out for the practice.) And therefore the Fourier Series representation of
$f(x)$ is $\displaystyle f(x)$
$\displaystyle=\frac{1}{L}\left\\{1+2\sum_{j=1}^{\infty}\left[\cos\left(\frac{j\pi
x_{0}}{L}\right)\cos\left(\frac{j\pi x}{L}\right)+\sin\left(\frac{j\pi
x_{0}}{L}\right)\sin\left(\frac{j\pi x}{L}\right)\right]\right\\}$
$\displaystyle=\frac{1}{L}\left\\{1+2\sum_{j=1}^{\infty}\cos\left[\frac{j\pi(x-x_{0})}{L}\right]\right\\}$
$\displaystyle=\frac{1}{L}\left\\{-1+2\sum_{j=0}^{\infty}\cos\left[\frac{j\pi(x-x_{0})}{L}\right]\right\\}.$
Here we have made use of the cosine-of-a-sum formula $\cos(a\pm b)=\cos a\cos
b\mp\sin a\sin b.$

The important thing to notice about this example is that we took a
representation that was perfectly localized, $\delta(x-x_{0})$, but we got a
representation out that does not look to be localized in the slightest,
especially not when it comes to which angular wavenumbers,
$\kappa_{j}=j\pi/L$, are preferred. In fact, when we take the limit
$x_{0}\rightarrow 0^{+}$, we find

$\displaystyle\lim_{x_{0}\rightarrow
0^{+}}f(x)=\frac{1}{L}\left[-1+2\sum_{j=0}^{\infty}\cos\left(\frac{j\pi
x}{L}\right)\right]=\frac{1}{L}\left[-1+2\sum_{j=0}^{\infty}1\cdot\cos\left(\frac{j\pi
x}{L}\right)\right].$

I chose to emphasize that 1 in front of the $\cos$ in the last equality to
show you that every single coefficient in the sum is weighted equally (with
the exception of $a_{0}$ which is only $1/2$ of the others). Remember that if
this were a particle with a position $x$, then our perfectly localized
particle has zero uncertainty associated with where it is located. Meanwhile,
if we recall from quantum mechanics that the momentum is $p=\hbar\kappa$, then
we conclude that we have an infinite uncertainty associated with what momentum
our particle has, since it seems to require all possible momenta equally
($0\leq j<\infty$) to perfectly localize it at $x_{0}\rightarrow 0^{+}$. In
other words, if we know position with perfect precision, we know nothing about
the momentum — i.e. we just found an example of the Heisenberg Uncertainty
Principle!

Now is this really the uncertainty principle, or is it simply a coincidence?
It turns out that this “loss of perfect localization” is a mathematical
consequence of having non-commuting linear operators which essentially just
means that perfect localization cannot exist for both operator
representations666For anybody interested, this means that the _eigenstates_ of
the non-commuting operators are not identical. Since linear operators are not
strictly physical things, but they are still mathematical objects, then we
conclude that it is the property of the mathematics we use which can produce
the Heisenberg Uncertainty Principle. However, as soon as we drop in the
$\hbar$ in front of all the $\kappa$, we find quantum mechanics — an
absolutely experimentally-verified physical thing. So this relationship is
definitely non-accidental, at least in terms of how we currently know the
universe behaves. The beauty of this example is it shows that this behavior
can be seen mathematically as well as experimentally!

###  4.3 Fourier Transforms

Now that we have the tools to talk about reciprocal representations of any
integrable function, we will try to generalize them further so that we may
call upon Fourier Analysis often when we study the laws that govern the
universe. This section is build based a lot on the framework developed in the
last section, so be sure to refer back to it if you get stuck.

####  4.3.1 The Case of the Infinite Domain

The great thing about the Fourier Series is that it converges exactly to any
piece-wise continuous function $f(x)$ over its domain of definition
[POMN_ConvergenceFourierSeries]777The Fourier Series converges to the average
of the left and right limits at each discontinuity
[POMN_ConvergenceFourierSeries].. However, the Fourier Series is a function of
sines and cosines — both of those are defined over an infinite domain. Thus,
the Fourier Series itself is defined over an infinite domain, even if our
original function $f(x)$ is not. Furthermore, the fact that sines and cosines
are $2\pi$-periodic, then our Fourier Series will also necessarily be
periodic; the Fourier Series over $0\leq x\leq L$ in Eq. 4.11 is $2L$-period
in $x$ as $j\pi(x+2L)/L=j\pi x/L+2\pi j$. Sometimes though, we need to talk
about $f(x)$ without these periodic extensions. In other words, sometimes we
must talk about $f(x)$ over an infinite domain otherwise we might misrepresent
our data or include non-physical elements in our work. Hence we seek a linear
combination of oscillating functions similar to Eq. 4.11 to represent $f(x)$
but over $-\infty\leq x\leq\infty$ instead of just $0\leq x\leq L$.

####  4.3.2 The Complex Fourier Series

Let’s start again with Eq. 4.6, but this time let’s assume that we can have
negative frequencies. We actually could have used negative frequencies before,
but based on the integrals for the coefficients $a_{j}$ and $b_{j}$, the
negative frequencies would not yield _linearly independent_ coefficients, so
nothing is lost by ignoring them in the full Fourier Series in Eq. 4.11.
Unfortunately, as they stand on their own, the sine and cosine function hide
negative frequencies as they are functions with odd and even symmetry,
respectively. In other words,

$\displaystyle\sin(-\theta)$ $\displaystyle=-\sin(\theta),$ (4.15)
$\displaystyle\cos(-\theta)$ $\displaystyle=\cos(\theta),$ (4.16)

which is a property that can be seen directly from comparing clockwise
($-\theta$) and counter-clockwise ($+\theta$) rotations around a unit circle.
To handle the negative frequencies explicitly, we make use of the Euler
identity for imaginary exponentials:

$\displaystyle\textrm{e}^{i\beta}=\cos\beta+i\sin\beta.$ (4.17)

(Quick little exercise: use the symmetry properties of the sinusoids to show
that $\textrm{e}^{-i\beta}=\cos\beta-i\sin\beta$.) We can use
$\textrm{e}^{i\beta}$ and its complex conjugate $\textrm{e}^{-i\beta}$ to
derive an expression for cosine and sine (its real and imaginary parts,
respectively). We will do cosine out and then I recommend you follow the same
steps to find the sine.

$\displaystyle\textrm{e}^{i\beta}+\textrm{e}^{-i\beta}=\cos\beta+i\sin\beta+\cos\beta-i\sin\beta=2\cos\beta\Rightarrow\cos\beta=\frac{\textrm{e}^{i\beta}+\textrm{e}^{-i\beta}}{2}.$
(4.18)

By almost identical reasoning,

$\displaystyle\sin\beta=\frac{\textrm{e}^{i\beta}-\textrm{e}^{-i\beta}}{2i}.$
(4.19)

We now substitute these into Eq. 4.6 to find a suitable linear combination for
$f(x)$:

$\displaystyle f(x)=\sum_{\kappa}a_{\kappa}\left(\frac{\textrm{e}^{i\kappa
x}+\textrm{e}^{-i\kappa
x}}{2}\right)+b_{\kappa}\left(\frac{\textrm{e}^{i\kappa
x}-\textrm{e}^{-i\kappa x}}{2i}\right).$ (4.20)

Combining like terms in the positive frequency term $\textrm{e}^{i\kappa x}$
and the negative frequency term $\textrm{e}^{-i\kappa x}$, we have

$\displaystyle
f(x)=\sum_{\kappa}\left(\frac{a_{\kappa}-ib_{\kappa}}{2}\right)\textrm{e}^{i\kappa
x}+\left(\frac{a_{\kappa}+ib_{\kappa}}{2}\right)\textrm{e}^{-i\kappa x}.$
(4.21)

So now the Fourier coefficients for the imaginary exponentials are linear
combinations of those for the sinusoids. Recall that any complex number $z$
can be written as $z=x+iy$, where $x$ and $y$ are real numbers. Likewise, the
complex conjugate $z^{\ast}=x-iy$. We will then define the complex Fourier
coefficient $c_{\kappa}$ as

$\displaystyle
c_{\kappa}=\lambda\left(\frac{a_{\kappa}-ib_{\kappa}}{2}\right),$ (4.22)

where $\lambda$ is a real, positive number. I included $\lambda$ here because
I will soon take its limit to infinity. When we divide $c_{\kappa}$ by
$\lambda$, we have the Fourier coefficients in Eq. 4.21. Thus we can write
$f(x)$ as

$\displaystyle f(x)$
$\displaystyle=\frac{1}{\lambda}\sum_{\kappa}c_{\kappa}\textrm{e}^{i\kappa
x}+c_{\kappa}^{\ast}\textrm{e}^{-i\kappa x}$
$\displaystyle=\frac{1}{\lambda}\sum_{+\kappa}c_{+\kappa}\textrm{e}^{i(+\kappa)x}+\frac{1}{\lambda}\sum_{-\kappa}c_{-\kappa}\textrm{e}^{i(-\kappa)x}$
$\displaystyle=\frac{1}{\lambda}\sum_{\pm\kappa}c_{\kappa}\textrm{e}^{i\kappa
x}.$ (4.23)

Here the sum means that we must account for both positive and negative
$\kappa$, so we have recovered a linear combination that does explicitly show
the negative frequencies. Before we dive in to the case where
$-\infty<x<\infty$, let’s first consider the case when
$-\lambda/2<x<\lambda/2$ since the length of the entire domain is $\lambda$.

Suppose $f(x)$ is integrable on $-\lambda/2<x<\lambda/2$. Then we choose to
construct $f(x)$ from its harmonic modes, where $L=\lambda/2$. Then

$\displaystyle+\kappa_{j}=\frac{2\pi j}{\lambda},\;\;-\kappa_{j}=-\frac{2\pi
j}{\lambda},\;\;j=0,1,2,3,\dots$ (4.24)

But the expression for $-\kappa_{j}$ above is mathematically clunky, so we
decide to absorb the negative sign into $j$ itself. Thus,

$\displaystyle\kappa_{j}=\frac{2\pi j}{\lambda},\;\;j=0,\pm 1,\pm 2,\pm
3,\dots$

That means we need to sum over all integers for $j$. Remember that the
exponential function $\textrm{e}^{\alpha}=\exp{\alpha}$. Then Eq. 4.23 becomes

$\displaystyle
f(x)=\frac{1}{\lambda}\sum_{j=-\infty}^{\infty}c_{j}\exp\left(\frac{2\pi
ijx}{\lambda}\right),\;\;-\frac{\lambda}{2}<x<\frac{\lambda}{2}.$ (4.25)

This representation is called the Complex Fourier Series of $f(x)$. To find
$c_{j}$, we use the orthogonality relationship:

$\displaystyle\int_{-\lambda/2}^{\lambda/2}\exp\left(\frac{2\pi
ijx}{\lambda}\right)\exp\left(-\frac{2\pi ikx}{\lambda}\right)\,\textrm{d}x$
$\displaystyle=\int_{-\lambda/2}^{\lambda/2}\exp\left[\frac{2\pi
i(j-k)x}{\lambda}\right]\,\textrm{d}x$ $\displaystyle=\frac{\lambda}{2\pi
i(j-k)}\left[]\textrm{e}^{i\pi(j-k)}-\textrm{e}^{-i\pi(j-k)}\right]$
$\displaystyle=\lambda\frac{\sin[\pi(j-k)]}{\pi(j-k)}$
$\displaystyle=\begin{cases}0,\;\;\textrm{if }j\neq k\\\
\lambda,\;\;\textrm{if }j=k\end{cases}.$ (4.26)

This orthogonality relationship holds since $j$ and $k$ are two integers, and
so $j-k$ is an integer and the sine of any integer multiple of $\pi$ vanishes,
meanwhile

$\displaystyle\lim_{t\rightarrow 0}\frac{\sin t}{t}=1,$ (4.27)

which can be derived from L’Hôpital’s rule for evaluating indeterminate
limits. We could have also derived this orthogonality relationship from the
ones between sines and cosines, but that has more steps and is less pretty to
look at. Then, to find $c_{j}$, we have

$\displaystyle
c_{j}=\frac{1}{\lambda}\int_{-\lambda/2}^{\lambda/2}f(x)\exp\left(-\frac{2\pi
ijx}{\lambda}\right)\,\textrm{d}x,$ (4.28)

where we used the same techniques as we did to find Eqs. 4.12, 4.13, and 4.14.
Now, with this background in place, we take $\lambda\rightarrow\infty$.

####  4.3.3 The Fourier Transform

Consider again the Complex Fourier Series given by Eq. 4.25. If we have
function that is defined on $-\infty<x<\infty$, then we must allow
$\lambda\rightarrow\infty$. Before we do that though, let’s rewrite Eq. 4.25
so it is a little uglier.

$\displaystyle f(x)$
$\displaystyle=\frac{1}{\lambda}\sum_{j=-\infty}^{\infty}c_{j}\exp\left(\frac{2\pi
ijx}{\lambda}\right)$
$\displaystyle=\frac{1}{2\pi}\sum_{j=-\infty}^{\infty}\frac{2\pi}{\lambda}c_{j}\exp\left(\frac{2\pi
ijx}{\lambda}\right)$
$\displaystyle=\frac{1}{2\pi}\sum_{j=-\infty}^{\infty}\frac{2\pi}{\lambda}c_{j}\exp\left(\frac{2\pi
ijx}{\lambda}\right)$
$\displaystyle=\frac{1}{2\pi}\sum_{j=0}^{\infty}\frac{2\pi}{\lambda}c_{j}\exp\left(\frac{2\pi
ijx}{\lambda}\right)+\frac{1}{2\pi}\sum_{j=1}^{\infty}\frac{2\pi}{\lambda}c_{-j}\exp\left(-\frac{2\pi
ijx}{\lambda}\right).$ (4.29)

Now let’s define $\Delta\kappa_{\lambda}=2\pi/\lambda$. Then we have

$\displaystyle f(x)$
$\displaystyle=\frac{1}{2\pi}\sum_{j=0}^{\infty}\Delta\kappa_{\lambda}c_{j}\exp\left[i(0+j\Delta\kappa_{\lambda})x\right]+\frac{1}{2\pi}\sum_{j=1}^{\infty}\Delta\kappa_{\lambda}c_{-j}\exp\left[i(0-j\Delta\kappa_{\lambda})x\right]$
(4.30)

But these are precisely the expression for Riemann sums as long as the
function $c(\kappa)$ is integrable. Here the integration variable is $\kappa$
and $\Delta\kappa_{\lambda\rightarrow\infty}=\textrm{d}\kappa$. Thus,

$\displaystyle
f(x)=\frac{1}{2\pi}\int_{0}^{\infty}c(\kappa)\textrm{e}^{i\kappa
x}\,\textrm{d}\kappa+\frac{1}{2\pi}\int_{-\infty}^{0}c(\kappa)\textrm{e}^{i\kappa
x}\,\textrm{d}\kappa,$ (4.31)

or to be cleaner,

$\displaystyle
f(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}F(\kappa)\textrm{e}^{i\kappa
x}\,\textrm{d}\kappa,\;\;-\infty<x<\infty.$ (4.32)

This is called the Inverse Fourier Transform, and what it says is that a
function $f(x)$ can be written as a continuous, linear combination of angular
frequencies or angular wavenumbers described by its Fourier Transform
$F(\kappa)$. It is conventional to write the transform of a function $g(x)$ as
$G(x)$, so that’s why I switched from $c(\kappa)$ to $F(\kappa)$. The only
requirements that $f(x)$ has to meet is that it must be piece-wise continuous,
and it must at least be bound at $\pm\infty$, but it should preferably be zero
at $\pm\infty$. The latter may seem strict, but it turns out that most888As
far as I know, all physical signals are finite, but I am giving myself room
for error. physical signals are finite, so the Fourier Transform is actually
incredibly useful and it is a go-to tool for physicists. Now we must find out
what the coefficients $F(\kappa)$ are.

What do you know, we employ another orthogonality relationship! In fact, we
use the same one we already have derived, Eq. 4.26, but we again must take the
limit as $\lambda\rightarrow\infty$.

$\displaystyle\int_{-\lambda/2}^{\lambda/2}\exp\left(\frac{2\pi
ijx}{\lambda}\right)\exp\left(-\frac{2\pi ikx}{\lambda}\right)\,\textrm{d}x$
$\displaystyle=\int_{-\lambda/2}^{\lambda/2}\exp\left(ij\Delta\kappa_{\lambda}x\right)\exp\left(-ik\Delta\kappa_{\lambda}x\right)\,\textrm{d}x$
$\displaystyle=\int_{-\lambda/2}^{\lambda/2}\exp\left[i(j\Delta\kappa_{\lambda}-k\Delta\kappa_{\lambda})x\right]\,\textrm{d}x$
$\displaystyle=\begin{cases}0,\;\;\textrm{if }j\neq k\\\
\lambda,\;\;\textrm{if }j=k\end{cases}.$ (4.33)

Define $\kappa=j\Delta\kappa_{\lambda\rightarrow\infty}$ and
$\kappa_{0}=k\Delta\kappa_{\lambda\rightarrow\infty}$. Then we obtain the
continuous version of the orthogonality relationship

$\displaystyle\int_{-\infty}^{\infty}\textrm{e}^{i(\kappa-\kappa_{0})x}\,\textrm{d}x=\begin{cases}0,\;\;\;\;\textrm{if
}\kappa\neq\kappa_{0}\\\ \infty,\;\;\textrm{if }\kappa=\kappa_{0}\end{cases}.$
(4.34)

Hopefully this type of expression looks familiar, but just to be certain, we
will integrate the above expression over all possible $\kappa$.

$\displaystyle\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\textrm{e}^{i(\kappa-\kappa_{0})x}\,\textrm{d}x\,\textrm{d}\kappa$
$\displaystyle=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\textrm{e}^{i(\kappa-\kappa_{0})x}\,\textrm{d}\kappa\,\textrm{d}x$
$\displaystyle=\int_{-\infty}^{\infty}\lim_{K\rightarrow\infty}\int_{-K}^{K}\textrm{e}^{i(\kappa-\kappa_{0})x}\,\textrm{d}(\kappa-\kappa_{0})\,\textrm{d}x$
$\displaystyle=\int_{-\infty}^{\infty}\lim_{K\rightarrow\infty}\frac{\textrm{e}^{iKx}-\textrm{e}^{-iKx}}{ix}\,\textrm{d}x$
$\displaystyle=2\lim_{K\rightarrow\infty}\int_{-\infty}^{\infty}\frac{\sin
Kx}{x}\,\textrm{d}x$ $\displaystyle=2\lim_{K\rightarrow\infty}\pi$
$\displaystyle=2\pi.$ (4.35)

To be clear, the evaluation of the Sine Integral above is not straightforward
at all, and it will distract us from our job here. However, it is known to be
$\pi K/|K|=\pi$, since $K>0$. For anyone interested in knowing how to do it
out, follow a similar procedure given in [stackSincFunction]. Anyway, what we
have found is that

$\displaystyle\frac{1}{2\pi}\int_{-\infty}^{\infty}\textrm{e}^{i(\kappa-\kappa_{0})x}\,\textrm{d}x$
$\displaystyle=\begin{cases}0,\;\;&\textrm{if }\kappa\neq\kappa_{0}\\\
\infty,\;\;&\textrm{if }\kappa=\kappa_{0}\end{cases},$ (4.36)
$\displaystyle\frac{1}{2\pi}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\textrm{e}^{i(\kappa-\kappa_{0})x}\,\textrm{d}x\,\textrm{d}\kappa$
$\displaystyle=1.$ (4.37)

Thus, we have found that

$\displaystyle\frac{1}{2\pi}\int_{-\infty}^{\infty}\textrm{e}^{i(\kappa-\kappa_{0})x}\,\textrm{d}x=\delta(\kappa-\kappa_{0}).$
(4.38)

To find $F(\kappa)$ then, we (finally) multiply both sides of Eq. 4.32 by
$\textrm{e}^{-i\kappa_{0}x}\,\textrm{d}x$ and integrate.

$\displaystyle\int_{-\infty}^{\infty}f(x)\textrm{e}^{-i\kappa_{0}x}\,\textrm{d}x$
$\displaystyle=\frac{1}{2\pi}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}F(\kappa)\textrm{e}^{i\kappa
x}\,\textrm{d}\kappa\,\textrm{e}^{-i\kappa_{0}x}\,\textrm{d}x$
$\displaystyle=\frac{1}{2\pi}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}F(\kappa)\textrm{e}^{i(\kappa-\kappa_{0})x}\,\textrm{d}x\,\textrm{d}\kappa$
$\displaystyle=\int_{-\infty}^{\infty}F(\kappa)\left[\frac{1}{2\pi}\int_{-\infty}^{\infty}\textrm{e}^{i(\kappa-\kappa_{0})x}\,\textrm{d}x\right]\,\textrm{d}\kappa$
$\displaystyle=\int_{-\infty}^{\infty}F(\kappa)\delta(\kappa-\kappa_{0})\,\textrm{d}\kappa$
$\displaystyle=F(\kappa_{0}).$ (4.39)

In the last equality we used the Sifting Property of the Dirac-$\delta$. By
relabeling the points of interest from $\kappa_{0}$ back to $\kappa$, we
recover the Fourier Transform

$\displaystyle F(\kappa)=\int_{-\infty}^{\infty}f(x)\textrm{e}^{-i\kappa
x}\,\textrm{d}x,\;\;-\infty<\kappa<\infty.$ (4.40)

Sometimes people prefer to “split up” the factor of $2\pi$ equally to create
the “Normalized” Fourier Transform and “Normalized” Inverse Fourier Transform,
given below.

$\displaystyle F(\kappa)$
$\displaystyle=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(x)\textrm{e}^{-i\kappa
x}\,\textrm{d}x,\;\;-\infty<\kappa<\infty,$ (4.41) $\displaystyle f(x)$
$\displaystyle=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}F(\kappa)\textrm{e}^{i\kappa
x}\,\textrm{d}\kappa,\;\;-\infty<x<\infty.$ (4.42)

These functions have the same shape though as the non-normalized versions,
they just have different scaling. In physics, we are almost always concerned
with the characteristic frequencies or wavelengths of a particular signal
which are completely unaffected by the rescaling by $\sqrt{2\pi}$. In other
words, the functional form of the transform are what we physicists usually
find important, not necessarily the numerical values of the tranforms
themselves. So it usually is just a matter of personal preference. However, in
quantum mechanics, the normalization is very important as the Fourier
transform represents a continuous transformation between position
representation and momentum representation. The $\sqrt{2\pi}$ makes sure that
probability is conserved between the two representations. But more on that
when you get to it in your first quantum mechanics class.

Fourier Transforms have some really interesting properties, most of which we
will be unable to cover in this chapter. However, the examples below are
designed to go over a few basic and important properties of the Fourier
Transform for applications in physics.

####  4.3.4 Some Important Examples

##### Uniqueness and Duality

In this first example, we want to find out if there are multiple reciprocal
space representations of the same function $f(x)$ defined on an infinite
interval. Let’s start by assuming there are at least two Inverse Fourier
Transforms that exist for $f(x)$, namely $F_{1}(\kappa)$ and $F_{2}(\kappa)$.
By the definition given by Eq. 4.32 for the Inverse Fourier Transform, then

$\displaystyle f(x)$
$\displaystyle=\frac{1}{2\pi}\int_{-\infty}^{\infty}F_{1}(\kappa)\textrm{e}^{i\kappa
x}\,\textrm{d}\kappa,\;\;-\infty<x<\infty,$ $\displaystyle f(x)$
$\displaystyle=\frac{1}{2\pi}\int_{-\infty}^{\infty}F_{2}(\kappa)\textrm{e}^{i\kappa
x}\,\textrm{d}\kappa,\;\;-\infty<x<\infty.$

If we subtract the first equation from the second, then the left-hand side is
zero since $f(x)-f(x)=0$. Therefore, right-hand side must also vanish.

$\displaystyle
0=\frac{1}{2\pi}\int_{-\infty}^{\infty}\left[F_{2}(\kappa)-F_{1}(\kappa)\right]\textrm{e}^{i\kappa
x}\,\textrm{d}\kappa,\;\;-\infty<x<\infty.$

In this step I brought the subtraction inside of the integral and factored out
the common $\textrm{e}^{i\kappa x}$. For those of you who may already see how
to proceed, continue ahead. For those who do not, please allow me to get a
little more mathy. Remember that we can write the integral above as a Riemann
sum

$\displaystyle
0=\frac{1}{2\pi}\lim_{\lambda\rightarrow\infty}\sum_{j=-\infty}^{\infty}\left[F_{2}\left(\frac{2\pi
j}{\lambda}\right)-F_{1}\left(\frac{2\pi
j}{\lambda}\right)\right]\exp\left(\frac{2\pi ijx}{\lambda}\right)\frac{2\pi
j}{\lambda},\;\;-\infty<x<\infty.$

What this statement says is that this summation is zero for every single value
of $x$ while $\lambda$ gets larger and larger. While it is true that $2\pi
j/\lambda$ does approach zero, it never explicitly reaches it for any finite
$\lambda$. So the only possible “source” of the zero is the difference between
the functions $F_{2}$ and $F_{1}$. It is true that the whole sum must vanish,
but we found this out without specifying any relationship at all between the
individual values of $F_{2}$ and $F_{1}$. For example, we do not know if
$F_{2}(0)=-F_{2}(1)$ or any other possible combination. The only thing we do
know is that the whole summation must be zero as $\lambda$ gets larger and
larger. Thus, we conclude that

$\displaystyle 0=\lim_{\lambda\rightarrow\infty}\left[F_{2}\left(\frac{2\pi
j}{\lambda}\right)-F_{1}\left(\frac{2\pi j}{\lambda}\right)\right],$

for every single value of $j$ as $\lambda$ gets larger and larger. In other
words,

$\displaystyle 0=F_{2}(\kappa)-F_{1}(\kappa)\Rightarrow
F_{2}(\kappa)=F_{1}(\kappa),\;\;-\infty<\kappa<\infty.$

Thus these two reciprocal space representations of $f(x)$ are exactly
identical for every single value of $\kappa$. We conclude that there really
only is one reciprocal space representation of $f(x)$ — i.e. the Inverse
Fourier Transform of $f(x)$ is unique.

I leave it to you to show that for any $F(\kappa)$, there exists a unique
real-space representation of it given by $f(x)$ using an almost identical
argument to the one above. The reason why this is important is that it shows
that the Fourier Transform and Inverse Fourier Transform change one function
into exactly one other function and vice versa. Thus, if we know the transform
of one function in some context, then the transform of the same function in
another context is exactly the same. Furthermore, we can show that the Fourier
Transform of $F(x)$ is $2\pi f(-\kappa)$ if

$\displaystyle f(x)$
$\displaystyle=\frac{1}{2\pi}\int_{-\infty}^{\infty}F(\kappa)\textrm{e}^{i\kappa
x}\,\textrm{d}\kappa,\;\;-\infty<x<\infty,$ $\displaystyle F(\kappa)$
$\displaystyle=\int_{-\infty}^{\infty}f(x)\textrm{e}^{-i\kappa
x}\,\textrm{d}x,\;\;-\infty<\kappa<\infty.$

Let’s take the first equation, and multiply it by $2\pi$. We are also going to
change $x\rightarrow-x$.

$\displaystyle 2\pi f(-x)$
$\displaystyle=\int_{-\infty}^{\infty}F(\kappa)\textrm{e}^{i\kappa(-x)}\,\textrm{d}\kappa,\;\;-\infty<x<\infty,$
$\displaystyle=\int_{-\infty}^{\infty}F(\kappa)\textrm{e}^{-i\kappa
x}\,\textrm{d}\kappa,\;\;-\infty<x<\infty.$

Now we are going to change the labels of the variables $x\rightarrow\kappa$
and $\kappa\rightarrow x$.

$\displaystyle 2\pi f(-\kappa)$
$\displaystyle=\int_{-\infty}^{\infty}F(x)\textrm{e}^{-ix\kappa}\,\textrm{d}x,\;\;-\infty<\kappa<\infty,$
$\displaystyle=\int_{-\infty}^{\infty}F(x)\textrm{e}^{-i\kappa
x}\,\textrm{d}x,\;\;-\infty<\kappa<\infty.$

But this last expression fits the original definition of the Fourier Transform
of a function, but in this case, $F(x)$. We note that there is no extra factor
of $2\pi$ for the Normalized Transforms (prove this for yourself if you do not
see it at first glance). This property is called the Duality Property of
Fourier Transform Pairs. Thus a function and its Fourier Transform are bound
together through the Fourier Transform mapping.

Let’s consider a time-dependent signal this time, so all $x\rightarrow t$ and
all $\kappa\rightarrow\omega$. We will consider a fairly basic signal for
practice, but this signal also has some pretty interesting physics we can
study. So without further ado, consider a Morse code style dash, given by
$\displaystyle f(t)=\begin{cases}0,\;\;&\textrm{if }-\infty<t<-\tau\\\
A,\;\;&\textrm{if }-\tau\leq t\leq\tau\\\ 0,\;\;&\textrm{if
}\tau<t<\infty\end{cases}$ where $A$ is a fixed value and $\tau>0$. This
signal could be, for example, a pulse of DC current of value $A$ that lasts
for $2\tau$ time. We seek the Fourier Transform of $f(t)$ given by Eq. 4.40.
$\displaystyle F(\omega)$
$\displaystyle=\int_{-\infty}^{\infty}f(t)\textrm{e}^{-i\omega
t}\,\textrm{d}t,\;\;-\infty<\omega<\infty$
$\displaystyle=\int_{-\infty}^{-\tau}0\textrm{e}^{-i\omega
t}\,\textrm{d}t+\int_{-\tau}^{\tau}A\textrm{e}^{-i\omega
t}\,\textrm{d}t+\int_{\tau}^{\infty}0\textrm{e}^{-i\omega t}\,\textrm{d}t$
$\displaystyle=\left.\frac{A\textrm{e}^{-i\omega
t}}{-i\omega}\right|_{-\tau}^{\tau}$
$\displaystyle=-\frac{A}{i\omega}\left(\textrm{e}^{-i\omega\tau}-\textrm{e}^{i\omega\tau}\right)$
$\displaystyle=\frac{2A}{\omega}\left(\frac{\textrm{e}^{i\omega\tau}-\textrm{e}^{-i\omega\tau}}{2i}\right)$
$\displaystyle=\frac{2A}{\omega}\sin\omega\tau$
$\displaystyle=2A\tau\,\left(\frac{\sin\omega\tau}{\omega\tau}\right).$ The
$\sin(u)/u$ function is pretty common in science and engineering and is called
the _sine cardinal_ and written as $\textrm{sinc}(u)=\sin(u)/u$ (pronounced
sink).

What we want to look at now is what happens to $F(\omega)$ as we change the
value of $\tau$. We will focus on the sine part mostly. Recall that

$\displaystyle\lim_{u\rightarrow 0}\frac{\sin u}{u}=1,$

After this point, all the zeros of the sine capital function are the zeros of
the sine function. Thus we look for when

$\displaystyle\omega\tau=\pm
j\pi\Rightarrow\omega=\pm\frac{j\pi}{\tau},\;\;j=1,2,3,\dots$

This statement says that if $\tau$ is small, then the $\omega$ for which
$F(\omega)=0$ are farther apart, but if $\tau$ is large, then the zeros of
$F(\omega)$ are closer together, as can be seen in Fig. 4.1. Physically, these
cases correspond to a shorter signal pulse and a longer signal pulse,
respectively. Thus, shorter signals need a wider range of frequencies to make
them up, while longer wider signals need shorter ranges of frequencies to make
them up! Again, we see behavior that is reminiscent of the Heisenberg
Uncertainty Principle, but these behaviors occur outside of quantum mechanics,
too!

FIG. 4.1: A few capital sines
$\mathrm{sinc}(\omega\tau)=\sin\omega\tau/\omega\tau$ plotted as functions of
$\omega$ for varying values of $\tau$. Notice that as $\tau\rightarrow 0$, the
whole function $\mathrm{sinc}(\omega\tau)$ gets wider, i.e. the Fourier
Transform needs more values of $\omega$ to account for the smaller pulse.

Before we move onto the next example, I want to show how to use the Duality
Property to evaluate potentially difficult integrals. For example, let’s say
we want to find the Fourier Transform of $g(t)=\sin(t)/t,\;-\infty<t<\infty$.
Thus we would need to evaluate

$\displaystyle G(\omega)=\int_{-\infty}^{\infty}\frac{\sin
t}{t}\textrm{e}^{-i\omega t}\,\textrm{d}t,\;\;-\infty<\omega<\infty.$

This is doable, although it is gross. Instead let’s use what we found above.
We know that the Fourier Transform of the DC pulse of length $\tau$, $f(t)$,
defined above is

$\displaystyle
F(\omega)=2A\tau\,\left(\frac{\sin\omega\tau}{\omega\tau}\right)=F(\omega\tau).$

Therefore, $g(\omega\tau)$ is

$\displaystyle
g(\omega\tau)=\frac{\sin\omega\tau}{\omega\tau}=\frac{F(\omega\tau)}{2A\tau}.$

So if we make the substitution, $\theta=\omega\tau$, then

$\displaystyle
g(\theta)=\frac{\sin\theta}{\theta}=\frac{F\left(\theta\right)}{2A\tau}.$

Then the expression for $G(\omega)$ becomes

$\displaystyle G(\omega)$
$\displaystyle=\int_{-\infty}^{\infty}\frac{\sin\theta}{\theta}\textrm{e}^{-i\omega\theta}\,\textrm{d}\theta,\;\;-\infty<\omega<\infty,$
$\displaystyle=\frac{1}{2A\tau}\int_{-\infty}^{\infty}F\left(\theta\right)\textrm{e}^{-i\omega\theta}\,\textrm{d}\theta,\;\;-\infty<\omega<\infty.$

But this integral (not the fraction in front) has the same form of the duality
property when we now switch the label $\theta\rightarrow t$. Hence,

$\displaystyle G(\omega)=\frac{1}{2A\tau}\cdot 2\pi f(-\omega)=\frac{\pi
f(-\omega)}{A\tau},\;\;-\infty<\omega<\infty.$

Now we substitute in the definition of $f(t)$, but change every
$t\rightarrow-\omega$.

$\displaystyle G(\omega)$
$\displaystyle=\frac{\pi}{A\tau}\begin{cases}0,\;\;&\textrm{if
}-\infty<-\omega<-\tau\\\ A,\;\;&\textrm{if }-\tau\leq-\omega\leq\tau\\\
0,\;\;&\textrm{if }\tau<-\omega<\infty\end{cases}$
$\displaystyle=\frac{\pi}{\tau}\begin{cases}0,\;\;&\textrm{if
}\infty>\omega>\tau\\\ 1,\;\;&\textrm{if }\tau\geq\omega\geq-\tau\\\
0,\;\;&\textrm{if }\tau>\omega>-\infty\end{cases}$
$\displaystyle=\begin{cases}0,\;\;&\textrm{if }\infty>\omega>1\\\
\pi,\;\;&\textrm{if }1\geq\omega\geq-1\\\ 0,\;\;&\textrm{if
}1>\omega>-\infty\end{cases}$

We note that this could have also been evaluated using Dirac-$\delta$
functions by evaluating the integral directly, but the Duality Property
converts an otherwise advanced calculus problem into a relatively simple
algebra problem, where we really just needed to be careful with the labels we
chose as our variable names.

In this example, we will check out how wiggly functions behave under Fourier
Transformations. Remember, a Fourier Transform is a linear combination of
wiggly bits, so the Transform should tell us that the sinusoid
$f(t)=A\cos(\omega_{0}t+\phi)$ is wiggly too. Let us consider a real signal
though, perhaps from an LC-circuit, given by $\displaystyle
f(t)=\begin{cases}0,\;\;&\textrm{if }-\infty<t<-\tau\\\
A\cos(\omega_{0}t+\phi),\;\;&\textrm{if }-\tau\leq t\leq\tau\\\
0,\;\;&\textrm{if }\tau<t<\infty\end{cases},$ where $A$ and $\phi$ are just
numbers, and $\tau>0$ as in the last example. We seek the Fourier Transform of
$f(t)$ from Eq. 4.40, where $x\rightarrow t$ and $\kappa\rightarrow\omega$.
$\displaystyle F(\omega)$
$\displaystyle=\int_{-\infty}^{\infty}f(t)\textrm{e}^{-i\omega
t}\,\textrm{d}t,\;\;-\infty<\omega<\infty$
$\displaystyle=\int_{-\infty}^{-\tau}0\textrm{e}^{-i\omega
t}\,\textrm{d}t+\int_{-\tau}^{\tau}A\cos(\omega_{0}t+\phi)\textrm{e}^{-i\omega
t}\,\textrm{d}t+\int_{\tau}^{\infty}0\textrm{e}^{-i\omega t}\,\textrm{d}t$
$\displaystyle=\int_{-\tau}^{\tau}A\left[\frac{\textrm{e}^{i(\omega_{0}t+\phi)}+\textrm{e}^{-i(\omega_{0}t+\phi)}}{2}\right]\textrm{e}^{-i\omega
t}\,\textrm{d}t$
$\displaystyle=\frac{A}{2}\left[\int_{-\tau}^{\tau}\textrm{e}^{i\phi}\textrm{e}^{i(\omega_{0}-\omega)t}\,\textrm{d}t+\int_{-\tau}^{\tau}\textrm{e}^{-i\phi}\textrm{e}^{-i(\omega_{0}+\omega)t}\,\textrm{d}t\right]$
$\displaystyle=\frac{A\textrm{e}^{i\phi}}{2}\left[\frac{\textrm{e}^{i(\omega_{0}-\omega)t}}{i(\omega_{0}-\omega)}+\textrm{e}^{-2i\phi}\,\frac{\textrm{e}^{-i(\omega_{0}+\omega)t}}{-i(\omega_{0}+\omega)}\right|_{-\tau}^{\tau}$
$\displaystyle=A\tau\textrm{e}^{i\phi}\left[\frac{\sin(\omega_{0}-\omega)\tau}{(\omega_{0}-\omega)\tau}+\textrm{e}^{-2i\phi}\,\frac{\sin(\omega_{0}+\omega)\tau}{(\omega_{0}+\omega)\tau}\right].$
Again, since the sine cardinal approaches unity as its argument goes to zero,
then this Transform is maximized at when $\omega=\omega_{0}$ or
$\omega=-\omega_{0}$, for arbitrary $\tau$. This means that there are two
frequencies that dominate the linear combination: $+\omega_{0}$ and
$-\omega_{0}$. This is exactly what we expect for an even signal wiggling
around at frequency $\omega_{0}$.

It is important to note that as $\tau\rightarrow\infty$, both sine cardinals
approach Dirac-$\delta$ function like behavior, since the zeros of the sine
part get closer together, with the exception of when the argument is zero (see
the example above for elaboration). Furthermore, as $\tau$ gets larger, the
whole Transform increases in magnitude at $\omega=\pm\omega_{0}$. Thus, as
$\tau$ gets larger, we need fewer and fewer frequency components to describe
$f(t)$. And that is exactly what we expect for a sinusoidal function known for
an infinite domain! What we can interpret from this is that all of the
frequencies other than $\pm\omega_{0}$ work to completely cancel our signal to
exactly $f(t)=0$ along $-\infty<t<-\tau$ and $\tau<t<\infty$!

##### Derivatives in Different Representations

There are many other properties of Fourier Transforms that are important and
insightful. However, as I’m sure you can tell by now, it is pretty easy to
keep writing and writing about Fourier Analysis. If you would like to know
more about some of the properties, please see [wang_moore_2009,
thefouriertransform]. The last property I want to talk about are the
derivative properties of Fourier Transforms.

Let’s start with derivatives. Suppose I gave you $f(x)$ and its Fourier
Transform $F(\kappa)$ given by Eq. 4.40. What is the Fourier Transform of
$\textrm{d}f/\textrm{d}x$, namely $G(\kappa)$? We calculate it directly, but
using the Inverse Transform instead (Eq. 4.32).
$\displaystyle\frac{\textrm{d}}{\textrm{d}x}f(x)$
$\displaystyle=\frac{\textrm{d}}{\textrm{d}x}\left[\frac{1}{2\pi}\int_{-\infty}^{\infty}F(\kappa)\textrm{e}^{i\kappa
x}\,\textrm{d}\kappa,\right]\;\;-\infty<x<\infty,$
$\displaystyle=\frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{\partial}{\partial
x}\left[F(\kappa)\textrm{e}^{i\kappa
x}\right]\,\textrm{d}\kappa,\;\;-\infty<x<\infty,$
$\displaystyle=\frac{1}{2\pi}\int_{-\infty}^{\infty}i\kappa
F(\kappa)\textrm{e}^{i\kappa x}\,\textrm{d}\kappa,\;\;-\infty<x<\infty,$
$\displaystyle=\frac{1}{2\pi}\int_{-\infty}^{\infty}G(\kappa)\textrm{e}^{i\kappa
x}\,\textrm{d}\kappa,\;\;-\infty<x<\infty.$ Thus $G(\kappa)=i\kappa
F(\kappa)$. The reason why this works is because all of the $x$-dependence is
in the $\textrm{e}^{i\kappa x}$ part since $F(\kappa)$ only depends on
$\kappa$. Furthermore, if we subract the middle two lines from each other, we
have $\displaystyle 0$
$\displaystyle=\frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{\partial}{\partial
x}\left[F(\kappa)\textrm{e}^{i\kappa
x}\right]\,\textrm{d}\kappa-\frac{1}{2\pi}\int_{-\infty}^{\infty}i\kappa
F(\kappa)\textrm{e}^{i\kappa x}\,\textrm{d}\kappa,\;\;-\infty<x<\infty,$
$\displaystyle=\frac{1}{2\pi}\int_{-\infty}^{\infty}\left[\frac{\partial}{\partial
x}-i\kappa\right]F(\kappa)\textrm{e}^{i\kappa
x}\,\textrm{d}\kappa,\;\;-\infty<x<\infty.$ Since this integral must always be
zero as long as $f(x)$ is differentiable, then we conclude that the difference
in the integral must vanish since nothing about $F(\kappa)$ has been specified
(see the first example on uniqueness for details). In other words, with
respect to Fourier Transforms, $\partial/\partial x$ is interchangeable with
$i\kappa$. I leave it to you to show that $\partial^{n}/\partial x^{n}$ is
interchangeable with $i^{n}\kappa^{n}$, where $n$ is a nonnegative integer,
under Fourier Transformations. Furthermore, using the same argument, show that
the real-space representation of $\textrm{d}^{n}F/\textrm{d}\kappa^{n}$ is
$(-ix)^{n}$ which implies that $\partial/\partial\kappa$ is interchangeable
with $-ix$ under Fourier Transformations.

###  4.4 Concluding Remarks

As I have said in this document a few times, this chapter by no means can
possibly cover all there is to Fourier Analysis. For example, we have only
talked about Fourier Analysis of one independent variable (to see Fourier
Analysis in multiple dimensions check out [MultidimensionalTransform]). It is
my hope, however, that the material we have covered will serve as a sufficient
introductory resource for you in Fourier Analysis that you may have not
otherwise covered in an undergraduate career in physics. I have included Table
4.1 with a bunch of useful formulas that you can refer to later on, as well as
when I talk about them in the text. Remember, in physics our job is to
describe the laws of nature as we observe them. Although Fourier Analysis may
look a little mathematically messy, it is ultimately a tool to help us look at
different physical phenomena from a new angle, and we can do so just by adding
up contributions from distinct wiggling components.

TABLE 4.1: A bunch of useful and general formulas for Fourier analysis and
their text references.

Equation Description | Equation Formula | Text Reference
---|---|---
Linear Combination of Frequencies | $f(x)=\sum_{\kappa}a_{\kappa}\cos(\kappa x)+b_{\kappa}\sin(\kappa x)$ | Eq. 4.6
Cosine Orthogonality | $\dfrac{2}{L}\int_{0}^{L}\cos\left(\dfrac{j\pi x}{L}\right)\cos\left(\dfrac{k\pi x}{L}\right)\,\textrm{d}x=\begin{cases}1,&\textrm{if }j=k\neq 0\\\ 2,&\textrm{if }j=k=0\\\ 0,&\textrm{if }j\neq k\end{cases}$ | Eq. 4.8
Sine Orthogonality | $\dfrac{2}{L}\int_{0}^{L}\sin\left(\dfrac{j\pi x}{L}\right)\sin\left(\dfrac{k\pi x}{L}\right)\,\textrm{d}x=\begin{cases}1,&\textrm{if }j=k\neq 0\\\ 0,&\textrm{if }j\neq k\end{cases}$ | Eq. 4.9
Sine-Cosine Orthogonality | $\dfrac{2}{L}\int_{0}^{L}\sin\left(\dfrac{j\pi x}{L}\right)\cos\left(\dfrac{k\pi x}{L}\right)\,\textrm{d}x=0$ | Eq. 4.10
Fourier Constant Coefficient | $a_{0}=\dfrac{1}{L}\int_{0}^{L}f(x)\,\textrm{d}x,\;\;0\leq x\leq L$ | Eq. 4.13
Fourier Cosine Coefficients | $a_{j\geq 1}=\dfrac{2}{L}\int_{0}^{L}f(x)\cos\left(\dfrac{j\pi x}{L}\right)\,\textrm{d}x,\;\;0\leq x\leq L$ | Eq. 4.14
Fourier Sine Coefficients | $b_{j}=\dfrac{2}{L}\int_{0}^{L}f(x)\sin\left(\dfrac{j\pi x}{L}\right)\,\textrm{d}x,\;\;0\leq x\leq L$ | Eq. 4.12
Discrete Complex Orthogonality | $\int_{-\lambda/2}^{\lambda/2}\exp\left(\frac{2\pi ijx}{\lambda}\right)\exp\left(-\frac{2\pi ikx}{\lambda}\right)\,\textrm{d}x=\begin{cases}0,\;\;\textrm{if }j\neq k\\\ \lambda,\;\;\textrm{if }j=k\end{cases}$ | Eq. 4.26
Complex Fourier Coefficients | $c_{j}=\dfrac{1}{\lambda}\int_{-\lambda/2}^{\lambda/2}f(x)\exp\left(-\dfrac{2\pi ijx}{\lambda}\right)\,\textrm{d}x$ | Eq. 4.28
The Fourier Transform | $F(\kappa)=\int_{-\infty}^{\infty}f(x)\textrm{e}^{-i\kappa x}\,\textrm{d}x,\;\;-\infty<\kappa<\infty$ | Eq. 4.40
The Inverse Fourier Transform | $f(x)=\dfrac{1}{2\pi}\int_{-\infty}^{\infty}F(\kappa)\textrm{e}^{i\kappa x}\,\textrm{d}\kappa,\;\;-\infty<x<\infty$ | Eq. 4.32
Normalized Fourier Transform | $F(\kappa)=\dfrac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(x)\textrm{e}^{-i\kappa x}\,\textrm{d}x,\;\;-\infty<\kappa<\infty$ | Eq. 4.41
Normalized Inverse Fourier Transform | $f(x)=\dfrac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}F(\kappa)\textrm{e}^{i\kappa x}\,\textrm{d}\kappa,\;\;-\infty<x<\infty$ | Eq. 4.42

## References

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11institutetext: Department of Computer Science, Hong Kong Baptist University
22institutetext: NVIDIA AI Technology Center, NVIDIA
22email<EMAIL_ADDRESS>{chcheung<EMAIL_ADDRESS><EMAIL_ADDRESS>
# Protecting NeRFs’ Copyright via Plug-And-Play Watermarking Base Model

Qi Song 1122 Ziyuan Luo 1122 Ka Chun Cheung 22
Simon See 22 Renjie Wan Corresponding author. This work was done at Renjie’s
Research Group at the Department of Computer Science of Hong Kong Baptist
University.11

###### Abstract

Neural Radiance Fields (NeRFs) have become a key method for 3D scene
representation. With the rising prominence and influence of NeRF, safeguarding
its intellectual property has become increasingly important. In this paper, we
propose NeRFProtector, which adopts a plug-and-play strategy to protect NeRF’s
copyright during its creation. NeRFProtector utilizes a pre-trained
watermarking base model, enabling NeRF creators to embed binary messages
directly while creating their NeRF. Our plug-and-play property ensures NeRF
creators can flexibly choose NeRF variants without excessive modifications.
Leveraging our newly designed progressive distillation, we demonstrate
performance on par with several leading-edge neural rendering methods. Our
project is available at: https://qsong2001.github.io/NeRFProtector.

###### Keywords:

NeRF Copyright protection Plug-and-play

##  1 Introduction

Neural Radiance Fields (NeRFs) [29, 4, 31, 12] mark a new way for 3D scene
representation. As individuals might use radiance fields to represent their
chosen 3D scenes and share these representations with the public, it becomes
increasingly critical to establish a convenient copyright framework tailored
to these novel representations during their creations. Then, once created, the
copyright of these scene representations can be easily claimed in the event of
an ownership breach.

While CopyRNeRF [25] is developed to uphold copyright protection by embedding
binary watermarks into NeRF models and then extracting watermarks from
rendered images, its effectiveness is lessened by its weaknesses. First, a
delay exists between creating the NeRF model and inserting ownership messages
because binary messages are embedded into the NeRF model post-creation through
model fine-tuning. Should malicious users acquire NeRF models in this delay,
they could exploit them for nefarious intentions. Second, NeRF creators are
required to jointly train a message extractor during message embedding, which
makes the entire watermarking process exceedingly time-intensive and complex.
NeRF creators or owners may abandon the watermarking due to its excessive
complexity and difficulty.

If NeRF creators are reluctant to use it, any fabulous designs for
watermarking become meaningless. Therefore, in addition to the primary
evaluation criteria for watermarking systems - robustness and invisibility of
embedded binary messages [7] \- we introduce NeRFProtector, a NeRF
watermarking framework with a plug-and-play property. Such a plug-and-play
property ensures that binary messages can be conveniently incorporated into
existing NeRF variants with minimal modifications.

Figure 1: Proposed scenario of our method. NeRF creators can obtain a pre-
trained watermarking extractor from a third party (_e.g_., open-source
library) or train a message extractor separately via standard pipelines, as
the watermarking base model $\mathcal{F}$. Once obtained, this base model is
considered “plug-and-play” in our scenario. Throughout the NeRF creation
process, creators can readily use this base model to embed watermarks in their
NeRF. After the optimization of NeRF is complete, they obtain a watermarked
NeRF. When this watermarked NeRF is distributed publicly, the creators can
then use the same base model to retrieve binary messages from newly rendered
views, thereby asserting their ownership.

As shown in Fig. 1, we use a watermarking base model to achieve the above
goals. The watermarking base model can be obtained from a third party (_e.g_.,
some open-resource message extractors111https://github.com/ando-
khachatryan/HiDDeN) or trained separately via standard pipelines. NeRF
creators simply need to select one watermarking base model and integrate it
with their chosen NeRF variants (_e.g_., Instant-NGP [31], TensorRF [4], and
so on), and then message embedding and NeRF optimization can proceed
concurrently. Once optimization is complete, NeRF owners can use the base
model to retrieve binary messages from the rendered contents of NeRF. This
approach spares NeRF creators from extra efforts for modifying NeRF
structures. Crucially, message embedding occurs during the creation of NeRF,
leaving no window of opportunity for malicious attackers. Based on our
investigation, the message extractor from conventional image watermarking
frameworks [54, 14] can directly be our watermarking base model. These message
extractors, through their training, have acquired knowledge of copyright
messages and inherently possess the capability to retrieve such messages. Once
NeRF creators acquire this watermarking base model, they can watermark their
NeRF by distilling message knowledge from the extractor into their NeRF. Then,
we only need to consider how to distill such established knowledge of binary
messages into NeRF during creation.

Some methods like CLIP-NeRF [43] leverage knowledge from external CLIP model
[37] to execute shape and color manipulation [5]. However, these methods
necessitate modifications to NeRF’s core structure for representation, which
is inconsistent with our aim of ensuring watermarking compatibility with
existing NeRF frameworks. To maintain simplicity, our proposed scenario opts
for a simple Progressive Global Rendering (PGR) in place of NeRF’s typical
volume rendering. NeRF’s volume rendering typically involves rendering a
random subset of pixels, referred to as local rendering here, during
optimization to manage computational demands. This limited pixel rendering is
insufficient for embedding messages effectively within NeRFs. Rather than only
rendering a small subset of pixels each time, our progressive global rendering
renders all pixels across various image resolutions. This approach allows for
more effective message embedding within NeRF’s representation.

The framework of NeRFProtector is shown in Fig. 2. In our designs, we do not
change the fundamental representation of NeRF. NeRF creators only need to use
a pre-trained message extractor as the base model to finish both message
embedding and extraction. Despite the plug-and-play property of this base
model, our framework can also satisfy the key criteria for robustness and
invisibility. Our key contribution can be concluded as follows:

  * •

A plug-and-play mode that NeRF creators easily watermark their NeRFs during
the creation process, leaving no delays between NeRF creation and ownership
message embedding.

  * •

The utilization of a watermarking base model for efficient message embedding
and extraction.

  * •

Progressive global rendering is proposed to effectively distill message
knowledge from the watermarking base model into NeRFs by exploring relations
between rendering strategies and watermark embedding.

The plug-and-play property of NeRFProtector ensures that minimal modifications
are required to the architecture of NeRF representation, and the whole
watermarking process can be conducted conveniently. Extensive experiments are
conducted to verify the performance of our method, and the potential threats
are also analyzed.

##  2 Related work

Digital watermarking for 2D. Digital watermarking serves as a method for
embedding copyright messages, referred to as watermarks, into various digital
assets. These assets include images [54, 40, 18, 45], videos [24, 8, 2], and
certain generative models [50, 11]. Researchers also investigate techniques
that aim to ensure the watermark’s robustness against various attacks [39, 1,
9, 34] and common image processing operations such as compression, cropping,
and resizing [54, 14]. The embedded copyright can be used for various
purposes, including copyright protection and intellectual property protection.
Our work leverages the prior knowledge of 2D watermarking and achieves
copyright information protection for NeRF.

Neural Radiance Fields. Neural Radiance Fields (NeRF) [29, 31, 52, 42, 53] is
a method for 3D scene representation. It can represent a scene as a continuous
3D function that models the radiance at every point in space. In recent years,
NeRF has sparked a lot of research interest and has been widely adopted in the
computer vision and graphics communities. NeRF has been extended to
incorporate additional functionalities, such as text-to-3D [33, 19, 27],
image-to-3D [22, 41, 48], scene editing [23, 49], and 3D model copyright
protection [25, 16] tasks. These extensions further showcase the versatility
and potential applications of NeRF in various domains. As the popularity and
impact of NeRF continue to grow, the need for protecting its intellectual
property becomes crucial.

Digital watermarking for 3D. Traditional 3D watermarking approaches focus on
watermarking on polygonal meshes [32, 34, 38] and point clouds [13, 21] with
explicit structures. A deep-learning-based approach [47] embeds binary
messages in 3D meshes and extracts them from 2D rendered images. For implicit
3D models such as NeRF, SteagaNeRF [16] introduces a technique for embedding
data within NeRF. Specifically, CopyRNeRF [25] suggests safeguarding NeRF’s
copyright by embedding binary messages into NeRF and subsequently extracting
watermarks from the images rendered. CopyRNeRF [25] also investigates several
settings in the ownership claim of NeRF representation. However, CopyRNeRF
[25] is limited to embedding binary ownership messages through model fine-
tuning post-creation of NeRF models, a process that potentially leaves room
for exploitation by malicious individuals. Furthermore, its complex usage
might discourage NeRF creators from adopting it. In contrast, our
NeRFProtector enables the watermarking of NeRF models right at the point of
creation, utilizing an easy-to-use plug-and-play watermarking base model.

##  3 Problem statement

NeRF. NeRF [29] takes in a 3D location $\mathbf{x}\in\mathbb{R}^{3}$ and
viewing direction $\mathbf{d}\in\mathbb{R}^{2}$, then outputs a color value
$\mathbf{c}\in\mathbb{R}^{3}$ and a density value $\sigma\in\mathbb{R}^{+}$.
To produce a pixel’s color in a 2D image, samples of these values can be
composited along a ray according to volume rendering. Formally, $N_{p}$ points
are sampled along a camera ray $r$ with color and density values
$\\{(\mathbf{c}_{r}^{i},\sigma_{r}^{i})\\}_{i=1}^{N}$. The corresponding RGB
color value $\hat{\mathbf{C}}(r)$ is obtained using alpha composition as

$\hat{\mathbf{C}}(r)=\sum_{i=1}^{N_{p}}T_{r}^{i}(1-\exp\left(-\sigma_{r}^{i}\delta_{r}^{i}\right))\mathbf{c}_{r}^{i},$
(1)

where
$T_{r}^{i}=\prod_{j=1}^{i-1}\left(\exp\left(-\sigma_{r}^{i}\delta_{r}^{i}\right)\right)$,
and $\delta_{r}^{i}$ is the distance between adjacent sample points. The
weights of NeRF can be optimized by minimizing a reconstruction loss between
observations $\mathbf{C}$ and predictions $\hat{\mathbf{C}}$ as

$\mathcal{L}_{\text{content}}=\frac{1}{N_{r}}\sum_{m=1}^{N_{r}}\|\hat{\mathbf{C}}(r_{m})-\mathbf{C}(r_{m})\|_{2}^{2},$
(2)

where $N_{r}$ is the number of sampling pixels.

Our scenario. From the scenario shown in Fig. 1, NeRF creators can acquire the
off-the-shelf watermarking base models from a third party or train a base
model separately via standard pipelines. Then, they can directly combine the
base model with their NeRF optimization. Once the optimization has concluded,
the message embedding also concludes. Then, the base model can be further used
to extract binary messages for ownership claims.

Our approach diverges from conventional watermarking techniques [25] for NeRF
models. Rather than embedding the ownership messages after the creation of
NeRF, we ensure NeRF models are protected right from their outset,
substantially narrowing the window of opportunity for malicious actors.
Additionally, this method permits NeRF creators to train their models with
minimal deviation from standard procedures, avoiding the extra workload of
training supplementary modules.

##  4 Proposed NeRFProtector

Figure 2: Our plug-and-play method to watermark NeRF during its creation. (1)
Building watermarking base model: The watermarking base model $\mathcal{F}$
can be sourced from a third party (Sec. 4.1). We implement a HiDDeN [54]
framework to get the pre-trained message extractor as our watermarking base
model $\mathcal{F}$. During the training, the encoder $\mathcal{E}$ encodes a
randomly selected 48-kit message $\mathbf{m}$ and cover image $\mathbf{I}_{o}$
and outputs a watermarked image $\mathbf{I}_{en}$. Then the message extractor
$\mathcal{F}$ extracts embedded message $\hat{\mathbf{m}}$ from the
watermarked image. Message encoder $\mathcal{E}$ is discarded after building
watermarking base model $\mathcal{F}$. (2) NeRF creation with message
distillation: NeRF creators first fix a copyright message $\mathbf{m}$, then
employ this base model $\mathcal{F}$ to embed selected watermarks to NeRFs
during the creation process via Progressive Global Rendering (PGR) and message
distillation (Sec. 4.2). When the optimization of NeRF is finalized, creators
immediately obtain a watermarked NeRF. (3) Extracting watermark: Subsequently,
they can utilize the base model $\mathcal{F}$ to retrieve binary watermarks
from the rendered images, asserting their ownership.

Our NeRFProtector aims at embedding k-bit binary messages
$\mathbf{m}\in\\{0,1\\}^{k}$ during the creation of NeRF. As our scenario
relies on an important base model for watermarking, we first introduce it in
Sec. 4.1. Then, we introduce how to distill knowledge from this base model
into NeRF via a newly designed progressive distillation in Sec. 4.2.

###  4.1 Building watermarking base model

The essence of our approach is to switch out the current processes of message
embedding and extraction with a watermarking base model. We envision this base
model to possess a plug-and-play property, enabling NeRF creators to
seamlessly incorporate it into their NeRFs with only minor modifications to
established architecture. Additionally, this base model is designed to
withstand common image-level distortions, ensuring robust message extraction
even from distorted rendered images.

For simplicity, we choose the extractor from HiDDeN [54], a well-established
deep-learning-based watermarking framework, to serve as our base model. As a
message extractor for image watermarking, the base model naturally owns the
capability to extract binary messages from watermarked images. Besides, we can
easily make the extraction more robust by employing established studies in
image watermarking [54, 14]. At last, the base model has learned the message
pattern during its training, and we can then distill such learned knowledge
into the representation of NeRF during its optimization.

Due to the requirement for plug-and-play, we refrain from making additional
modifications to HiDDeN [54]. In general, HiDDeN [54] jointly optimizes the
watermark encoder $\mathcal{E}$ and the watermarking base $\mathcal{F}$ to
robustly embed k-bit binary messages into images. Specifically, during
optimization, the encoder $\mathcal{E}$ takes a cover image $\mathbf{I}_{o}$
and a message $\mathbf{m}\in\\{0,1\\}^{k}$ as inputs and output a watermarked
image $\mathbf{I}_{en}$ as follows,

$\mathbf{I}_{en}=\mathcal{E}(\mathbf{I}_{o},\mathbf{m}).$ (3)

Binary message $\hat{\mathbf{m}}$ is then extracted from distorted images
using a watermark extraction network $\mathcal{F}$ as
$\hat{\mathbf{m}}=\mathcal{F}(T(\mathbf{I}_{en}))$. Then, the whole network is
optimized by minimizing the Binary Cross Entropy (BCE) loss between the
original binary message $\mathbf{m}$ and extracted binary message
$\hat{\mathbf{m}}$ as:

$\mathcal{L}_{\text{m}}=\text{BCE}(\mathbf{m},\hat{\mathbf{m}}),$ (4)

where
$\text{BCE}(\mathbf{m},\hat{\mathbf{m}})=-\left(\mathbf{m}\cdot\log(\tau(\hat{\mathbf{m}}))+(1-\mathbf{m})\cdot\log(1-\tau(\hat{\mathbf{m}})\right)$
and $\tau$ is a sigmoid operation. After the training, we discard the encoder
$\mathcal{E}$ in Eq. 3 and only use the extractor $\mathcal{F}$ as the
watermarking base model. To improve the robustness, we follow HiDDeN [54] and
employ a randomly selected transformation layer $T$ during training to ensure
that the hidden messages are robust to common image distortions.

###  4.2 Distilling watermarks into NeRF progressively

The base model has encapsulated all necessary knowledge about watermarking. We
further distill the knowledge patterns into NeRF as they are being created.
Then, the binary messages can be robustly extracted from rendered images
regardless of the viewpoints. Although various approaches exist for distilling
external knowledge into the representations of NeRF [10, 44], they necessitate
changes to the essential structures of NeRF, conflicting with our need for a
plug-and-play property. We achieve the watermark distillation by only making
alternations to the rendering scheme of NeRF, the key obstacles for our
distillation.

As displayed in Eq. 2, NeRF hinges on a random and small subset of pixels
($N_{r}$ pixels in Eq. 2) for rendering. Despite its efficiency in preventing
the computational demand from being too high, the message patterns can only be
embedded into random positions during distillation if only such local
rendering is used. Then, as shown in Fig. 5, when the optimization settles
down, the message pattern cannot effectively form meaningful patterns that can
be identified and extracted by the base model. As shown in Fig. 2, we
substitute such conventional local rendering in established NeRF with a
progressive global rendering that is capable of rendering all pixels across
various image resolutions. This modification does not alter the representation
of NeRF, ensuring that our mechanism remains adaptable and can be easily
applied to a range of NeRF variants [12, 31, 4].

Progressive Global Rendering (PGR). For our PGR, as displayed in Fig. 2,
rather than only sampling a random subset of pixels, during rendering, we
sample rays corresponding to all pixels across various scales to gain a
comprehensive understanding of global information during optimization. By
embedding messages on a global scale, they become deeply integrated into the
scene’s representation. This ensures the consistent properties of binary
watermarks from various viewpoints, allowing message extraction irrespective
of rendering viewpoints. Additionally, since 3D information often exhibits
distinct characteristics in 2D projections at varying scales [35], progressive
rendering accentuates these resolution-dependent properties, aiding in message
distillation. Furthermore, the globally rendered images are all with reduced
resolution. Then, our PGR can achieve comparable results without necessitating
full-resolution images for optimization and make the computational cost
acceptable.

As shown in Fig. 2, we consider rendering $N_{k}$ progressive views
$\hat{\mathbf{I}}_{set}$ at different scales denoted as follows:

$\hat{\mathbf{I}}_{set}=\\{\hat{\mathbf{I}}_{n}\\}_{n=1}^{N_{k}}=\\{\hat{\mathbf{I}}_{1},\hat{\mathbf{I}}_{2},...,\hat{\mathbf{I}}_{N_{k}}\\},$
(5)

where $N_{k}$ and $\hat{\mathbf{I}}_{n}$ denote the number of layers and the
cascade rendered view at different scales. Each
$\hat{\mathbf{I}}_{n}\in\mathbb{R}^{\frac{W}{2^{n}}\times\frac{H}{2^{n}}\times
3}$ from progressive views set $\hat{\mathbf{I}}_{set}$ is obtained by
rendering global rays of at corresponding resolution. In our setting, we
empirically set $N_{k}=3$, a value that can well balance efficiency and
performance.

Distilling messages. We then utilize the base model $\mathcal{F}$ and
progressive global rendering to distill message patterns into NeRF. For
$N_{k}$ rendered cascade views $\hat{\mathbf{I}}_{set}$ at different scales,
we use the watermarking base model $\mathcal{F}$ to extract the embedded
message $\hat{\mathbf{m}}_{set}$ as follows:

$\hat{\mathbf{m}}_{set}=\mathcal{F}(\hat{\mathbf{I}}_{set})\\\
=\\{\hat{\mathbf{m}}_{1},\hat{\mathbf{m}}_{2},...,\hat{\mathbf{m}}_{N_{k}}\\}.$
(6)

Then a distillation loss $\mathcal{L}_{\text{dis}}$ is computed by calculating
the BCE loss between the predicted message set $\hat{\mathbf{m}}_{set}$ and
the ground truth message $\mathbf{m}$:

$\mathcal{L}_{\text{dis}}=\sum_{i=1}^{N_{k}}\alpha_{i}\cdot\text{BCE}(\mathbf{m},\hat{\mathbf{m}}_{i}),$
(7)

where $N_{k}$ is the number of cascade layers, and $\alpha_{i}$ represents the
weight of $i$ layer’s loss function. During the creation, minimizing this
equation can distill messages into NeRF.

To make the distilled message pattern invisible, we enforce the high
similarity between the rendering image and its corresponding ground truth as
follows:

$\mathcal{L}_{\text{inv}}=\|\hat{\mathbf{I}}_{1}-\mathbf{I}_{1}\|_{2}^{2},$
(8)

where $\hat{\mathbf{I}}_{1}$ denotes the cascade rendered images with the
highest resolution, and $\mathbf{I}_{1}$ is its corresponding ground truth.
The ground truth is with a reduced resolution to be compatible with
$\hat{\mathbf{I}}_{1}$.

Besides progressive global rendering, we still use local rendering for
building radiance fields. During optimization, the local rendering randomly
selects $N_{r}$ pixels for rendering, which ensures the local pattern can be
effectively considered during the message distillation. Thus, our whole loss
functions for creating NeRF with a plug-and-play watermarking base model can
be concluded as follows:

$\mathcal{L}=\lambda_{1}\mathcal{L}_{\text{local}}+\lambda_{2}\mathcal{L}_{\text{inv}}+\lambda_{3}\mathcal{L}_{\text{dis}},$
(9)

where $\lambda_{1}$, $\lambda_{2}$, and $\lambda_{3}$ are hyperparameters that
balance the performance between embedding watermarking and reconstruction, and
the $\mathcal{L}_{\text{local}}$ is same to Eq. 2. The network setup described
above can be effectively adapted for embedding copyright messages into NeRF.
PGR is only used during the optimization. Once NeRF is created, users can
still render images via established ways [29].

###  4.3 Implementation details

For NeRF, we implement our method with Instant-NGP [31]. Instant-NGP takes a
3D grid as input and encodes it using a grid encoder. It then predicts density
values $\sigma$ using two-layer MLP with 64 channels. For the color branch, a
three-layer MLP with 64 channels is used to predict color $c$ and $\sigma$
from view direction $(\theta,\phi)$. As for the rendering, the number of
cascade layers $N_{k}$ is set to 3, and the rendered pixel count $N_{r}$ for
the local rendering is set to $4096$, respectively. For base model, we
implement our method with HiDDeN [54]. The watermark encoder $\mathcal{E}$ and
extractor $\mathcal{F}$ in HiDDeN [54] are jointly trained on COCO [20]
dataset. During its training process, encoder $\mathcal{E}$ encodes a randomly
selected 48-kit message $\mathbf{m}$ and cover image $\mathbf{I}_{o}$ and
predicts a watermarked image $\mathbf{I}_{en}$. Then extractor $\mathcal{F}$
extracts embedded messages from watermarked image $\mathbf{I}_{en}$. The whole
network is optimized with Adam [15] optimizer in $100K$ iterations.

During NeRF creation, the NeRF creator chooses one preferable message
$\mathbf{m}$ as copyright messages. We fix the weight of watermarking base
model $\mathcal{F}$ and minimize the distance between the chosen message
$\mathbf{m}$ and extracted message $\hat{\mathbf{m}}_{set}$. Hyperparameters
in Eq. 9 are set as $\lambda_{1}=0.01$ and $\lambda_{3}=0.001$. We use the
Adam [15] optimizer with setting the values of $\beta_{1}=0.9$,
$\beta_{2}=0.999$, and a learning rate of 0.01. The learning rate decays
following the exponential scheduler during optimization. We train our model
for each scene on a single NVIDIA A100 GPU.

##  5 Experiments

###  5.1 Experiment settings

Dataset. We conduct experiments on the Blender [29] and LLFF [28] dataset. The
Blender dataset comprises 8 synthetic object images captured from different
viewpoints, while the LLFF dataset consists of 8 real-world scenes captured in
a forward-facing manner. We choose these datasets based on previous research
[29, 31] and use them to assess the bit accuracy and reconstruction
performance of NeRF. For each scene, we render testing views and calculate
their average values for our analysis.

Baselines. We compare four strategies to guarantee a fair comparison: 1)
HiDDeN [54] \+ NeRF [31]: processing images with classical 2D watermarking
method HiDDeN [54] before training the NeRF model; 2) MBRS [14] \+ NeRF [31]:
processing images with a recently proposed 2D watermarking method MBRS [14]
before training the NeRF model; 3) CopyRNeRF [25]: a state-of-the-art method
for protecting the copyright of NeRF models via digital watermarking. To
evaluate the impact of watermarks on reconstruction, we also compare the
results with the non-watermarked version of NeRF; 4) NeRF [31] w/o watermark:
we also compare with NeRF [31] w/o watermark to evaluate the reconstruction
quality. To further illustrate our plug-and-play properties, we implement our
method with other NeRF variants [12, 4] and base model [9] as demonstrated in
Sec. 5.2.

Figure 3: Visual quality comparisons with CopyRNeRF [25]. The texts below the
images show the results of PSNR and bit accuracy. We show the differences
$\times 10$ between the rendered views and the ground truth for both methods.
Our NeRFProtector exhibits better consistency across multiple viewpoints,
achieving a well-balanced trade-off between reconstruction quality and bit
accuracy.

Evaluation methodology. We assess our method against other baselines using the
established digital watermarking standards of invisibility and robustness [7].
For invisibility, we measure the performance by employing metrics including
PSNR, SSIM [46], and LPIPS [51] to compare the visual fidelity of the output
images after binary message embedding. For robustness, we conduct experiments
to determine if the embedded messages can be accurately extracted by assessing
the bit accuracy under different types of image distortions. We consider
various distortions [11, 36] for message extraction, including JPEG
compression, image crop, image scaling, contrast changes, and text overlay. We
directly set the bit length used in our experiments as $48$ bits, the largest
length used in previous methods for watermarking 3D models [25, 47]. We
further discuss potential intended watermarking removal technologies in Sec.
5.3.

Table 1: Quantitative results of visual representation and watermarking
extraction performance. We report the results of watermarking robustness when
rendered images meet common distortions (_e.g_., Crop, Resize, and JPEG
compression). “None” indicates that no distortion has been applied. “N/A”
denotes Not Applicable as NeRF w/o watermark does not involve message
embedding.

Dataset | Method | PSNR $\uparrow$ | SSIM $\uparrow$ | LPIPS $\downarrow$ | Bit accuracy %
---|---|---|---|---|---
None | Crop | Resize | JPEG
Blender-test | NeRF w/o watermark | 30.62 | 0.9579 | 0.0343 | N/A
HiDDeN [54]+NeRF | 28.30 | 0.9219 | 0.0495 | 50.29 | 50.03 | 52.32 | 50.84
MBRS [14]+NeRF | 24.17 | 0.8461 | 0.2967 | 50.53 | 49.64 | 51.72 | 49.81
CopyRNeRF [25] | 25.50 | 0.9073 | 0.0885 | 62.15 | 56.63 | 57.32 | 58.41
NeRFProtector | 29.26 | 0.9393 | 0.0483 | 92.69 | 92.95 | 91.87 | 78.62
LLFF-test | NeRF w/o watermark | 26.37 | 0.8352 | 0.1013 | N/A
HiDDeN [54]+NeRF | 25.70 | 0.8300 | 0.1096 | 51.69 | 50.48 | 50.53 | 51.11
MBRS [14]+NeRF | 25.44 | 0.8198 | 0.1238 | 50.84 | 51.55 | 50.85 | 49.08
CopyRNeRF [25] | 25.80 | 0.8302 | 0.1035 | 63.72 | 60.45 | 55.34 | 54.11
NeRFProtector | 26.82 | 0.8569 | 0.0834 | 96.99 | 93.57 | 80.53 | 76.26

Invisibility against Bit accuracy. As shown in Tab. 1 and Fig. 3, we first
investigate the invisibility of our watermarks. This is to show whether the
embedded messages undermine the visual quality. Similar to previous studies
[25], for HiDDeN [54]/MBRS [14] \+ NeRF, binary messages cannot be effectively
extracted from rendered images, though they all achieve acceptable visual
quality. CopyRNeRF [25] achieves the second-best results while it is still
largely below our values for bit accuracy. Only our approach can effectively
balance visual quality (PSNR/SSIM/LPIPS) and message embedding (Bit accuracy).
Besides, from Fig. 3, the viewpoints have a higher impact on the bit accuracy
of message extraction, while our method can achieve more consistent
performances across different viewpoints.

Is it robust to common distortion? Our method is robust to common image-level
distortions since we have considered their impact during building watermarking
base model $\mathcal{F}$. As shown in Tab. 1 and 4, when image operations are
applied to the rendered image, our base model can still effectively extract
the embedded binary messages on both evaluation datasets.

Figure 4: Image-level distortion experiment on common image distortions on
rendered views to determine if the extractor can still extract the message.
Operations like sharpness, cropping, contrast, and text overlay have little
impact on message extraction. Even after severe operations (JPEG and Resize),
our method still maintains satisfactory accuracy ($\geq 70\%$).

Effectiveness. We compare the training time of our method and CopyRNeRF [25]
with same message length setting. In contrast to CopyRNeRF, our approach costs
only around $50$m. As CopyRNeRF [25] jointly trains NeRF and the watermarking
extractor at each step with a random message, it significantly extends the
time required to embed the watermark ($30$h).

###  5.2 Ablation study

Is progressive global rendering good? We propose a progressive distillation to
effectively distill message patterns from the base model to NeRF. We compare
three settings in Tab. 2: 1) Local rendering only, the rendering strategy used
in vanilla NeRF; 2) Single-scale global rendering: we replace the progressive
global rendering with a single-scale rendering; 3) Our complete progressive
global rendering.

Table 2: Quantitative results for evaluating progressive global rendering.
Local denotes the rendering strategy used in vanilla NeRF. Single-scale
represents single-scale global rendering from progressive rendering.
Progressive denotes our proposed progressive global rendering.

Settings | PSNR $\uparrow$ | SSIM $\uparrow$ | LPIPS $\downarrow$ | Bit acc. %
---|---|---|---|---
\small1⃝ Local | 30.38 | 0.9521 | 0.0360 | 45.99
\small2⃝ Single-scale | 29.57 | 0.9402 | 0.0449 | 87.27
\small3⃝ Progressive | 29.26 | 0.9394 | 0.0483 | 92.69

As shown in Tab. 2, the local rendering in current mechanisms of NeRF cannot
guarantee successful message embedding, where the bit accuracy is kept at a
very low level. Besides, we also show the heat maps obtained from our
progressive global rendering and classical local rendering in Fig. 5. From the
results for “Single-scale global rendering”, even with a single-scale global
rendering, the accuracy of embedding bits is further enhanced while ensuring
the reconstruction quality. Our experiments have demonstrated that the
progressive rendering approach more effectively incorporates distillation
information into the NeRF model.

Figure 5: Visual heat map comparisons of local rendering and progressive
global rendering. The text below denotes PSNR/Bit accuracy. The colors
represent the area’s impact on message extraction. Our PGR can distill
messages into NeRF with high accuracy.

Can proposed method adapt to other NeRF variants? Our base model for
watermarking can also be easily adapted to other NeRF variants. As displayed
in Tab. 4, besides Instant-NGP [31] used in our experiments, we further
combine the base model with TensorRF [4], and Plenoxels [12]. From the
experimental results, our base model for watermarking can also achieve
comparable performance on different NeRF variants.

Can the base model be other extractors? In our default setting, we use the
extractor of HiDDeN [54] as our base model. However, we can also consider
using other extractors trained in different settings [14] as our base model.
In Tab. 4, we show that message extractor from MBRS [14] can also be used as
the base model, and they also achieve comparable performance.

Table 3: Quantitative results on combining the same watermarking base model
with different NeRF variants. The extractor of HiDDeN [54] is employed as the
watermarking base model $\mathcal{F}_{1}$. We combine $\mathcal{F}_{1}$ with
Instant-NGP [31], TensorRF [4], and Plenoxels [12] to see whether the base
model can be adapted to NeRF variants in a plug-and-play manner. N/A denotes
not applicable as message embedding is not involved.

Method | PSNR $\uparrow$ | SSIM $\uparrow$ | LPIPS $\downarrow$ | Bit acc.%
---|---|---|---|---
Instant NGP [31] | 33.42 | 0.9709 | 0.0157 | N/A
Instant NGP + $\mathcal{F}_{1}$ | 32.92 | 0.9697 | 0.0161 | 91.96
TensorRF [4] | 34.96 | 0.9786 | 0.0105 | N/A
TensorRF + $\mathcal{F}_{1}$ | 32.73 | 0.9743 | 0.0125 | 89.35
Plenoxels [12] | 35.80 | 0.9872 | 0.0102 | N/A
Plenoxels + $\mathcal{F}_{1}$ | 34.19 | 0.9730 | 0.0127 | 97.92

Table 4: Quantitative results on combining different watermarking base model.
$\mathcal{F}_{1}$ and $\mathcal{F}_{2}$ denote HiDDeN [54] and MBRS [14]
extractors. We combine a different base model from MBRS [14] with Instant-NGP
[31] to see whether the backbone of NeRF can be adapted to other base models.
N/A denotes not applicable as message embedding is not involved.

Method | PSNR $\uparrow$ | SSIM $\uparrow$ | LPIPS $\downarrow$ | Bit acc.%
---|---|---|---|---
Instant-NGP [31] | 33.42 | 0.9709 | 0.0157 | N/A
Instant-NGP + $\mathcal{F}_{1}$ | 32.92 | 0.9697 | 0.0161 | 91.96
Instant-NGP + $\mathcal{F}_{2}$ | 31.71 | 0.9635 | 0.0152 | 89.13

###  5.3 Analyzing potential threats

Our target is to propose a framework that is able to utilize the robustness
within the base model to defend some common image-level threats. The
experiments also demonstrate the advantages of our approach. We further
examine the embedded watermark’s robustness to more potential intentional
tampering and threats.

Threat via neural compression. Malicious users may alter the image to remove
the watermark with deep learning techniques, like methods used for image
compression with neural auto-encoders [6, 30, 3]. We evaluate the robustness
of the watermark against neural auto-encoders at different compression rates
(Fig. 7). Specifically, when the bit accuracy decreases to $50\%$, a threshold
indicating watermarking failure, the image quality significantly deteriorates.
This suggests that our watermarking method is resilient to the effects of
auto-encoder models. Even in extreme cases where bit accuracy reaches $50\%$,
the image quality is so compromised that the images are no longer suitable for
sharing or distribution purposes.

Threat via watermarking base model (white-box attack).  In this scenario,
attackers obtain the watermarking base model used by the NeRF creator. Then,
malicious users can implement adversarial attacks [26] with a leaked
watermarking base model to remove copyright messages in rendered images. We
launch a Projected gradient descent (PGD) attack [26], a classic adversarial
attack on rendered images. As depicted in Fig. 7, experimental results show
attackers can remove messages with minimal distortion to the rendered content.
This highlights the importance of keeping the watermarking base models
confidential [17, 11].

Figure 6: Attacks on rendered images. We evaluate neural compression methods
including VAE-Hyperior [3], JointARH [30] and GM-Att [6]. PGD-attack [26] is
also applied to attack the base model. Attacks using image compression methods
can only succeed when there is a significant decrease in image quality.
However, it is possible to remove watermarks with PGD attack [26] on the base
model without obvious distortion.

Figure 7: In an extreme situation, attackers might gain access to the images
utilized in creating NeRF. With these unwatermarked images, they could then
fine-tune the model to eliminate the watermarks. In this example, as the fine-
tuning progresses, the PSNR keeps increasing while the detected bit accuracy
decreases.

Threat on the representation of NeRF. If malicious users have obtained NeRF
models, they may choose to eliminate the watermarks by finetuning the models
with no-watermark images. This process involves finetuning NeRF only via the
content loss in Eq. 2 without the message embedding process.

We assume that malicious users can obtain the original images corresponding to
the scenes represented by NeRF. Then, from the results shown in Fig. 7, the
bit accuracy indeed drops, highlighting the need to securely manage both the
original data and the NeRF models. However, if malicious users cannot obtain
the images originally used for creating the NeRF, launching the attack becomes
difficult. For example, if they directly use some images unrelated to the
scenes represented by NeRF, it may significantly change the information stored
in NeRF, making the model sharing unfeasible (as shown in Fig. 8).

Figure 8: Attackers might fine-tune watermarked NeRF with unrelated scenes to
remove the watermarks. However, the model quality significantly decreases
after fine-tuning NeRF via unrelated images. The text below the image shows
the results of PSNR and bit accuracy.

##  6 Conclusion

This paper introduces a plug-and-play method for watermarking NeRF during
their creation. Utilizing a base watermarking model, NeRF creators can embed
binary messages directly into their models as they are being developed. We
propose a progressive global rendering to integrate message patterns into
NeRF. The base model inherently offers basic robustness to its users. Our
experimental results showcase the effectiveness of our approach. Our base
model is compatible with NeRF variants, offering a more versatile watermarking
solution.

Limitations. Though we have provided an effective technical solution for
protecting the copyright of NeRF, copyright protection should be a multi-
pronged effort involving various stakeholders. The legitimate enforcement of
copyright requires a comprehensive strategy that goes beyond technological
solutions. Legislative measures from relevant parties are also important to
establish a regulatory framework that safeguards intellectual property rights.

Acknowledgements. Renjie’s Research Group is supported by the National Natural
Science Foundation of China under Grant No. 62302415, Guangdong Basic and
Applied Basic Research Foundation under Grant No. 2022A1515110692,
2024A1515012822, and the Blue Sky Research Fund of HKBU under Grant No.
BSRF/21-22/16.

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