Improving image quality and detection

#### Window functions

Window functions are commonly used in signal processing. They are defined for a sample length and are usually of symmetric shape with a peak in the middle and tapering off to the sides. This can be used to control targets' sidelobe levels. This helps with target detection but might also reduce the resolution by broadening the main lobe. A window function can be applied to $\mathbf{F}$ by element wise multiplication with a window matrix $\mathbf{W}$ before the periodogram algorithm. The window matrix $\mathbf{W}$ is calculated by multiplying the outer product of two window vectors with a normalization factor \begin{equation} \mathbf{W}=\frac{1}{||w_n ||^2||w_m||^2}w_n^T\otimes w_m, w_n \in\mathbb{R}^{1\times N}, w_m \in\mathbb{R}^{1\times M}. \end{equation}

The CFAR detection is a commonly used target detection method in radar. CFAR is an acronym for constant false alarm rate and describes the basic principle of the target detector. A false alarm happens when the target detector identifies a non-existent target in a received frame. The probability of a false alarm in a frame when only noise is present is fixed to the desired value $p_\mathrm{FA}$. A threshold $\eta$ can be calculated from $p_\mathrm{FA}$. Periodogram bins with a power above the threshold are identified as targets . The probabability of the power $Z$ of a bin being above the threshold while only white noise is present is \begin{align} p_{FA, bin}&=\int_{\eta}^{\infty} f_z(z|H_0)dz=1-F_Z(\eta|H_0)\\ &=e^{\frac{-\eta}{\sigma^{2}_{N}}}. \end{align} For a false alarm rate $p_\mathrm{FA, bin}$ the threshold $\eta$ can be determined to be \begin{equation} \eta=-\sigma^{2}_{N}\ln p_\mathrm{FA, bin}. \end{equation} The desired false alarm rate $p_\mathrm{FA}$ for a full frame can be attained with \begin{equation} p_\mathrm{FA}=1-(1- p_\mathrm{FA, bin})^{MN}. \end{equation} Therefore the required threshold $\eta$ for a desired $p_\mathrm{FA}$ with noise power $\sigma^{2}_\mathrm{N}$ can be determined by \begin{equation} \eta=-\sigma^{2}_\mathrm{N}\ln{(1-(1- p_\mathrm{FA})^{\frac{1}{MN}})}. \end{equation} For the following demonstrations $p_\mathrm{FA}$ is set to 0.0001.