Fix typos <noupdate>

src/year1/image-processing-and-computer-vision/module1/sections/_instance_obj_detection.tex

@@ -97,7 +97,7 @@ Edge-based template matching that works as follows:
             \nabla \tilde{I}_{i,j}(\tilde{P}_k) = \begin{pmatrix} \partial_x \tilde{I}_{i,j}(\tilde{P}_k) \\ \partial_y \tilde{I}_{i,j}(\tilde{P}_k) \end{pmatrix} \hspace{2em} 
             \tilde{\vec{u}}_k(\tilde{P}_k) = \frac{\nabla \tilde{I}_{i,j}(\tilde{P}_k)}{\Vert \nabla \tilde{I}_{i,j}(\tilde{P}_k) \Vert} 
         \]
-    \item Compute the similarity as the mean of the cosine similarities of each pair of gradients:
+    \item Compute the similarity as the mean cosine similarity of each pair of gradients:
         \[ S(T, \tilde{I}_{i,j}) = \frac{1}{n} \sum_{k=1}^{n} \vec{u}_k(P_k) \cdot \tilde{\vec{u}}_k(\tilde{P}_k) = \frac{1}{n} \sum_{k=1}^{n} \cos \theta_k \in [-1, 1] \]
         $S(T, \tilde{I}_{i,j}) = 1$ when the gradients perfectly match. A minimum threshold $S_\text{min}$ is used to determine if there is a match.
 \end{enumerate}

src/year1/image-processing-and-computer-vision/module1/sections/_local_features.tex

@@ -143,7 +143,7 @@ By developing the error function into matrix form, we obtain the following:
         $\matr{R}^T$ is the rotation matrix that aligns the image to the eigenvectors of $\matr{M}_w$, 
         while the eigenvalues remain the same for any rotation of the same patch.
 
-        Therefore, the eigenvalues $\lambda_1^{(w)}, \lambda_2^{(w)}$ of $\matr{M}_w$ allow to detect intensity changes along the shift directions:
+        Therefore, the eigenvalues $\lambda_1^{(w)}, \lambda_2^{(w)}$ of $\matr{M}_w$ allow to detect intensity changes along the shift direction:
         \[
             \begin{split}
                 E(\Delta x, \Delta y) &= \begin{pmatrix} \Delta x & \Delta y \end{pmatrix} 
@@ -450,7 +450,7 @@ After finding the keypoints, a descriptor of a keypoint is computed from the pix
 Given a keypoint, SIFT detector works as follows:
 \begin{enumerate}
     \item Center on the keypoint a $16 \times 16$ grid divided into $4 \times 4$ regions.
-    \item Compute for each region its orientation histogram with eight bins (i.e. bins of size $45^\circ$).
+    \item Compute, for each region, its orientation histogram with eight bins (i.e. bins of size $45^\circ$).
         The Gaussian weighting function is centered on the keypoint and has $\sigma$ equal to half the grid size.
     \item The descriptor is obtained by concatenating the histograms of each region. 
         This results in a feature vector with $128$ elements ($(4 \cdot 4) \cdot 8$).

src/year1/image-processing-and-computer-vision/module1/sections/_spatial_filtering.tex

@@ -408,7 +408,7 @@ where $\tilde{I}(p)$ is the real information.
         \end{descriptionlist}
 
         Given a pixel $p$, its neighborhood $\mathcal{N}(p)$ and the variances $\sigma_s$, $\sigma_r$ of two Gaussians,
-        the bilateral filter applied on $p$ is computes as follows:
+        the bilateral filter applied on $p$ is computed as follows:
         \[ 
             \begin{split}
                 O(p) &= \sum_{q \in \mathcal{N}(p)} H(p, q) \cdot \texttt{intensity}(q) \\