Obtaining high resolution 3D structure of protein is the key to Structure Biology. Biologist can make out protein functions by analyzing its structure. Medical engineer can design new medicine according to targets founded in virus structure. The countless examples can stand this point. Thus, it is necessary to improve its precision as high as possible. Among all methods in this area, ISAF method is the efficient method to reach atomic resolution. Some experiments also certify it. The icosahedral symmetry characteristics of ISAF function can describe icosahedral protein naturally. In addition, ISAF method fits for spherical protein because it works in spherical coordinate system.

But this method has some weaknesses. On the one hand, it is proposed not long. The study in this aspect is a little. Thus, the advantage in precision is far from being exploited. On the other hand, its time consumption is huge, which restricts its application seriously. Facing this problem, I try to improve ISAF method in aspects of precision and speed. My contribution is as follows.

**(1) The improvement in precision**

The first is CTF model based on sine and Gaussian modulation and spline interpolation. This strategy modulates the CTF curve to reduce oscillation and noise in high frequency.

The second is high definition sampling method in 2D Cryo-EM image. The traditional bilinear interpolation method is not fit for sampling frequency domain information because FFT signal is not continuous. This method use sliding window global interpolation method to solve this problem.

**(2) The improvement in speed**

The first is fast computation model of ISAF basis function. The natural logarithm method is used for solving the large number computation problem in the course of generating ISAF combination coefficients. The two-level index is built for all combination coefficients in memory to improve the addressing-speed. The dynamic programming is used for computing spherical harmonic function. This model is built based on the above solutions. Its speedup could grow up with the increase of the resolution requirement and the number of images.

The second is fast mapping method based on sixty-symmetry characteristic. The traditional method works in cylindrical coordinate and one image need to rotate 60 times to enhance SNR. But ISAF method works in spherical coordinate, the 60 mapped positions in asymmetrical triangle are all the same. It is no use for SNR. So, the 60 symmetrical operations can be reduced.

The third is fast sampling method based on rotated angle invariability of radial sampling points. The traditional method use uniform arc sampling strategy. The sampling points in every ring have no relation. Every point must be mapped into 3D space with matrix-vector multiplication. Thus, time consumption is a lot. The uniform angle sampling strategy can solve this problem. The polar angles of radial sampling points are all the same. So, this method only calculates the polar angles in one ring and the one in other rings can be deduced. Therefore, the speedup in this stage is \(S=\frac{R_m + 1}{2}\) (\(R_m\) is the maximum Fourier radius) and the whole speed of ISAF algorithm also increases to some extent.

The fourth is fast calculation strategy of density function. The main bottleneck of ISAF method lies in the calculation of density function as it occupies 90% running time. This strategy can solve this problem and is composed of three components: the fast calculation method of density function of mesh point in spherical coordinate system, the transformation method of density function of mesh point from spherical coordinate system to Cartesian coordinate system and the fast two-phase mapping method. The time complexity of calculating density function is decreased from \(O\left[ {{{\left( {{L_M}} \right)}^8}} \right]\) to \(O\left[ {{{\left( {{L_M}} \right)}^7}} \right]\) with this strategy.

We have developed the ISAF reconstruction suite by combining the above strategies and some existent architecture. It is called ISAFICT. It works well and has reconstructed some protein density maps.

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