  In this paper, we consider using total variation minimization to recover
signals whose gradients have a sparse support, from a small number of
measurements. We establish the proof for the performance guarantee of total
variation (TV) minimization in recovering \emph{one-dimensional} signal with
sparse gradient support. This partially answers the open problem of proving the
fidelity of total variation minimization in such a setting \cite{TVMulti}. In
particular, we have shown that the recoverable gradient sparsity can grow
linearly with the signal dimension when TV minimization is used. Recoverable
sparsity thresholds of TV minimization are explicitly computed for
1-dimensional signal by using the Grassmann angle framework. We also extend our
results to TV minimization for multidimensional signals. Stability of
recovering signal itself using 1-D TV minimization has also been established
through a property called "almost Euclidean property for 1-dimensional TV
norm". We further give a lower bound on the number of random Gaussian
measurements for recovering 1-dimensional signal vectors with $N$ elements and
$K$-sparse gradients. Interestingly, the number of needed measurements is lower
bounded by $\Omega((NK)^{\frac{1}{2}})$, rather than the $O(K\log(N/K))$ bound
frequently appearing in recovering $K$-sparse signal vectors.
