge images demonstrate that the pro- posed reconstruction scheme is more precise with Signal-to-Noise Ratio (SNR) levels larger than 4 dB. This paper is organized as follows: Section 2 briefly reviews the existing step-edge reconstruction approaches. Section 3 presents our proposed method based on the 2D FRI principle. Comparisons on both synthetic and real step-edge images between the proposed method and the existing approaches are given in Section 4. Finally, Section 5 concludes this paper. 2. EXISTING STEP-EDGE RECONSTRUCTION METHODS The Hough transform [7] is a traditional step-edge reconstruction approach which includes two steps, i.e., an edge detection step and a voting step for parameter estimation. Popovici et al. [8] devel- oped a step-edge reconstruction method by the Custom-built mo- ments which use a testing function in an integral to find the edge parameters. The 1D FRI principle [1] has been developed to retrieve the sig- nal parameters from its sampled version. Recently, Baboulaz et al. [4] treated a step-edge as rows of 1D FRI signal and reconstructed it in a sampling framework using the B-spline kernel. First, the 1D moments were obtained by a weighted sum of the differentiated sam- ples in each row. The step-edge parameters in each row were then found in terms of the 1D moments. Finally, the estimation process was iterated row by row along an edge. The estimated step-edge parameters were obtained by averaging over edge points that have similar parameters. However, only the 1D moments from two con- secutive rows were considered at a time in the estimation process, and the estimation results from different rows can have large vari- ations under noisy condition. The edge points from the same step- edge cannot be identified precisely by the similarity measure stated in [4]. Even when the estimation results are averaged over multi- ple rows, the errors are still significant. Such limitation can also be found in Hirabayashi’s approaches [5, 6]. 3. PROPOSED RECONSTRUCTION SCHEME BASED ON THE 2D FRI PRINCIPLE 3.1. The Sampling Setup The sampling setup that is considered in this paper can be described by g(m, n)= 1 T 2  f (x, y)β(x/T - m, y/T - n)dxdy = 1 T 2  f (x, y),β(x/T - m, y/T - n)  (1) 3833 978-1-4673-0046-9/12/$26.00 ©2012 IEEE ICASSP 2012