IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 19, NO. 3, MARCH 2010 561 A Fast Nonparametric Noncausal MRF-Based Texture Synthesis Scheme Using a Novel FKDE Algorithm Arnab Sinha and Sumana Gupta Abstract—In this paper, a new algorithm is proposed for fast kernel density estimation (FKDE), based on principal direction divisive partitioning (PDDP) of the data space. A new framework is also developed to apply FKDE algorithms (both proposed and existing), within nonparametric noncausal Markov random field (NNMRF) based texture synthesis algorithm. The goal of the proposed FKDE algorithm is to use the finite support property of kernels for fast estimation of density. It has been shown that hyperplane boundaries for partitioning the data space and prin- cipal component vectors of the data space are two requirements for efficient FKDE. The proposed algorithm is compared with the earlier algorithms, with a number of high-dimensional data sets. The error and time complexity analysis, proves the efficiency of the proposed FKDE algorithm compared to the earlier algorithms. Due to the local simulated annealing, direct incorporation of the FKDE algorithms within the NNMRF-based texture synthesis algorithm, is not possible. This work proposes a new methodology to incorporate the effect of local simulated annealing within the FKDE framework. Afterward, the developed texture synthesis algorithms have been tested with a number of different natural textures, taken from a standard database. The comparison in terms of visual similarity and time complexity, between the pro- posed FKDE based texture synthesis algorithm with the earlier algorithms, show the efficiency. Index Terms—Kernel density estimation (KDE), Markov random field (MRF), nonparametric density estimation, principal component analysis, texture synthesis, vector quantization. I. INTRODUCTION N ATURAL texture synthesis is an important problem in image processing and computer vision. Texture synthesis models can be broadly classified within two domains, spatial and transformed domains, respectively. Within spatial-domain models, there are two basic categories; Linear, [1]–[3] and non- linear models [4], [5]. All the models within image-domain, share a common property, i.e., the pixel random variable is mod- eled as a function of neighborhood pixel random variables. In the transformed-domain, the synthesis algorithms can be clas- sified into two categories, the first one [6] is based upon the Manuscript received May 01, 2009; revised September 01, 2009. First published November 20, 2009; current version published February 18, 2010. This work was supported by the BSNL Telecom Centre of Excellence (Project Number EE/BSNL/20080033). The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Arun Abraham Ross. The authors are with the Department of Electrical Engineering, Indian Insti- tute of Technology, Kanpur, UP 208016, India (e-mail: arnabiitk@gmail.com; sumana@iitk.ac.in). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIP.2009.2036685 modeling of the transform domain with respect to a nonlinear model, and the second one [7]–[9] is based upon modeling the transformed domain with respect to some nonlinear constraints. There are also other models as described in [10]–[12], which use both spatial and transformed domain informations. There is also another sub-class, named as, intuitive models. Within this sub-class one can find spatial-domain [13]–[17], mixed-do- main [18] and transformed-domain [19] algorithms. All the spa- tial-domain intuitive models have been derived from the earlier work of [5]. We have considered the texture synthesis algorithm of [5] as our basis. The advantage of these intuitive algorithms is faster speed, but at the cost of probable convergence to a local optimum. Nonparametric, noncausal, Markov random field (NNMRF) model [5] provides a theoretical background for synthesizing textures from a broad range of natural textures. The sampling process for synthesizing texture depends upon repetitive estima- tion of Markovian conditional density. This process has a large computational complexity [20], which does not favor real-time application. Recently, a number of algorithms have been pro- posed for fast kernel density estimation (FKDE) in multivariate data analysis field, [21]–[23]. Direct application of FKDE al- gorithms within NNMRF framework for texture synthesis is not possible, due to the evolving nature of conditional density. This evolving nature is required for modeling local simulated annealing, which in turn provides faithful texture synthesis re- sult [5]. In this paper, we have proposed a new FKDE algo- rithm and compared its performance with existing algorithms, in terms of the computational complexity, time complexity, and error precision. It is concluded from the discussion and analysis that, the proposed algorithm is more efficient than the earlier algorithms for the same error bound. Furthermore, we have de- veloped a new texture synthesis algorithm to apply the FKDE algorithms within the NNMRF framework for fast texture syn- thesis. Results are shown for a broad range of textures from sto- chastic to near-regular, and are compared in terms of visual sim- ilarity and time complexity. The results provide the efficiency of the proposed FKDE algorithm and its application to model evolving nature of conditional density for fast texture synthesis. An abridged version of this work has been published in [24]. In this paper, a detailed discussion of the proposed algorithms and a comparative analysis of the proposed algorithms with the ear- lier works is provided. In Section II, we give the basic description of NNMRF model for texture synthesis. The computational complexity analysis of texture synthesis through NNMRF model is provided in Section II-B. Section III introduces the problem definition of FKDE and describes the existing approaches along with their 1057-7149/$26.00 © 2010 IEEE