Conversion of rotation matrix to quaternion We use unit length Quaternion to represent rotations. Following is a general rotation matrix obtained ${\begin{bmatrix}m_{00} & m_{01}&m_{02} \\ m_{10} & m_{11}&m_{12}\\  m_{20} & m_{21}&m_{22}\end{bmatrix}}_{3\times 3}\tag 1 $. 
How do I accurately calculate quaternion $q = q_1i+q_2j+q_3k+q_4$for this matrix?Means   how can we write $q_i$s interms of given $m_{ij}$  accurately? 

The axis and angle are directly coded in this matrix.
Compute the unit eigenvector for the eigenvalue $1$ for this matrix (it must exist!) and call it $u=(u_1,u_2,u_3)$. You will be writing it as $u=u_1i+u_2j+u_2k$ from now on. This is precisely the axis of rotation, which, geometrically, all nonidentity rotations have.
You can recover the angle from the trace of the matrix: $tr(M)=2\cos(\theta)+1$. This is a consequence of the fact that you can change basis to an orthnormal basis including the axis you found above, and the rotation matrix will be the identity on that dimension, and it will be a planar rotation on the other two dimensions. That is, it will have to be of the form 
$$\begin{bmatrix}\cos(\theta)&-\sin(\theta)&0\\\sin(\theta)&\cos(\theta)&0\\0&0&1\end{bmatrix}$$
Since the trace is invariant between changes of basis, you can see how I got my equation.
Once you've solved for $\theta$, you'll use it to construct your rotation quaternion $q=\cos(\theta/2)+u\sin(\theta/2)$.
