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content/CSE559A/CSE559A_L5.md

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+# CSE559A Lecture 5
+
+## Continue on linear interpolation
+
+- In linear interpolation, extreme values are at the boundary.
+- In bicubic interpolation, extreme values may be inside.
+
+`scipy.interpolate.RegularGridInterpolator`
+
+### Image transformations
+
+Image warping is a process of applying transformation $T$ to an image.
+
+Parametric (global) warping: $T(x,y)=(x',y')$
+
+Geometric transformation: $T(x,y)=(x',y')$ This applies to each pixel in the same way. (global)
+
+#### Translation
+
+$T(x,y)=(x+a,y+b)$
+
+matrix form:
+
+$$
+\begin{pmatrix}
+x'\\y'
+\end{pmatrix}
+=
+\begin{pmatrix}
+1&0\\0&1
+\end{pmatrix}
+\begin{pmatrix}
+x\\y
+\end{pmatrix}
++
+\begin{pmatrix}
+a\\b
+\end{pmatrix}
+$$
+
+#### Scaling
+
+$T(x,y)=(s_xx,s_yy)$ matrix form:
+
+$$
+\begin{pmatrix}
+x'\\y'
+\end{pmatrix}
+=
+\begin{pmatrix}
+s_x&0\\0&s_y
+\end{pmatrix}
+\begin{pmatrix}
+x\\y
+\end{pmatrix}
+$$
+
+#### Rotation
+
+$T(x,y)=(x\cos\theta-y\sin\theta,x\sin\theta+y\cos\theta)$
+
+matrix form:
+
+$$
+\begin{pmatrix}
+x'\\y'
+\end{pmatrix}
+=
+\begin{pmatrix}
+\cos\theta&-\sin\theta\\\sin\theta&\cos\theta
+\end{pmatrix}
+\begin{pmatrix}
+x\\y
+\end{pmatrix}
+$$
+
+To undo the rotation, we need to rotate the image by $-\theta$. This is equivalent to apply $R^T$ to the image.
+
+#### Affine transformation
+
+$T(x,y)=(a_1x+a_2y+a_3,b_1x+b_2y+b_3)$
+
+matrix form:
+
+$$
+\begin{pmatrix}
+x'\\y'
+\end{pmatrix}
+=
+\begin{pmatrix}
+a_1&a_2&a_3\\b_1&b_2&b_3
+\end{pmatrix}
+\begin{pmatrix}
+x\\y\\1
+\end{pmatrix}
+$$
+
+Taking all the transformations together.
+
+#### Projective homography
+
+$T(x,y)=(\frac{ax+by+c}{gx+hy+i},\frac{dx+ey+f}{gx+hy+i})$
+
+$$
+\begin{pmatrix}
+x'\\y'\\1
+\end{pmatrix}
+=
+\begin{pmatrix}
+a&b&c\\d&e&f\\g&h&i
+\end{pmatrix}
+\begin{pmatrix}
+x\\y\\1
+\end{pmatrix}
+$$
+
+### Image warping
+
+#### Forward warping
+
+Send each pixel to its new position and do the matching.
+
+- May cause gaps where the pixel is not mapped to any pixel.
+
+#### Inverse warping
+
+Send each new position to its original position and do the matching.
+
+- Some mapping may not be invertible.
+
+#### Which one is better?
+
+- Inverse warping is better because it usually more efficient, doesn't have a problem with holes.
+- However, it may not always be possible to find the inverse mapping.
+
+## Sampling and Aliasing
+
+### Naive sampling
+
+- Remove half of the rows and columns in the image.
+
+Example:
+
+When sampling a sine wave, the result may interpret as different wave.
+
+#### Nyquist-Shannon sampling theorem
+
+- A bandlimited signal can be uniquely determined by its samples if the sampling rate is greater than twice the maximum frequency of the signal.
+
+- If the sampling rate is less than twice the maximum frequency of the signal, the signal will be aliased.
+
+#### Anti-aliasing
+
+- Sample more frequently. (not always possible)
+- Get rid of all frequencies that are greater than half of the new sampling frequency.
+  - Use a low-pass filter to get rid of all frequencies that are greater than half of the new sampling frequency. (eg, Gaussian filter)
+
+```python
+import scipy.ndimage as ndimage
+def down_sample(height, width, image):
+    # Apply Gaussian blur to the image
+    im_blur = ndimage.gaussian_filter(image, sigma=1)
+    # Down sample the image by taking every second pixel
+    return im_blur[::2, ::2]
+```
+
+## Nonlinear filtering
+
+### Median filter
+
+Replace the value of a pixel with the median value of its neighbors.
+
+- Good for removing salt and pepper noise. (black and white dot noise)
+
+### Morphological operations
+
+Binary image: image with only 0 and 1.
+
+Let $B$ be a structuring element and $A$ be the original image (binary image).
+
+- Erosion: $A\ominus B = \{p\mid B_p\subseteq A\}$, this is the set of all points that are completely covered by $B$.
+- Dilation: $A\oplus B = \{p\mid B_p\cap A\neq\emptyset\}$, this is the set of all points that are at least partially covered by $B$.
+- Opening: $A\circ B = (A\ominus B)\oplus B$, this is the set of all points that are at least partially covered by $B$ after erosion.
+- Closing: $A\bullet B = (A\oplus B)\ominus B$, this is the set of all points that are completely covered by $B$ after dilation.
+
+Boundary extraction: use XOR operation on eroded image and original image.
+
+Connected component labeling: label the connected components in the image. _use prebuild function in scipy.ndimage_
+
+## Light,Camera/Eyes, and Color
+
+### Principles of grouping and Gestalt Laws
+
+- Proximity: objects that are close to each other are more likely to be grouped together.
+- Similarity: objects that are similar are more likely to be grouped together.
+- Closure: objects that form a closed path are more likely to be grouped together.
+- Continuity: objects that form a continuous path are more likely to be grouped together.
+
+### Light and surface interactions
+
+A photon's life choices:
+
+- Absorption
+- Diffuse reflection (nice to model) (lambertian surface)
+- Specular reflection (mirror-like) (perfect mirror)
+- Transparency
+- Refraction
+- Fluorescence (returns different color)
+- Subsurface scattering (candles)
+- Photosphorescence
+- Interreflection
+
+#### BRDF (Bidirectional Reflectance Distribution Function)
+
+$$
+\rho(\theta_i,\phi_i,\theta_o,\phi_o)
+$$
+
+- $\theta_i$ is the angle of incidence.
+- $\phi_i$ is the azimuthal angle of incidence.
+- $\theta_o$ is the angle of reflection.
+- $\phi_o$ is the azimuthal angle of reflection.