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+# Lecture 4
+
+## Practical issues with filtering
+
+$$
+h[m,n]=\sum_{k=0}^{m-i}\sum_{l=0}^{n-i}g[k,l]f[m+k,n+l]
+$$
+
+Loss of information on edges of image
+
+- The filter window falls off the edge of the image
+- Need to extrapolate
+- Methods:
+  - clip filter
+  - wrap around (extend the image periodically)
+  - copy edge (extend the image by copying the edge pixels)
+  - reflect across edge (extend the image by reflecting the edge pixels)
+
+## Convolution vs Correlation
+
+- Convolution:
+  - The filter is flipped and convolved with the image
+
+$$
+h[m,n]=\sum_{k=i}^{m}\sum_{l=i}^{n}g[k,l]f[m-k,n-l]
+$$
+
+- Correlation:
+  - The filter is not flipped and convolved with the image
+
+$$
+h[m,n]=\sum_{k=0}^{m-i}\sum_{l=0}^{n-i}g[k,l]f[m+k,n+l]
+$$
+
+does not matter for deep learning
+
+```python
+scipy.signal.convolve2d(image, kernel, mode='same')
+scipy.signal.correlate2d(image, kernel, mode='same')
+```
+
+but pytorch uses correlation for convolution, the convolution in pytorch is actually a correlation in scipy.
+
+## Frequency domain representation of linear image filters
+
+TL;DR: It can be helpful to think about linear spatial filters in terms fro their frequency domain representation
+
+- Fourier transform and frequency domain
+- The convolution theorem
+
+Hybrid image: More in homework 2
+
+Human eye is sensitive to low frequencies in far field, high frequencies in near field
+
+### Change of basis from an image perspective
+
+For vectors:
+
+- Vector -> Invertible matrix multiplication -> New vector
+- Normally we think of the standard/natural basis, with unit vectors in the direction of the axes
+
+For images:
+
+- Image -> Vector -> Invertible matrix multiplication -> New vector -> New image
+- Standard basis is just a collection of one-hot images
+
+Use `im.flatten()` to convert an image to a vector
+
+$$
+Image(M^{-1}GMVec(I))
+$$
+
+- M is the change of basis matrix, $M^{-1}M=I$
+- G is the operation we want to perform
+- Vec(I) is the vectorized image
+
+#### Lossy image compression (JPEG)
+
+- JPEG is a lossy compression algorithm
+- It uses the DCT (Discrete Cosine Transform) to transform the image to the frequency domain
+- The DCT is a linear operation, so it can be represented as a matrix multiplication
+- The JPEG algorithm then quantizes the coefficients and entropy codes them (use Huffman coding)
+
+## Thinking in frequency domain
+
+### Fourier transform
+
+Any univariate function can be represented as a weighted sum of sine and cosine functions
+
+$$
+X[k]=\sum_{n=N-1}^{0}x[n]e^{-2\pi ikn/N}=\sum_{n=0}^{N-1}x[n]\left[\sin\left(\frac{2\pi}{N}kn\right)+i\cos\left(\frac{2\pi}{N}kn\right)\right]
+$$
+
+- $X[k]$ is the Fourier transform of $x[n]$
+- $e^{-2\pi ikn/N}$ is the basis function
+- $x[n]$ is the original function
+
+Real part:
+
+$$
+\text{Re}(X[k])=\sum_{n=0}^{N-1}x[n]\cos\left(\frac{2\pi}{N}kn\right)
+$$
+
+Imaginary part:
+
+$$
+\text{Im}(X[k])=\sum_{n=0}^{N-1}x[n]\sin\left(\frac{2\pi}{N}kn\right)
+$$
+
+Fourier transform stores the magnitude and phase of the sine and cosine function at each frequency
+
+- Amplitude: encodes how much signal there is at a particular frequency
+- Phase: encodes the spacial information (indirectly)
+- For mathematical convenience, this is often written as a complex number
+
+Amplitude: $A=\pm\sqrt{\text{Re}(\omega)^2+\text{Im}(\omega)^2}$
+
+Phase: $\phi=\tan^{-1}\left(\frac{\text{Im}(\omega)}{\text{Re}(\omega)}\right)$
+
+So use $A\sin(\omega+\phi)$ to represent the signal
+
+Example:
+
+$g(t)=\sin(2\pi ft)+\frac{1}{3}\sin(2\pi (3f)t)$
+
+### Fourier analysis of images
+
+Intensity image and Fourier image
+
+Signals can be composed.
+
+![jpeg basis](https://static.notenextra.trance-0.com/CSE559A/8x8_DCT_basis.png)
+
+Note: frequency domain is often visualized using a log of the absolute value of the Fourier transform
+
+Blurring the image is to delete the high frequency components (removing the center of the frequency domain)
+
+## Convolution theorem
+
+The Fourier transform of the convolution of two functions is the product of their Fourier transforms
+
+$$
+F[f*g]=F[f]F[g]
+$$
+
+- $F$ is the Fourier transform
+- $*$ is the convolution
+
+Convolution in spatial domain is equivalent to multiplication in frequency domain
+
+$$
+g*h=F^{-1}[F[g]F[h]]
+$$
+
+- $F^{-1}$ is the inverse Fourier transform
+
+### Is convolution invertible?
+
+- Redo the convolution in the image domain is division in the frequency domain
+
+$$
+g*h=F^{-1}\left[\frac{F[g]}{F[h]}\right]
+$$
+
+- This is not always possible, because $F[h]$ may be zero and we may not know the filter
+
+Small perturbations in the frequency domain can cause large perturbations in the spatial domain and vice versa
+
+Deconvolution is hard and a active area of research
+
+- Even if you know the filter, it is not always possible to invert the convolution, requires strong regularization
+- If you don't know the filter, it is even harder
+
+## 2D image transformations
+
+### Array slicing and image wrapping
+
+Fast operation for extracting a subimage
+
+- cropped image `image[10:20, 10:20]`
+- flipped image `image[::-1, ::-1]`
+
+Image wrapping allows more flexible operations
+
+#### Upsampling an image
+
+- Upsampling an image is the process of increasing the resolution of the image
+
+Bilinear interpolation:
+
+- Use the average of the 4 nearest pixels to determine the value of the new pixel
+
+Other interpolation methods:
+
+- Bicubic interpolation: Use the average of the 16 nearest pixels to determine the value of the new pixel
+- Nearest neighbor interpolation: Use the value of the nearest pixel to determine the value of the new pixel
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