Exploring Texture Ensembles by Efficient Markov Chain Monte Carlo-Toward a 'Trichromacy' Theory of Texture

I E E E Trans on P A M ! , (accepted), 1999. Exploring Texture Ensembles by Efficient Markov Chain Monte Carlo - T o w a r d s a T r i e h r o m a e y T h e o r y of Texture Song C h u n Z h u , X i u Wen Liu, Ying Nian W u Abstract T h i s article presents a m a t h e m a t i c a l definition of texture - the Julesz ensemble 0(h), w h i c h is the set of a l l images (defined on Z ) that share identical statistics h. T h e n texture m o d e l i n g is posed as a n inverse problem: given a set of images sampled from a n u n k n o w n Julesz ensemble fi(h*), we search for the statistics h* w h i c h define the ensemble. A Julesz ensemble fi(h) has a n associated p r o b a b i l i t y d i s t r i b u t i o n g(I;h), w h i c h is uniform over the images i n the ensemble a n d has zero p r o b a b i l i t y outside. I n a c o m p a n i o n paper[32], q(I; h) is shown to be the limit distribution of the F R A M E ( F i l t e r , R a n d o m F i e l d , A n d M i n i m a x E n t r o p y ) model[35] as the image lattice A -~ 7r. T h i s conclusion establishes the intrinsic link between the scientific definition of texture o n Z models of texture o n finite lattices. a n d the m a t h e m a t i c a l It brings two advantages to computer vision. T h e engineering practice of synthesizing texture images by m a t c h i n g statistics has been put o n a m a t h e m a t i c a l foundation. 2). W e are released from the b u r d e n of learning the expensive F R A M E m o d e l i n feature pursuit, m o d e l selection a n d texture synthesis. In this paper, a n efficient M a r k o v chain M o n t e C a r l o a l g o r i t h m is proposed for s a m p l i n g Julesz ensembles. T h e a l g o r i t h m generates r a n d o m texture images by m o v i n g along the directions of filter coefficients a n d thus extends the t r a d i t i o n a l single site G i b b s sampler. T h i s paper also compares four p o p u l a r statistical measures i n the literature, namely, moments, rectified functions, m a r g i n a l histograms a n d joint histograms of linear filter responses i n terms of their descriptive abilities. O u r experiments suggest that a s m a l l number of bins i n m a r g i n a l histograms are sufficient for c a p t u r i n g a variety of texture patterns. W e illustrate our theory a n d a l g o r i t h m by successfully synthesizing a number of n a t u r a l textures. Keywords: G i b b s ensemble, Julesz ensemble, texture modeling, texture synthesis, M a r k o v chain M o n t e C a r l o . Song C h u n Z h u a n d X i u W e n L i u are w i t h the Department of C o m p u t e r a n d Informa- t i o n Sciences, T h e O h i o State University, C o l u m b u s , O H 43210. Y i n g N i a n W u is w i t h the Department of Statistics, U n i v e r s i t y of C a l i f o r n i a , Los Angeles, C A 90095.