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Theoretical aspects of a technique for target detection and texture segmentation in synthetic aperture radar (SAR) imagery using a wavelet frame are presented. Texture measures consist of multiscale local estimates of the following: 1) normalized second moment of the backscattered intensity and 2) variance of the wavelet-frame coefficients. This work is an extension of a method proposed in the image-processing literature. Novel issues, which are considered in the passage to radar imagery, are the influence of speckle on texture measures afforded by the wavelet frame and their dependence on polarization states (polarimetric texture). Regarding speckle, estimators that decouple the influence of speckle over texture are introduced and characterized by their expected value and variance. The response of the wavelet frame to discontinuities, which is an important issue in target detection problems, is addressed in terms of signal-to-speckle-noise ratio. The notion of polarimetric texture is revisited, providing a theoretical model that explains the dependences of texture measures on the polarization states. For one-point statistics, such model calls for a mixture of diverse polarimetric scattering mechanisms within the texture estimator support. For two-point statistics, the difference in spatial correlation properties among the polarimetric channels is called into play. To analyze these effects in polarimetric SAR data, a novel tool is introduced that is called the Wavelet Polarimetric Signature. The tool encapsulates, in graphical form, the dependence on scale and polarization state of the texture measure afforded by the wavelet frame. The theory exposed here underpins a method that has been proven successful and computationally attractive in a selected number of SAR thematic applications. It also sets the stage for the exploitation of novel target detection and textural segmentation capabilities based on polarimetric diversity.