# SNR Analysis for Quantitative Comparison of Line Detection Methods
## 3 Method--SNR of ED and SD for Line Detection
### Measure of Noise Strengths
#### 3.3.1. Correlation of Noises in Smoothed Images  
First, the amount of noise at an arbitrary location (\(r\),\(c\)) in a noise image is denoted by \(n\) as follows:  
\[n=\mathbf{n}(r,c). \tag{15}\]  
Noise \(n\) is assumed to be symmetrically distributed around zero, and its expectation \(E\{n\}\) satisfies the following equation:  
\[E\{n\}=0. \tag{16}\]  
Then, the dispersion or variance of the noise at a single pixel \(\sigma_{n}^{2}\) is defined as follows:  
\[\sigma_{n}^{2}=D\{n\}=E\{\left[n-E\{n\}\right]^{2}\}=E\{n^{2}\}. \tag{17}\]  
By extrapolating the expectation and dispersion of the noise to the noises of all the pixels in an image, the expectation and dispersion of the vector containing all noises can be written as follows:  
\[E\{\text{vec}(\mathbf{n})\}=\mathbf{0}, \tag{18}\]  
where \(\text{vec}(\cdot)\) is an operation converting a matrix into a column vector and \(\mathcal{I}_{\text{c}}\) is the continuous version of the identity matrix defined as \(\mathcal{I}_{\text{c}}(a,b)=1\) if \(a=b\), else \(\mathcal{I}_{\text{c}}(a,b)=0\). The noise amount remaining after smoothing at an arbitrary location (\(r\),\(c\)) can be modeled as follows:  
\[(\mathbf{n}*\mathbf{s})(r,c)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \mathbf{n}(r-u,c-v)\mathbf{s}(u,v)dudv. \tag{19}\]  
Furthermore, the expectation of the vectors of the smoothed noises is derived as follows:  
\[E\{\text{vec}(\mathbf{n}*\mathbf{s})\}=\text{vec}(E\{\mathbf{n}\}*\mathbf{s}) =\mathbf{0}. \tag{20}\]  
Moreover, the dispersion of the vectors of the smoothed noises is derived as follows:  
\[D\{\text{vec}(\mathbf{n}*\mathbf{s})\} =E\{\left[\text{vec}(\mathbf{n}*\mathbf{s})-E\{\text{vec}( \mathbf{n}*\mathbf{s})\}\right]\left[\text{vec}(\mathbf{n}*\mathbf{s})-E\{ \text{vec}(\mathbf{n}*\mathbf{s})\}\right]^{T}\} \tag{21}\] \[=E\{\left[\text{vec}(\mathbf{n}*\mathbf{s})\right]\left[\text{vec }(\mathbf{n}*\mathbf{s})\right]^{T}\}\]  
To quantify the correlation between the noises remaining after the smoothing, the covariance between them is first derived for two arbitrary pixels at (\(r\),\(c\)) and (\(r-\alpha\),\(c-\beta\)) as follows:  
\[\sigma_{\mathbf{n}*\mathbf{s}}|_{(\alpha,\beta)} =\text{cov}\{(\mathbf{n}*\mathbf{s})|_{(r,c)},(\mathbf{n}*\mathbf{ s})|_{(r-\alpha,c-\beta)}\} \tag{22}\] \[=E\{(\mathbf{n}*\mathbf{s})|_{(r,c)}(\mathbf{n}*\mathbf{s})|_{(r -\alpha,c-\beta)}\}\] \[E\{n\}=0. \tag{16}\]0 \[E\{n\}=0. \tag{16}\]1 \[E\{n\}=0. \tag{16}\]2 \[E\{n\}=0. \tag{16}\]3If we let \(d\) be the distance between the two positions (\(r\),\(c\)) and (\(r-\alpha\),\(c-\beta\)), then \(d\) is calculated as follows:  
\[E\{n\}=0. \tag{16}\]4  
Now, according to Equation (22), the covariance between the two pixels can be calculated as follows:  
\[E\{n\}=0. \tag{16}\]5  
Then, the dispersion \(D\{\text{vec}(\mathbf{n}*\mathbf{s})\}\), which is the auto-covariance of the remaining noise at (\(r\),\(c\)), is derived from Equation (24) as follows:  
\[E\{n\}=0. \tag{16}\]6  
Therefore, the correlation among noises remaining after the convolution with a smoothing function \(\mathbf{s}\) for two arbitrary pixels separated by a distance \(d\) is derived from Equations (24) and (25) as follows:  
\[E\{n\}=0. \tag{16}\]7