\(x(t)\rightarrow\text{Analog to digital converter (ADC)}\rightarrow x[\underbrace{n}_{\text{index}}]=x(n\underbrace{T_s}_{\text{period}})=x(\underbrace{t_n}_{\text{time}})\)
Sampling converts an analog signal into a set of values specifying the signal amplitude over pre-set intervals.
Quantization converts the signal amplitude into one of a discrete set of values. (Quantization Error to estimate info loss)
A distortion or artifact that results when the signal reconstructed from samples different from the original continuous signal.
Nyquist Sampling Theorem: \(f_s\geq 2f_{max}\) to achieve anti-aliasing.
Properties: Linearity, Scaling and Convolution Theorem [conv in f then mult in t, conv in t then mult in f].
Frequency domain
$$\hat{f}(\omega)=\int^\infty_{-\infty}f(t)e^{-i\omega t}dt$$Time domain
$$f(t)=\int^\infty_{-\infty}\hat{f}(\omega)e^{i\omega t}d\omega$$Properties: commutativity, associativity, Distributive, Scaling, Identity and Derivatives.
$$h(t)=f\otimes g=f*g=\int^\infty_{-\infty}f(\tau)g(t-\tau)d\tau$$Sifting for multiplication, Shifting for Shifting.
$$\int^\infty_{-\infty}\delta(t)dt = 1$$where \(\delta(t)=\frac{1}{a\sqrt{\pi}}e^{-t^2/a^2} \text{,}a\rightarrow 0 \)
1D Convolution
$$y[n]=x[n]\otimes y[n]=\sum^\infty_{k=-\infty}x[k]h[n-k]$$2D Convolution
$$X\otimes G=\sum^\infty_{k=-\infty}\sum^\infty_{l=-\infty}X[i-k,j-l]G[k,l]$$Equivalence of linear filtering and convolution: \(Y=f[X;G]=X\otimes G_{flip}\).
Example: Identity filter \(\begin{pmatrix}0&0&0\\0&1&0\\0&0&0\end{pmatrix}\), Blurring filter \(\frac{1}{9}\begin{pmatrix}1&1&1\\1&1&1\\1&1&1\end{pmatrix}\), sharpening filter \(\frac{1}{9}\begin{pmatrix}1&1&1\\1&17&1\\1&1&1\end{pmatrix}\), sobel filter (for edge detection) \(\begin{pmatrix}1&2&1\\0&0&0\\-1&-2&-1\end{pmatrix}\).
Choose how pixel values are defined outside the image during convultion. -- constant: zero padding/ replication, varying: mirror/periodic.
Parts 3-5: black screen comp, Desktop/3rd year/Computing in Medicine/General/DSIP