@article {6328,
	title = {Optimal noise reduction in 3D reconstructions of single particles using a volume-normalized filter.},
	journal = {Journal of Structural Biology},
	volume = {180},
	year = {2012},
	month = {2012 Oct},
	pages = {26-38},
	abstract = {<p>The high noise level found in single-particle electron cryo-microscopy (cryo-EM) image data presents a special challenge for three-dimensional (3D) reconstruction of the imaged molecules. The spectral signal-to-noise ratio (SSNR) and related Fourier shell correlation (FSC) functions are commonly used to assess and mitigate the noise-generated error in the reconstruction. Calculation of the SSNR and FSC usually includes the noise in the solvent region surrounding the particle and therefore does not accurately reflect the signal in the particle density itself. Here we show that the SSNR in a reconstructed 3D particle map is linearly proportional to the fractional volume occupied by the particle. Using this relationship, we devise a novel filter (the "single-particle Wiener filter") to minimize the error in a reconstructed particle map, if the particle volume is known. Moreover, we show how to approximate this filter even when the volume of the particle is not known, by optimizing the signal within a representative interior region of the particle. We show that the new filter improves on previously proposed error-reduction schemes, including the conventional Wiener filter as well as figure-of-merit weighting, and quantify the relationship between all of these methods by theoretical analysis as well as numeric evaluation of both simulated and experimentally collected data. The single-particle Wiener filter is applicable across a broad range of existing 3D reconstruction techniques, but is particularly well suited to the Fourier inversion method, leading to an efficient and accurate implementation.</p>
},
	keywords = {Algorithms, Capsid, Cryoelectron Microscopy, Data Interpretation, Fourier Analysis, Imaging, Kinesin, Models, Molecular, Papillomaviridae, Protein Structure, Quaternary, Signal-To-Noise Ratio, Software, Statistical, Tertiary, Three-Dimensional},
	issn = {1095-8657},
	doi = {10.1016/j.jsb.2012.05.005},
	author = {Sindelar, Charles V and Grigorieff, Nikolaus}
}
@article {6331,
	title = {An adaptation of the Wiener filter suitable for analyzing images of isolated single particles.},
	journal = {Journal of Structural Biology},
	volume = {176},
	year = {2011},
	month = {2011 Oct},
	pages = {60-74},
	abstract = {<p>The Wiener filter is a standard means of optimizing the signal in sums of aligned, noisy images obtained by electron cryo-microscopy (cryo-EM). However, estimation of the resolution-dependent ("spectral") signal-to-noise ratio (SSNR) from the input data has remained problematic, and error reduction due to specific application of the SSNR term within a Wiener filter has not been reported. Here we describe an adjustment to the Wiener filter for optimal summation of images of isolated particles surrounded by large regions of featureless background, as is typically the case in single-particle cryo-EM applications. We show that the density within the particle area can be optimized, in the least-squares sense, by scaling the SSNR term found in the conventional Wiener filter by a factor that reflects the fraction of the image field occupied by the particle. We also give related expressions that allow the SSNR to be computed for application in this new filter, by incorporating a masking step into a Fourier Ring Correlation (FRC), a standard resolution measure. Furthermore, we show that this masked FRC estimation scheme substantially improves on the accuracy of conventional SSNR estimation methods. We demonstrate the validity of our new approach in numeric tests with simulated data corresponding to realistic cryo-EM imaging conditions. This variation of the Wiener filter and accompanying derivation should prove useful for a variety of single-particle cryo-EM applications, including 3D reconstruction.</p>
},
	keywords = {Algorithms, Computer-Assisted, Cryoelectron Microscopy, Fourier Analysis, Image Processing, Models, Molecular, Protein Conformation, Signal-To-Noise Ratio},
	issn = {1095-8657},
	doi = {10.1016/j.jsb.2011.06.010},
	author = {Sindelar, Charles V and Grigorieff, Nikolaus}
}
