IGARSS2007 1 Abstract— An extension of the Iterated Constrained Endmembers (ICE) that incorporates sparsity promoting priors to find the correct number of endmembers and simultaneously select informative spectral bands is presented. In addition to solving for endmembers and endmember fractional maps, this algorithm attempts to autonomously determine the number of endmembers required for a particular scene. The number of endmembers is found by adding a sparsity-promoting term to ICE’s objective function. Additionally, hyperspectral band selection is performed by incorporating weights associated with each hyperspectral band. A sparsity promoting term for the band weights is added to the objective function to perform band selection. Index Terms—Sparsity Promotion, Endmember, Hyperspectral Imagery I. INTRODUCTION UTONOMOUS endmember detection is a difficult problem in hyperspectral imaging. Many endmember extraction algorithms have been formulated but the majority of these algorithms require the knowledge of the number of endmembers required for a scene. The problem of autonomously determining the number of required endmembers to a large extent has not been tackled. We provide an extension of the Iterated Constrained Endmembers (ICE) Algorithm [1] that provides better estimates of the number of endmembers required for a dataset. This extension adds a sparsity-promoting term to the ICE objective function and is, therefore, referred to as SPICE. This added term encourages the pruning of unnecessary endmembers. Band selection is also performed by incorporating band weights into the objective function. Band selection is performed simultaneously with endmember determination by adding a sparsity promoting term for the band weights into SPICE’s objective function. In Section II, we review the ICE algorithm and discuss the sparsity promoting extension for both endmember determination and band selection. It is assumed that the reader is familiar with the endmember detection and hyperspectral band selection problems. In Section III, we present results from artificial and real image data. Section IV is the conclusion. II. ICE WITH SPARSITY PROMOTION A. Review of ICE Algorithm The ICE Algorithm performs a least squares minimization of the residual sum of squares (RSS) based on the convex geometry model. The convex geometry model assumes that every pixel in a scene is a linear combination of the endmembers of the scene. The convex geometry model can be written as N i p M k i k ik i ,..., 1 1 = + = ∑ = ε E X (1) where N is the number of pixels in the image, M is the number of endmembers, ε i is an error term, p ik is the proportion of endmember k in pixel i, and E k is the k th endmember. The proportions satisfy the constraints M k p p ik M k ik ,..., 1 , 0 , 1 1 = ≥ = ∑ = . (2) By minimizing the RSS, subjected to the constraints in (2), the error between the pixel spectra and the pixel estimate found by the ICE algorithm for the endmembers and their proportions is minimized.         -         - = ∑ ∑ ∑ = = = M k k ik i N i T M k k ik i p p RSS 1 1 1 E X E X (3) As described in [1], the minimizer for RSS is not unique. Therefore, the ICE algorithm adds a sum of squared distances (SSD) term to the objective function. ( )( ) V M M SSD M k M k l l k T l k ) 1 ( 1 1 1 - = - - = ∑∑ - = + = E E E E (4) where V is the sum of variances (over the bands) of the simplex vertices. The second equality is shown in [1]. As done in [1], V is used in the objective function instead of M(M-1)V in an effort to make this term independent of the number of endmembers, M. This term is related to the size of the area bounded by the endmembers. Therefore, by adding this term to the objective function, the algorithm finds endmembers that provide a tight fit around the data. Therefore, the objective function used in the ICE algorithm is Sparsity Promoting Iterated Constrained Endmember Detection with Integrated Band Selection Alina Zare, Paul Gader Senior Member, IEEE A 1-4244-1212-9/07/$25.00 ©2007 IEEE. 4045