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tokens: Θ s ⊥ ( r ) are denoted as and . The pseudo s are linear combinations of the underlying s. The linear transform relating the two power spectra e.g. and , or, equivalently and , are encapsulated in a mode-mixing matrix. However, due to our assumption of factorizability of the mask we can write the mixing matrix as product of two different mode-mixing matrices. The radial and angular mode-mixing matrices denoted as W r and W a depends on the w r and w a , respectively and can be expressed as follows: For nearly all-sky coverage the mixing matrices can be inverted to recover the underlying s. However, binning may be required for partial sky coverage. The power spectra and can be dealt with in an exactly similar manner. The mixing matrices remain unchanged. It is possible to further simplify the expressions by using an approximate form for j l ( x ) valid at high l (LoVerde Afshordi 2008 ): A few comments are in order. The PCL based sub-optimal estimators for projected or 2D data sets were developed by Hivon et al. ( 2002 ) in the context of CMB power-spectrum estimation. It has also been generalized for computation of cross-spectra involving external data sets (Munshi et al. 2011a ). More recently, it has been extended for the 3D data by (Munshi et al. 2011b ) and was successfully used for analysing 3D weak lensing power spectrum using CFHTLenS 19 data (Kitching et al. 2014 ). Though PCL based estimators remain sub-optimal, suitable use of weights can improve their performance. The results presented here are generalization of the one presented in Munshi et al. ( 2011b ) (where the effect of radial selection function was not included; i.e. W r = 1.) and will have general applicability in cross-correlating arbitrary projected field (3D) against a 3D cosmological tracer. Similar results can be derived for higher-order spin fields by suitably modifying . Generalization to arbitrary spin. For the sake of completeness we also provide the expression for a generic 3D field with an arbitrary spin S α. We would like to cross-correlate it with another arbitrary projected field with spin S ′ say β. In this case the basis of expansion involves spinorial harmonics . The spin-harmonics are defined in terms of D matrices as (Penrose Rindler 1984, 1986 ; Varshalovich, Moskalev Khersonskii 1988 ). The forward and backward transformations are obtained by replacing in equations ( 11 ) and ( 12 ) with spin-weighted spherical harmonics or spinorial harmonics : We follow the same notation as before but we assume the 3D field is a spinorial field of spin-weight S and is a projected filed of spin S ′ . The harmonic components of these fields are now denoted as S α l m and . The suffix s denotes quantities defined in partial sky. The radial component of the mixing matrix given in equation ( A4 ) remains unchanged. However, the angular mode-mixing matrix depends on the spin of the specific field equation ( A5 ). The corresponding auto- and cross-spectrum are defined as