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        <a href="/#/topics" id="go2topics"><i class="icon-back-s"></i></a>
        <p class="p-font">&nbsp;&nbsp;&nbsp;2021-11-10</p>
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<h1 class="h1-font">Lanczos Interpolation</h1>
<h3 class="h3-font txt-yellow">The Lanczos filter</h3>
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<h3 class="h3-font">The <a href="https://www.oxfordlearnersdictionaries.com/us/" target="_blank">Oxford English Dictionary</a> defines interpolation as "the act of adding a value into a series by calculating it from surrounding known values".</h3>
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<h3 class="h3-font">For example, we have a set of values:</h3>
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\[
\begin{align}
&S = (2, 0, 1.5, 1)\\\\
&s_0 = 2,\quad s_1 = 0,\quad s_2 = 1.5,\quad s_3 = 1
\end{align}
\]
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<h3 class="h3-font">and we add a value at \(x=1.4\), between \(s_1\) and \(s_2\):</h3>
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\[
\begin{align}
&S = (2, 0, ?, 1.5, 1)\\\\
&s_1 = 0,\quad s_{1.4} =\:?,\quad s_2 = 1.5
\end{align}
\]
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<h3 class="h3-font">The interpolation value depends on the interpolation method. For example, with <a href="/linear-interpolation.html" target="_blank">linear interpolation</a> the value is \(0.6\):</h3>
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<h3 class="h3-font">To interpolate a value with <a href="https://en.wikipedia.org/wiki/Lanczos_resampling" target="_blank">Lanczos interpolation</a>, we do a <a href="https://en.wikipedia.org/wiki/Weight_function" target="_blank">weighted sum</a> of the surrounding values:</h3>
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\[
S(x) = \displaystyle \sum_{i=\lfloor x \rfloor - a + 1}^{\lfloor x \rfloor + a} s_i L(x - i)
\]
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<h3 class="h3-font">Where \(\lfloor x \rfloor\) is the <a href="https://en.wikipedia.org/wiki/Floor_and_ceiling_functions" target="_blank">floor function</a>; \(a\) is a possitive integer (2 throughout this example); \(L(x)\) is the weight function (the Lanczos filter):</h3>
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\[
L(x) =
\begin{cases}
1 & \quad \text{if } x = 0\\
\frac{a\:sin(\pi x) sin(\pi x / a)}{\pi^2 x^2} & \quad \text{if } -a \leq x < a\\
0 & \quad \text{otherwise.}
\end{cases}
\]
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<h3 class="h3-font">This function in <a href="https://www.python.org/" target="_blank">Python</a> would be like:</h3>
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<code class="language-python">from numpy import pi, sin

a = 2

def L(x):
    if x == 0: return 1
    elif -a <= x < a:
        return a*sin(pi*x)*sin(pi*x/a) / (pi**2 * x**2)
    else: return 0
</code>
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<h3 class="h3-font">For example, the value at \(1.4\) gives us:</h3>
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\[
\begin{align}
&S(1.4) = \displaystyle \sum_{i=0}^{3} s_i L(1.4 - i)\\
&S(1.4) = s_0 L(1.4) + s_1 L(0.4) + s_2 L(-0.6) + s_3 L(-1.6)\\
&S(1.4) \approx 0.4463
\end{align}
\]
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<h3 class="h3-font">Python code:</h3>
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<pre class="code"><code class="language-python">from numpy import pi, sin, floor

a = 2

def L(x):
    if x == 0: return 1
    elif -a <= x < a:
        return a*sin(pi*x)*sin(pi*x/a) / (pi**2 * x**2)
    else: return 0


# ys: y-coordinates
# x:  any value between 0 and len(ys) - 1
def lanczos_query(ys, x):
    n = ys.shape[0]

    if x < 0 or x > n-1:
        raise ValueError('value x must be between 0 and n-1')

    y = 0.0

    for i in range(int(floor(x)-a+1), int(floor(x)+a+1)):
        j = i

        if i < 0: j = 0
        elif i > n-1: j = n-1

        y += ys[j] * L(x - i)

    return y
</code></pre>
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