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33 changes: 16 additions & 17 deletions linear_image_filtering.qmd
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Expand Up @@ -18,7 +18,7 @@ that makes more difficult comparisons with previously learned visual
signals. Let's proceed by invoking the simplest mathematical processing
we can think of: a linear filter. Linear filters as computing structures
for vision have received a lot of attention because of their surprising
success in modeling some aspect of the processing carried our by early
success in modeling some aspect of the processing carried out by early
visual areas such as the retina, the lateral geniculate nucleus (LGN)
and primary visual cortex (V1) @Hubel62. In this chapter we will see how
far it takes us toward these goals.
Expand Down Expand Up @@ -158,7 +158,7 @@ E = \frac{1}{R} \int_{-\infty}^{\infty} v(t) ^2 dt
\end{equation}
:::

Signal are further classified as **finite energy** and **infinite
Signals are further classified as **finite energy** and **infinite
energy** signals. Finite length signals are finite energy and periodic
signals are infinite energy signals when measuring the energy in the
whole time axis.
Expand All @@ -167,7 +167,7 @@ If we want to compare two signals, we can use the squared Euclidean
distance (squared L2 norm) between them:
$$D^2 = \frac{1}{N} \sum_{n=0}^{N-1} \left| \ell_1 \left[n\right] - \ell_2 \left[n\right] \right| ^2$$
However, the euclidean distance (L2) is a poor metric when we are
interested in comparing the content of the two images and the building
interested in comparing the content of the two images and building
better metrics is an important area of research. Sometimes, the metric
is L2 but in a different representation space than pixel values. We will
talk about this more later in the book.
Expand Down Expand Up @@ -203,7 +203,7 @@ analytical characterization: linear systems.
**Linear systems** represent a very small portion of all the possible
systems one could implement, but we will see that they are capable of
creating very interesting image transformations. A function $f$ is
linear is it satisfies the following two properties:
linear if it satisfies the following two properties:

$$\begin{aligned}
f\left( \boldsymbol\ell_1+\boldsymbol\ell_2 \right) &=& f(\boldsymbol\ell_1)+ f(\boldsymbol\ell_2) \\ \nonumber
Expand Down Expand Up @@ -432,7 +432,7 @@ transformation each output pixel at location $n,m$ is the local average
of the input pixels in a local neighborhood around location $n,m$, and
this operation is the same regardless of what pixel output we are
computing. Therefore, it can be written as a convolution. The rotation
transformation (for a fix rotation) is a linear operation but it is not
transformation (for a fixed rotation) is a linear operation but it is not
translation invariant. As illustrated in @fig-transformationsquizz2 (b),
different output pixels require looking at input pixels in a way that is
location specific. At the top left corner, one wants to grab a pixel
Expand Down Expand Up @@ -570,9 +570,8 @@ Some typical choices for how to pad the input image are (see
consists of the following:
$$\ell_{\texttt{in}}\left[n,m\right] = \ell_{\texttt{in}}\left[(n)_P,(m)_Q\right]$$
where $(n)_P$ denotes the modulo operation and $(n)_P$ is the
reminder of $n/P$.

This padding transform the finite length signal into a periodic
remainder of $n/P$.
This padding transforms the finite length signal into a periodic
infinite length signal. Although this will introduce many artifacts,
it is a convenient extension for analytical derivations.

Expand All @@ -596,8 +595,8 @@ visual results.



But which one is the best boundary extension? Ideally, we would to get a
result that is a close as possible to the output we would get if we had
But which one is the best boundary extension? Ideally, we would like to get a
result that is as close as possible to the output we would get if we had
access to a larger image. In this case we know what the larger image
looks like as shown in the last column of @fig-boundaries. For each
boundary extension method, the final row shows the absolute value of the
Expand Down Expand Up @@ -662,7 +661,7 @@ strict sense) or the Dirac delta function.


:::{.column-margin}
The delta distribution is usually represented with an arrow of height 1, indicating that it has an finite value at that point, and a finite area equal to 1:
The delta distribution is usually represented with an arrow of height 1, indicating that it has an infinite value at that point, and a finite area equal to 1:

![](figures/linear_image_filtering/mn4.png){width="70%"}
:::
Expand Down Expand Up @@ -695,7 +694,7 @@ image $\ell_{\texttt{in}}$ and the kernel $h$ is written as follows:
$$\ell_{\texttt{out}}\left[n,m\right] = \ell_{\texttt{in}}\star h = \sum_{k,l=-N}^N \ell_{\texttt{in}}\left[n+k,m+l \right] h \left[k,l \right]$$
where the sum is done over the support of the filter $h$, which we
assume is a square $(-N,N)\times(-N,N)$. In the convolution, the kernel
is inverted left-right and up-down, while in the cross-correlation is
is inverted left-right and up-down, while in the cross-correlation it is
not. Remember that the convolution between image $\ell_{\texttt{in}}$
and kernel $h$ is written as follows:
$$\ell_{\texttt{out}}\left[n,m\right] = \ell_{\texttt{in}}\circ h = \sum_{k,l=-N}^N \ell_{\texttt{in}}\left[n-k,m-l \right] h \left[k,l \right]$$
Expand Down Expand Up @@ -730,7 +729,7 @@ outputs are identical when the kernel $h$ has central symmetry.

The goal of template matching is to find a small image patch within a
larger image. @fig-normcorr shows how to use template matching to
detect all the occurrences of the letter "a" with an image of text,
detect all the occurrences of the letter "a" within an image of text,
shown in @fig-normcorr (b). @fig-normcorr (a) shows the template of the
letter "a".

Expand Down Expand Up @@ -761,7 +760,7 @@ $$\ell_{\texttt{out}}\left[n,m\right]
\ell_{\texttt{in}}\left[n+k,m+l \right]
\hat{h} \left[k,l \right]$$

Note that this equation is similar to the correlation function, but the output is scaled by the local standard deviation, $\sigma \left[m,n \right]$, of the image patch
Note that this equation is similar to the correlation function, but the output is scaled by the local standard deviation, $\sigma \left[n,m \right]$, of the image patch
centered in $(n,m)$ and with support $(-N,N)\times(-N,N)$, where
$(-N,N)\times(-N,N)$ is the size of the kernel $h$. To compute the local
standard deviation, we first compute the local mean,
Expand Down Expand Up @@ -816,7 +815,7 @@ approximation to an impulse of sound), the echoes that you hear are very
close to the impulse response of the system formed by the acoustics of
the room. Any sounds that originates at the location of your clap will
produce a sound in the room that will be qualitatively similar to the
convolution of the acoustic signal with the echoes that you hear before
convolution of the acoustic signal with the echoes that you heard before
when you clapped. @fig-impulse_response_room_a2 shows a recording made
in a restaurant in Cambridge, Massachusetts.

Expand All @@ -838,8 +837,8 @@ $$h (t) = a_0 \delta (t)
The sound that reaches the listener
is the superposition of four copies of the sound emitted by the speaker.
This can be written as the convolution of the input sound,
$\ell_{\texttt{in}}(t)$, with room impulse response, $h (t)$:
$$\ell_{\texttt{out}}(t) = \ell_{\texttt{in}}(t) \circ h(t) = a_0 \ell_{\texttt{out}}(t)
$\ell_{\texttt{in}}(t)$, with room impulse response, $h(t)$:
$$\ell_{\texttt{out}}(t) = \ell_{\texttt{in}}(t) \circ h(t) = a_0 \ell_{\texttt{in}}(t)
+ a_1 \ell_{\texttt{in}}(t-T_1)
+ a_2 \ell_{\texttt{in}}(t-T_2)
+ a_3 \ell_{\texttt{in}}(t-T_3)$$ which is shown graphically in
Expand Down