$\operatorname{CF}^{*}(\underline{\Lambda}_{0},X;\mathfrak{a}_{0,X}^{R})\cong\operatorname{CF}^{*}_{\natural}(S_{\star},X;\mathfrak{a}_{X})\otimes\operatorname{CF}^{*}_{\natural}(\underline{\Lambda}_{0},U_{\star};\mathfrak{a}_{0}),$

that is, they are isomorphic as chain complexes). If $\mathfrak{a}_{0}$ is
chosen properly,
$\operatorname{CF}^{*}_{\natural}(\underline{\Lambda}_{0},U_{\star};\mathfrak{a}_{0})$
has rank 1 and is generated by the constant soliton $e_{q}$. This proves

$H(\operatorname{CF}^{*}(\underline{\Lambda}_{0};\mathfrak{a}_{0,X}^{R})))\cong\operatorname{HF}_{\natural}^{*}(S_{\star},X)\cong\operatorname{HF}_{\natural}^{*}(\Lambda_{0},X)\
(\text{by Proposition }\ref{FL.P.4})$

Finally, the cohomology group
$H(\operatorname{CF}^{*}(\underline{\Lambda}_{0},X;\mathfrak{a}_{0,X}^{R}))$
is independent of $R>\pi$. We can always pull back the solution (cf. (1.6))
and think of the Floer equation as defined on the same domain
$\mathbb{R}_{t}\times[0,1]_{s}$, but the Floer datum may change as $R$ varies.
Even in the worst case that $W(X)$ is only bounded in the direction of
$e^{i\beta_{X}}$, since $W(\Lambda_{0})$ is the ray $l_{q,\theta_{0}}$, one
may still use a cobordism datum on the strip $\mathbb{R}_{t}\times[0,1]_{s}$
to construct continuation maps (cf. Section 5.2) and verify this invariance.

In this paper the construction of Floer cohomology always relies on a
perturbation by the superpotential $W$. If one wishes to compare
$H(\operatorname{CF}^{*}_{\natural}(\underline{\Lambda}_{0},X;\mathfrak{a}_{0,X}^{R}))$
with the version without such a perturbation, extra assumptions on $(M,W)$ are
required to verify the compactness theorem, and we leave this task to
interested readers. One may also replace $X$ by a thimble $\Lambda_{1}$ to
obtain the following result.

######  Proposition 13.13.

Suppose that $\Lambda_{j}=\Lambda_{q,\theta_{j}},j=0,1$ has bounded geometry
and $\theta_{1}<\theta_{0}<\theta_{1}+2\pi$, then
$\operatorname{HF}_{\natural}^{*}(\Lambda_{0},\Lambda_{1})\cong\operatorname{HF}_{\natural}^{*}(\Lambda_{0},\underline{\Lambda}_{1})\cong\operatorname{HF}_{\natural}^{*}(\underline{\Lambda}_{0},\Lambda_{1})\cong\operatorname{HF}^{*}(\underline{\Lambda}_{0},\underline{\Lambda}_{1})$.

Moreover, for any Lagrangian submanifold $X=(X,h_{X},\beta_{X},\xi_{X}^{\\#})$
satisfying the conditions in Assumption 12.1 and with
$e^{i(\beta_{X}-\theta_{0})}>0$, we have
$\operatorname{HF}_{\natural}^{*}(\Lambda_{0},X)\cong\operatorname{HF}^{*}(\underline{\Lambda}_{0},X)$.
Here $\operatorname{HF}^{*}$ is defined by counting instantons on infinite
strips $\mathbb{R}_{t}\times[0,1]_{s}$, but the equation is still perturbed by
the superpotential $W$ using a suitable Floer datum.

###  Appendix A The Diameter Estimate

This appendix is devoted to the proof of the diameter estimate in Proposition
3.3. This result concerns only the almost Hermitian geometry of
$(M,J_{M},\omega_{M},g_{M})$ and has played an essential role in the proof of
the Local Compactness Lemma 3.1. When $P:B(0,1)\to M$ is $J$-holomorphic, this
estimate follows from [IS00]; we shall only explain the necessary adaptation.
In what follows, we shall always assume that $(M,J_{M},\omega_{M},g_{M})$ has
bounded geometry.

####  A.1. The first step

For any $z\in\mathbb{C}$, let $B(z,R)$ denote the disk centered at $z$ of
radius $t$. Fix some $r>2$. We begin with the small energy estimate.

######  Lemma A.1.

There exist $\delta_{1},C_{1}>0$ with the following property. Let
$B(z,R)\subset B(0,1)$. If $P:B(0,1)\to M$ is any smooth map such that
$\operatorname{diam}P(B(z,R))\leq\delta_{1}$, then

$\operatorname{diam}\big{(}P\big{(}B(z,\frac{R}{2})\big{)}\big{)}<C_{1}\big{(}\|dP\|_{L^{2}(B(z,R))}+\|\bar{\partial}_{J}P\|_{L^{r}(B(z,R))}\big{)}.$

###### Proof of A.1.

This follows from the proof of [IS00, Lemma 1.1, Step 2]. For $0<\delta_{1}\ll
1$, we may assume that the image $P(B(z,R))$ lies in a geodesic ball, then the
estimate follows from the case when $M=\mathbb{R}^{2n}$ and $J=J_{M}$ is
$C^{0}$-close to the standard complex structure. See [IS00, Lemma 1.2]. ∎

######  Lemma A.2.

There exist $\epsilon_{2},C_{2}>0$ with the following property. If
$P:B(0,1)\to M$ is any smooth map such that

(A.1) $\int_{B(0,1)}|dP|^{2}+|\bar{\partial}_{J}P|^{r}<\epsilon_{2},$

then

(A.2)
$\operatorname{diam}\big{(}P\big{(}B(0,\frac{1}{2})\big{)}\big{)}<C_{2}\big{(}\|dP\|_{L^{2}(B(0,1))}+\|\bar{\partial}_{J}P\|_{L^{r}(B(0,1))}\big{)}.$

###### Proof of A.2.

This follows from Lemma A.1 and by repeating the argument in [IS00, Lemma 1.1,
Step 3]. Alternatively, it can be organized as follows. Let $\delta_{1}$ be
the constant from Lemma A.1. Consider the continuous function

(A.3) $f:B(0,1)\to\mathbb{R}^{+},\ f(z)=\max\\{0\leq R\leq
1-|z|:\operatorname{diam}(P(B(z,R)))\leq\delta_{1}\\},$

and let

$\alpha_{0}\colonequals\min_{|z|<\frac{3}{4}}\frac{f(z)}{\frac{3}{4}-|z|}.$

We claim that if $0<\epsilon_{2}\ll 1$ is chosen sufficiently small, then
$\alpha_{0}\geq\frac{1}{2}$. If not, suppose that this minimum is attained at
$z_{0}\in B(0,\frac{3}{4})$, set $r_{0}\colonequals\frac{3}{4}-|z_{0}|$, and
consider the map

$g:B(0,1)\to B(z_{0},r_{0}),\ w\mapsto z_{0}+r_{0}w.$

Define $P_{0}(w)\colonequals P(g(w)):B(0,1)\to M$, and denote by $f_{0}$ the
function (A.3) associated to $P_{0}$. Then $f_{0}(0)=\alpha_{0}$. In general,
since

$f(g(w))\geq\alpha_{0}\bigg{(}\frac{3}{4}-|g(w)|\bigg{)}\geq\alpha_{0}\bigg{(}r_{0}-|g(w)-z_{0}|\bigg{)},$

we have
$f_{0}(w)=\min\\{f(g(w))/r_{0},1-|w|\\}\geq\min\\{\alpha_{0}(1-|w|),1-|w|\\}\geq\alpha_{0}(1-|w|)$.
Hence, for all $|w|\leq\alpha_{0}$, $f_{0}(w)\geq\alpha_{0}/2$. Moreover,
under this rescaling, the quantity (A.1) associated to $P_{0}$ gets smaller.
Since $B(0,\alpha_{0})$ can be covered by finitely many disks
$B(w_{i},\alpha_{0}/4)$ with each $|w_{i}|\leq\alpha_{0}$, the diameter
estimate from Lemma A.1 applied to each $B(w_{i},\alpha_{0}/2)$ shows that
$\operatorname{diam}(P_{0}(B(0,\alpha_{0})))<\delta_{1}$ if $\epsilon_{2}$ is
chosen sufficiently small. This contradicts our choice of $\alpha_{0}$.

We have verified that $\alpha_{0}\geq\frac{1}{2}$, and so
$f(z)\geq\frac{1}{8}$ for all $|z|\leq\frac{1}{2}$. Now we cover
$B(0,\frac{1}{2})$ by finitely many disks $B(z_{i},\frac{1}{16})$ with each
$|z_{i}|\leq\frac{1}{2}$. The estimate (A.2) now follows from Lemma A.1. ∎

####  A.2. The second step

Lemma A.2 is almost enough to deduce Proposition 3.3. But when the energy is
highly concentrated near a point, we need another lemma to control the
diameter over an annulus. Let $A(R_{1},R_{2})$ denote the annulus
$B(0,R_{2})\setminus B(0,R_{1})$ for $R_{1}<R_{2}$.

######  Lemma A.3.

There exists $\epsilon_{3},C_{3}>0$ with the following property. Suppose that
$P:B(0,1)\to M$ is any smooth map such that for some $0<R<\frac{1}{16}$,

(A.4) $\int_{A(R,1)}|dP|^{2}+|\bar{\partial}_{J}P|^{r}<\epsilon_{3},$

then

(A.5)
$\operatorname{diam}\big{(}P\big{(}A(2R,1/2)\big{)}\big{)}<C_{3}\big{(}\|dP\|_{L^{2}(A(R,1))}+\|\bar{\partial}_{J}P\|_{L^{r}(A(R,1))}\big{)}.$

The best way to understand Lemma A.3 is to think of cylinders instead of
annuli. Let $Z(a,b)$ denote the cylinder $[a,b]\times S^{1}$ equipped with the
product metric for $a<b$, and $Z_{n}\colonequals Z(n-1,n)$. The proof of
[IS00, Lemma 1.4] can be adapted to show the following.

######  Lemma A.4.

There exists $\epsilon_{4},C_{4}>0$ and $\gamma\in(0,1)$ with the following
property. Suppose that $P:Z(0,5)\to M$ is any smooth map with

$\|dP\|_{L^{2}(Z_{n})}+\|\bar{\partial}_{J}P\|_{L^{r}(Z_{n})}<\epsilon_{4}$

for all $1\leq n\leq 5$, then

$\|dP\|_{L^{2}(Z_{3})}^{2}\leq\frac{\gamma}{2}\bigg{(}\|dP\|_{L^{2}(Z_{2})}^{2}+\|dP\|_{L^{2}(Z_{2})}^{2}\bigg{)}+C_{4}\|\bar{\partial}_{J}P\|_{L^{r}(Z(1,4))}^{2}.$

We prove Lemma A.3 given Lemma A.4.

###### Proof of Lemma A.3.

Consider $P$ as a map on $(-\infty,0]_{\rho}\times S^{1}_{\theta}$ with
$z=\exp(\rho+i\theta)$. Define

$\displaystyle a_{k}$ $\displaystyle=\|dP\|_{L^{2}(Z_{-k})}^{2},$
$\displaystyle b_{k}$
$\displaystyle=\|\bar{\partial}_{J}P\|_{L^{r}(Z_{-k})}^{2}.$

If $\epsilon_{3}$ is sufficiently small, then Lemma A.4 implies that

$a_{k}\leq\frac{\gamma}{2}(a_{k+1}+a_{k-1})+C_{4}\cdot b_{k}$

for all $1\leq k\leq n-1$ with $n=\lfloor-\ln R\rfloor$. Let
$c_{0}\colonequals\|dP\|_{L^{2}(A(R,1))}+\|\bar{\partial}_{J}P\|_{L^{r}(A(R,1))}$.
Then $a_{k}\leq c_{0}^{2}$ and

$\displaystyle c_{0}^{r}$
$\displaystyle\geq\int_{A(R,1)}|\bar{\partial}_{J}P|^{r}dtds=\int_{[\ln
R,0]\times S^{1}}|\bar{\partial}_{J}P|^{r}e^{(2-r)\rho}d\rho
d\theta\geq\int_{Z_{-k}}e^{(r-2)k}|\bar{\partial}_{J}P|^{r}d\rho d\theta,$

which implies that $b_{k}=\|\bar{\partial}_{J}P\|_{L^{r}(Z_{-k})}^{2}\leq
c_{0}^{2}e^{-2k(r-2)/r}$, so $b_{k}$ decays exponentially as $k\to\infty$. Now
we can repeat the argument of [IS00, Corollary 1.5] to show that both $\sum
a_{k}^{1/2}$ and $\sum b_{k}^{1/2}$ are bounded by a constant multiple of
$c_{0}^{2}$. Now apply Lemma A.2 to each $1\leq k\leq n-1$ to obtain the
diameter bound. ∎

####  A.3. The third step

Finally, we prove Proposition 3.3. Let $\epsilon_{\star}$ be the minimum of
the constants $\epsilon_{2}$ and $\epsilon_{3}$ in the Lemma A.2. Then the
proof proceeds by induction for $C\leq n\epsilon_{\star}$. The base case when
$n=1$ is Lemma A.2. For any $n\geq 2$, consider the continuous function
$h:B(0,1)\to\mathbb{R}^{+}$

$h(z)=\max\\{0\leq R\leq
1-|z|:\int_{B(z,R)}|dP|^{2}+|\bar{\partial}_{J}P|^{2}\leq(n-1)\epsilon_{\star}\\},$

and let

$\beta_{0}=\min_{|z|<\frac{3}{4}}\frac{h(z)}{\frac{3}{4}-|z|}.$

If $\beta_{0}\geq\frac{1}{16}$, then for all $z\in B(0,\frac{1}{2})$,
$h(z)\geq 1/64$. Then one can cover $B(0,\frac{1}{2})$ using finitely many
disks $B(z_{k},\frac{1}{32})$ with each $|z_{k}|\leq\frac{1}{2}$ and use the
induction hypothesis to estimate $\operatorname{diam}(P(B(0,\frac{1}{2})))$.

If $\beta_{0}<\frac{1}{16}$, then the energy concentrates at a smaller scale.
Suppose that this minimum is attained at some $z_{0}\in B(0,\frac{3}{4})$. Let
$r_{1}=\frac{3}{4}-|z|$ and $r_{0}\colonequals h(z_{0})=\beta_{0}\cdot
r_{1}<r_{1}/16$. Consider the inclusion

$B(z_{0},r_{0})\subset B(z_{0},2r_{0})\subset B(z_{0},\frac{r_{1}}{2})\subset
B(z_{0},r_{1}).$

First, the diameter of $P$ on $B(z_{0},\frac{r_{1}}{2})\setminus
B(z_{0},2r_{0})$ can be estimated using Lemma A.3. Second, the region
$B(0,\frac{1}{2})\setminus B(z_{0},\frac{r_{1}}{2})$ can be covered using
finitely many disks of radius $\frac{1}{32}$ which are all disjoint from
$B(z_{0},r_{0})$, then the diameter of $P$ on $B(0,\frac{1}{2})\setminus
B(z_{0},\frac{r_{1}}{2})$ is estimated using Lemma A.2. Finally, for any $z\in
B(z_{0},2r_{0})$, we have

$h(z)\geq\beta_{0}\bigg{(}\frac{3}{4}-|z|\bigg{)}\geq
r_{0}\cdot\frac{r_{1}-2r_{0}}{r_{1}}\geq\frac{r_{0}}{2}.$

Now we cover $B(z_{0},2r_{0})$ by finitely many disks of radius
$\frac{r_{0}}{4}$, then the diameter of $P$ on $B(z_{0},2r_{0})$ can be
estimated using the induction hypothesis. This finishes the proof of
Proposition 3.3.

###  Appendix B A Well-Known but Technical Lemma

This appendix is devoted to a well-known but technical lemma used in the proof
of Lemma 11.16.

######  Lemma B.1.

Under the assumption of Lemma 11.16, we have the following pointwise estimate
for the $n$-th derivative of $\mathcal{F}_{0}$, $n\geq 1$. For any $c_{2}>0$,
there exists a constant $C>0$ such that for any smooth $\xi_{0},\xi_{j}\in
C^{\infty}(B(0,1);P^{*}TM)$, $1\leq j\leq n$ with $\|\xi_{0}\|_{\infty}<c_{2}$
, we have

(B.1)
$|(d_{n}\mathcal{F}_{0})(\xi_{0};\xi_{1},\cdots,\xi_{l})|<C(1+|dP|+|\nabla\xi_{0}|+\sum_{j=1}^{n}\frac{|\nabla\xi_{j}|}{|\xi_{j}|})\prod_{j=1}^{n}|\xi_{j}|.$

on $B(0,\frac{1}{2})$. This constant $C$ depends on $c_{2}$ and the
$C^{n}$-norm of $\mathcal{X}$ on a slightly larger compact subset of $M$
$($which depends on $c_{2})$.

Let $\mathcal{B}^{r}_{1}(S;M)$ denote the space of $L^{r}_{1}$-maps $P:S\to
M$, and consider the Banach bundle:

$\mathcal{E}^{r}\to\mathcal{B}^{r}_{1}(S;M),r>2,$

whose fiber at $P$ is $L^{r}(S;\Lambda^{0,1}S\otimes TM)$. If $M$ and $S$ are
compact, then Lemma B.1 implies that
$\mathfrak{F}_{0}(P)=(dP-\mathcal{X})^{0,1}$ defines a smooth section of
$\mathcal{E}^{r}$, and $\mathcal{F}_{0}$ is the expression of
$\mathfrak{F}_{0}$ in a local trivialization of $\mathcal{E}^{r}$.

###### Proof of Lemma B.1.

We decompose $\mathfrak{F}_{0}(P)$ as

$\frac{1}{2}(dP+J_{P}\circ dP\circ j_{S})+\mathcal{X}^{0,1}$

where $j_{S}$ is the complex structure on $S$. The estimate (B.1) is linear in
$\mathfrak{F}_{0}$, so one may prove (B.1) for each of the three pieces
separately. To start, suppose that $\mathfrak{F}_{0}(P)$ takes the form
$\mathcal{G}(P)\circ dP$, where $\mathcal{G}:TM\to TM$ is any smooth bundle
map. We are mostly interested in the case when $\mathcal{G}=\operatorname{Id}$
or $J_{M}$, but the proof below works for a general $\mathcal{G}$.

Let $\exp:TM\to M$ denote the exponential map and $\pi:TM\to M$ the project
map. Then the parallel transportation $\Psi$ can be viewed as a smooth bundle
isometry:

${\pi^{*}TM}$${\exp^{*}TM}$${TM}$$\scriptstyle{\Psi}$

Given any smooth sections $\xi_{0},\xi_{j}\in C^{\infty}(B(0,1);P^{*}TM)$,
$1\leq j\leq n$, consider a bundle map $\hat{P}$ which covers $P:B(0,1)\to M$:

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}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-39.15933pt}{-22.94385pt}\pgfsys@lineto{0.48486pt}{-22.94385pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-21.8704pt}{-20.59108pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\qquad\begin{array}[]{l}\hat{P}:S\times\mathbb{R}^{n+1}\to
TM,\\\ \\\
(z,t_{0},\cdots,t_{n})\mapsto(z,\sum_{j=0}^{n}t_{j}\xi_{j}).\end{array}$

Now we consider the differential of the top row of (B.2) along $S$ then apply
$\mathcal{G}$:

(B.3)
${TS\times\mathbb{R}^{n+1}}$${T(TM)}$${TM}$${TM}$${S\times\mathbb{R}^{n+1}}$${TM}$${M}$${M,}$$\scriptstyle{d\hat{P}}$$\scriptstyle{\pi}$$\scriptstyle{d\exp}$$\scriptstyle{\pi}$$\scriptstyle{\mathcal{G}}$$\scriptstyle{\pi}$$\scriptstyle{\hat{P}}$$\scriptstyle{\exp}$

where $d\hat{P}$ is given explicitly by the formula:

(B.4) $\displaystyle d\hat{P}:TS\times\mathbb{R}^{n+1}$ $\displaystyle\to
T(TM)=T^{h}(TM)\oplus T^{v}(TM)$ $\displaystyle(Y,z,t_{0},\cdots,t_{n})$
$\displaystyle\mapsto(dP(Y),\sum_{j=0}^{n}t_{j}\nabla_{Y}\xi_{j}),\ Y\in
T_{z}S$

Here the vertical and horizontal sub-bundles of $T(TM)$ are defined using the
Levi-Civita connection. One may view $d\exp$ as a bundle map
$T(TM)\to(\exp)^{*}TM$ over $TM$ and compose with $\Psi^{-1}$ to obtain
another digram

(B.5)
${TS\times\mathbb{R}^{n+1}}$${T(TM)}$${\exp^{*}(TM)}$${\exp^{*}(TM)}$${\pi^{*}TM}$${S\times\mathbb{R}^{n+1}}$${TM.}$$\scriptstyle{d\hat{P}}$$\scriptstyle{\pi}$$\scriptstyle{d\exp}$$\scriptstyle{\mathcal{G}}$$\scriptstyle{\Psi^{-1}}$$\scriptstyle{\hat{P}}$

The composition of the first row is the definition of
$\mathcal{F}_{0}(\sum_{j=0}^{n}t_{j}\xi_{j})$, whose output is a smooth
section of $P^{*}TM\to S$. Now fix some $(Y,z)\in TS$ and let $t_{0}=1$. We
shall differentiate $\mathcal{F}_{0}(\sum_{j=0}^{n}t_{j}\xi_{j})$ in the
variables $t_{1},\cdots,t_{n}$ and set them to be zero. In this case, the
image of $d\hat{P}$ lies in

$T_{\hat{P}(z,t_{0},\cdots,t_{n})}(TM)\cong
T_{\hat{P}(z,t_{0},\cdots,t_{n})}^{h}(TM)\oplus
T_{\hat{P}(z,t_{0},\cdots,t_{n})}^{v}(TM)$

which is isomorphic to $T_{P(z)}M\oplus T_{P(z)}M$ canonically, so
$\Psi^{-1}\circ d\exp$ is a smooth family of endomorphism $T_{z}M\oplus
T_{z}M\to T_{z}M$ parametrized by
$T_{\hat{P}(z,t_{0},\cdots,t_{n})}^{v}(TM)\cong T_{z}M$. Thus
$\mathcal{F}_{0}(\sum_{j=0}^{n}t_{j}\xi_{j})$ takes the form

(B.6) $(\Psi^{-1}\circ\mathcal{G}\circ
d\exp)(t_{0}\xi_{0}+\sum_{j=1}^{n}t_{j}\xi_{j})(dP(Y),t_{0}\nabla_{Y}\xi_{0}+\sum_{j=1}^{n}t_{j}\nabla_{Y}\xi_{j}).$

Since $\|\xi_{0}\|_{\infty}<c_{2}$, all $n$-th derivatives of
$(\Psi^{-1}\circ\mathcal{G}\circ d\exp)$ are bounded uniformly. It is clear
that if (B.6) is differentiated along $t_{j},1\leq j\leq n$ and evaluated at
the origin, the final result is bounded by

$C_{1}|Y|(|dP(z)|+|\nabla\xi_{0}(z)|+\sum_{j=1}^{n}\frac{|\nabla\xi_{j}(z)|}{|\xi_{j}(z)|})\prod_{j=1}^{n}|\xi_{j}(z)|.$

for some $C_{1}>0$. By taking $\mathcal{G}=\operatorname{Id}$ and $J_{M}$, we
obtain the desired estimate for $\mathfrak{F}_{0}(P)=dP$ and
$\mathfrak{F}_{0}(P)=J_{M}\circ dP\circ j_{S}$ respectively (the complex
rotation $j_{S}$ is irrelevant for this estimate).

Finally, if $\mathcal{F}=\mathcal{X}^{0,1}$, then the diagram (B.3) is
replaced by

(B.7)
${\pi_{1}^{*}TS}$${\pi_{1}^{*}TS}$${\pi_{2}^{*}TM}$${S\times\mathbb{R}^{n+1}}$${S\times
TM}$${S\times
M}$$\scriptstyle{\mathcal{X}^{0,1}}$$\scriptstyle{(\pi_{1},\hat{P})}$$\scriptstyle{(\pi_{1},\exp\circ\pi_{2})}$

where $\pi_{1}:S\times M\to S$, $\pi_{2}:S\times M\to M$ are projection maps.
The diagram (B.5) is replaced by

(B.8)
${\pi_{1}^{*}TS}$${\pi_{1}^{*}TS}$${(\exp\circ\pi_{2})^{*}TM}$${(\pi\circ\pi_{2})^{*}TM}$${S\times\mathbb{R}^{n+1}}$${S\times
TM}$$\scriptstyle{\mathcal{X}^{0,1}}$$\scriptstyle{\Psi^{-1}}$$\scriptstyle{(\pi_{1},\hat{P})}$

For any fixed $(Y,z)\in TS$, this diagram defines a smooth function
$\mathbb{R}^{n+1}\to T_{z}M$ which can be differentiated along $t_{j},1\leq
j\leq n$ and evaluated at the origin. This derivative is then bounded by

$C_{2}\prod_{j=1}^{n}|\xi_{j}(z)|.$

and this $C_{2}$ depends at most on the $C^{n}$-norm of $\mathcal{X}$ on a
slightly larger compact subset of $M$ (which depends on $c_{2}$). This
completes the proof of Lemma B.1. ∎

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11institutetext:  Pedro M. Silva 22institutetext: Federal Institute of
Education, Science and Technology of Espírito Santo
Rodovia Governador José Sete, nº 184 - Itacibá, CEP 29150-410, Cariacica, ES,
Brasil.
22email<EMAIL_ADDRESS>33institutetext: Joao B. Florindo 44institutetext:
Institute of Mathematics, Statistics and Scientific Computing - University of
Campinas
Rua Sérgio Buarque de Holanda, 651, Cidade Universitária ”Zeferino Vaz” -
Distr. Barão Geraldo, CEP 13083-859, Campinas, SP, Brasil
44email<EMAIL_ADDRESS>
# Fractal measures of image local features: an application to texture
recognition ††thanks: J. B. F. gratefully acknowledges the financial support
of São Paulo Research Foundation (FAPESP) (Grant #2016/16060-0) and from
National Council for Scientific and Technological Development, Brazil (CNPq)
(Grants #301480/2016-8 and #423292/2018-8).

Pedro M. Silva  Joao B. Florindo

(Received: date / Accepted: date)

###### Abstract

Here we propose a new method for the classification of texture images
combining fractal measures (fractal dimension, multifractal spectrum and
lacunarity) with local binary patterns. More specifically we compute the box
counting dimension of the local binary codes thresholded at different levels
to compose the feature vector. The proposal is assessed in the classification
of three benchmark databases: KTHTIPS-2b, UMD and UIUC as well as in a real-
world problem, namely the identification of Brazilian plant species (database
1200Tex) using scanned images of their leaves. The proposed method
demonstrated to be competitive with other state-of-the-art solutions reported
in the literature. Such results confirmed the potential of combining a
powerful local coding description with the multiscale information captured by
the fractal dimension for texture classification.

###### Keywords:

Fractal geometry Multifractals Texture classification Box counting Local
binary patterns.

##  1 Introduction

Texture analysis, and in particular texture recognition, has been one of the
most important tasks in computer vision. Despite most of the initial
applications of this area being focused on material sciences, during the last
decades it has been applied to problems in a wide range of research areas,
such as Medicine Krishnamoorthi and Chinnababu (2019), Biology Verma et al.
(2018), Engineering Zhang et al. (2019), Physics Zheng et al. (2018), and many
others.

Although texture images do not have a consensual formal definition, they are
usually associated to structures commonly found in nature and composed by
elements whose complexity makes them difficult to be described by the
classical Euclidean geometry. This observation naturally opened space for the
possibility of analyzing such objects by employing techniques developed in
fractal geometry Xu et al. (2009, 2010). Indeed, a fractal is a geometrical
set characterized by the self-repetition of basic patterns and by a high
degree of complexity. It was in fact described by Mandelbrot in his seminal
book Mandelbrot (1983) as the “geometry of nature”.

Despite the recent popularization of deep learning / neural networks methods
in texture analysis, the intrinsic connection between the way that natural
structures are composed and the self-similar replication of fractals allows
the development of specific purpose models, which present some advantages over
deep learning methods, like not requiring so much annotated data for training
and the possibility of more straightforward interpretation of the obtained
results. The difficulty in obtaining large amounts of annotated data and
importance of result interpretation are key aspects, for example, in the
analysis of medical images.

However, the theoretical interest in fractals still struggles with some
practical limitations. The first and most obvious one is that there is no
mathematical fractal in nature. Fractals are defined in infinite scales and
this is obviously infeasible in the real world. In this way we need to be
aware that the fractal geometry representation is an approximation, a modeling
of a real system. A second point is that fractal geometry deals with
geometrical sets embedded within a continuous $N$-dimensional space, whereas
the most practical way of analyzing a real world object is by inspecting its
digital image representation. This issue was addressed for example in Pentland
(1983), where a relation between the “fractality” of a surface and its
respective photography was established. That work also accomplished a
psychological experiment that established connections between the fractal
dimension and the natural visual perception of attributes such as smoothness
and roughness.

Since then, a number of works applying fractal geometry to texture recognition
have been presented in the most diverse areas of applications. Nevertheless, a
third point still remains without a convincing solution: this is the ideal
representation of the image to estimate measures like the fractal dimension.

Among the attempts we have seen the fractional Brownian motion (fBm) model
Zhang et al. (1990) (and similar variations like the Fourier-based fractal
dimension Russ (1994)) and the geometrical approaches Falconer (2004). fBm has
achieved considerable success in texture synthesis, but there is no relevant
success reported in recognition. A possible explanation for this is the high
degree of specialization of the model, which is the reason for its success in
texture synthesis, but at the same time also makes it highly prone to over-
fitting in texture recognition. On the other hand, the geometrical approaches
rely on interpreting the image as a surface (or cloud of points) embedded in
the three-dimensional Euclidean space. This approach has achieved interesting
results in texture classification Florindo et al. (2018). However, it clearly
lacks an effective explanation for using the pixel spatial domain (its
localization) and the intensity domain (its gray value) to compose a unique
three-dimensional space. Questions like “how a homogeneous resolution scale
can be defined in such conditions?” are definitely not well answered.

The multifractal theory Xu et al. (2009, 2010); Zaghloul et al. (2019)
represented a noticeable attempt to address this issue with an advanced
mathematical framework based on Measure Theory. It also achieved promising
results on classification problems Xu et al. (2009, 2010), even though not too
much of these results have been reported on more modern and challenging
texture databases.

Finally, a simple solution would be to take a family of images resulting from
thresholding the original texture image at different levels. The direct
application of this strategy to the texture image is not feasible as the
relation between the gray values of neighbor pixels is lost and this is known
to be a fundamental characteristic to distinguish textures of different
materials. The authors in Quan et al. (2014) elegantly solved this issue by
applying the threshold strategy to the local binary pattern (LBP) Ojala et al.
(2002) map of the texture. This is a representation well known to capture the
relations between neighbor pixels Liu et al. (2019). This allows the
preservation of such locality information even in the thresholded image.

Based on this context, we propose here the combination of numerical techniques
to estimate the fractal dimension of binary images with a well known local
encoding of texture images, namely, LBP. We use the fractal dimension of the
encoded image thresholded at a range of levels to compose a vector of image
descriptors, which are employed for texture classification. To compose the
descriptors we verify the use of box-counting and Bouligand-Minkowski fractal
dimension as well as the lacunarity measure and the multifractal spectrum.

The proposed descriptors are compared with other fractal-based texture
features, namely, invariant multifractals Xu et al. (2009), wavelet
multifractals Xu et al. (2010), and pattern lacunarity spectrum Quan et al.
(2014). Other classical and state-of-the-art texture descriptors are also
compared, such as VZ-MR8 Varma and Zisserman (2005), local binary patterns
Ojala et al. (2002), convolutional neural networks Cimpoi et al. (2016), and
others. Comparative tests were carried out over three well known texture
benchmark data sets, to know, KTH-TIPS2b, UIUC and UMD. The descriptors were
also applied to the identification of species of Brazilian plants (database
1200Tex). The proposed method outperformed the compared approaches in terms of
classification accuracy and the results confirmed the potential of such a
strategy to provide rich and meaningful texture descriptors.

##  2 Fractal geometry

Roughly speaking, a fractal is a mathematical object defined at infinite
scales and characterized by self-similarity, i.e., repetition of geometrical
patterns at different scales and high degree of complexity, which basically
means that a hypothetical observer would never see the same object at
different scales. It is also commonly distinguished from Euclidean elements
for presenting non-integer dimension. Its formal definition is in fact stated
in terms of a dimension, technically called Hausdorff dimension.

###  2.1 Hausdorff dimension

Given a set $X$, its Hausdorff measure is defined as:

$\mathcal{H}^{s}(X)=\lim_{\sigma\rightarrow 0}H^{s}_{\sigma}(X)\text{ where \
}H^{s}_{\sigma}(X)=\inf\left\\{\sum_{i=1}^{\infty}|X_{i}|^{s}:X_{i}\mbox{ is a
}\sigma\mbox{-cover of }X\right\\},$ (1)

where $|X_{i}|$ is the diameter of $X_{i}$ given by
$|X_{i}|=\sup\\{d(x,y):x,y\in X_{i}\\}$, being $d(x,y)$ a metric. We say that
a family of sets $\\{X_{i}\\}$ is a $\sigma$-cover of a set $X$ if:

$\left\\{\begin{array}[]{l}X\subset\bigcup_{i=1}^{\infty}X_{i}\\\
0<|X_{i}|<\sigma.\end{array}\right.$ (2)

If $X$ is a fractal structure, it can be demonstrated that there exists a real
and non-negative value $D$ such that $H^{s}_{\sigma}=\infty$ for $s<D$ and
$H^{s}_{\sigma}=0$ for $s>D$. Then $D$ is defined as the Hausdorff dimension
of $X$. Formally:

$D=\inf\\{s:\mathcal{H}^{s}(F)=0\\}=\sup\\{s:\mathcal{H}^{s}(F)=\infty\\}.$
(3)

The most widely accepted definition of fractal is that of a geometrical set
whose Hausdorff dimension strictly exceeds its Euclidean (topological)
dimension.

###  2.2 Numerical estimates in fractal geometry

Although objects with fractal characteristics can be easily found in nature,
such real-world structures differ from mathematical fractals in two crucial
points: first, they do not have infinite self-similarity; second, the rules
dictating the formation of the structure are usually not known. The analytical
calculation of the Hausdorff dimension in this scenario is infeasible
Mandelbrot (1983). In this context, a large number of numerical values were
developed to estimate to which extent these objects can be approximated by a
fractal set. Especial interest has been devoted to methods that estimate the
fractal dimension of these structures.

Essentially, the computation of the Hausdorff measure involves an infinite
covering by units with diameter smaller than $\sigma$. The diameters of these
elements are therefore raised to an exponent $s$ and summed up. A discrete
approximation of this operation can be accomplished with the aid of an
exponential function:

$M_{\sigma}\propto\sigma^{s},$ (4)

where $M_{\sigma}$ is a measure of the object at the scale $\sigma$, in such a
way that any detail larger than $\sigma$ is not counted. By changing the
definition of $M$ and $\sigma$, we have definitions of the fractal dimension
that are alternative to Hausdorff formulation. These alternative definitions
may (and often do) assume values that do not coincide with the Hausdorff
dimension, but they preserve the idea of measuring the complexity and spatial
occupation of the object: the most complex the structure, the highest the
dimension. Many of these alternative definitions are more suitable for
numerical computation. Among the most popular definitions possessing this
property we have box-counting and Bouligand-Minkowski dimension and lacunarity
Falconer (2004); Yang et al. (2016).

####  2.2.1 Box-counting

Mathematically, if $X$ is a non-empty bounded subset of $\mathbb{R}^{n}$ and
$N_{\delta}(X)$ is the smallest number of sets with diameter at most $\delta$
covering $X$, then we define the lower and upper box-counting dimension of $X$
respectively by

$\begin{array}[]{l}\underline{\mbox{dim}}_{B}(X)=\liminf\limits_{\delta\rightarrow
0}\frac{\log N_{\delta}(X)}{-\log\delta}\\\
\overline{\mbox{dim}}_{B}(X)=\limsup\limits_{\delta\rightarrow 0}\frac{\log
N_{\delta}(X)}{-\log\delta}.\end{array}$ (5)

If both limits have the same value, we simply call it the box-counting
dimension:

$\mbox{dim}_{B}(X)=\lim\limits_{\delta\rightarrow 0}\frac{\log
N_{\delta}(X)}{-\log\delta}.$ (6)

However, the most interesting equivalent definition of box-counting dimension
for practical purposes is that defined under a mesh of hyper-cubes. In
$\mathbb{R}^{n}$ we define a $\delta$-mesh of hypercubes by

$[m_{1}\delta,(m_{1}+1)\delta]\times\cdots\times[m_{n}\delta,(m_{n}+1)\delta],$
(7)

where $m_{1},\cdots,m_{n}$ are integer numbers. The measure $N_{\delta}$ in
(6) can be replaced by the number of $\delta$-mesh cubes intersecting $X$ as
they can be demonstrated to be equivalent measures Falconer (2004).

####  2.2.2 Bouligand-Minkowski

The first step in the calculus of the Bouligand-Minkowski dimension of
$X\in\mathbb{R}^{n}$ is to dilate $X$ by balls in $\mathbb{R}^{n}$ with radius
$\delta$ forming the structure $D(\delta)$ according to:

$D(\delta)=\\{p:\|p-p^{\prime}\|_{2}\leq\delta,\forall p^{\prime}\in X\\}.$
(8)

The hypervolume of $D(\delta)$ is obtained by simply counting the number of
points in $D(\delta)$, i.e,

$V(\delta)=\sum_{p\in\mathbb{R}^{n}}\chi_{D(\delta)}(p),$ (9)

where $\chi$ is the characteristic function: $\chi_{A}(x)=1$ if $x\in A$ and
$\chi_{A}(x)=0$ otherwise.

Finally, the dimension $\mbox{dim}_{M}$ is provided by

$\mbox{dim}_{M}(X)=n-\lim\limits_{\delta\rightarrow 0}\frac{\log
V(\delta)}{\log\delta}.$ (10)

####  2.2.3 Lacunarity

For the calculus of the lacunarity of $X$ a hypercube with side-length
$\delta$ glides through $X$ and a histogram $H_{\delta}(k)$ stores the number
of hypercubes intersecting exactly $k$ points of $X$. The lacunarity at scale
$\delta$ is computed by

$\Lambda_{\delta}(X)=\frac{E[(H_{\delta})^{2}]}{(E[H_{\delta}])^{2}},$ (11)

where $E$ is the expected value for the distribution represented by $H$.
Finally, a scale-independent lacunarity is provided by

$\Lambda(X)=\lim\limits_{\delta\rightarrow
0}\frac{\log\Lambda_{\delta}(X)}{\log\delta}.$ (12)

####  2.2.4 Multifractals

For the multifractal spectrum we follow the method in Posadas et al. (2001).
First, the image is partitioned into boxes of sizes $L$. Therefore a local
measure $\mu_{i}$ is defined for a range of $q$ values as

$\mu_{i}(q,L)=\frac{\mathfrak{p}_{i}^{q}(L)}{\sum_{i=1}^{N(L)}\mathfrak{p}_{i}^{q}(L)},$
(13)

where $\mathfrak{p}_{i}(L)$ is the ratio (probability) of white points (points
of interest of the analyzed object) falling inside the $i^{th}$ box of size
$L$ and $N(L)$ is the number of $L$-sized boxes.

The spectrum $f(q)$ is obtained by

$f(q)=\lim\limits_{L\rightarrow
0}\frac{\sum_{i=1}^{N(L)}\mu_{i}(q,L)\log[\mu_{i}(q,L)]}{\log L}.$ (14)

##  3 Proposed method

We propose here the combined use of box-counting, Bouligand-Minkowski,
lacunarity and multifractal spectrum to quantify the fractality of binary
images with a well known local encoding of texture images, namely, LBP codes.
We use the fractal measure of the encoded image thresholded at a range of
levels to compose a vector of image descriptors, which are employed for
texture classification.

In practice, for a binary image $I$ like that used here, $\delta$ values
follow an exponential sequence $\delta=2,4,8,16,\cdots,\log(M)$ where $M$ is
the smallest dimension of the image. In the calculus of box-counting
dimension, for each $\delta$ the number of squares intersecting the object of
interest is accumulated in the variable $N_{\delta}$ and the numerical
dimension is estimated by

$\mbox{dim}_{B}(I)=-\alpha,$ (15)

where $\alpha$ is the slope of the straight line fit by minimum least squares
to the curve $\log\delta\times\log N_{\delta}$. We also employ the linear
coefficient of the straight line to compose the feature vector. Figure 1
illustrates the procedure.

Figure 1: Box counting descriptors for LBP mappings. To simplify the
visualization we use $P=4$ and $R=1$ as LBP parameters. (a) Texture. (b) LBP
codes. (c) LBP thresholds. (d) Box-counting dimension.

The procedure to compute Bouligand-Minkowski dimension is similar and we use
Euclidean distance transform to optimize the calculus of $V(\delta)$ and we
employ a maximum radius of $9$, as recommended in Casanova et al. (2009).
Again, we use the slope and linear coefficient of a straight line fit to
$\log\delta\times\log V(\delta)$.

For the lacunarity, the radius $\delta$ ranges between $2$ and $14$, as
recommended in Quan et al. (2014) and we also employ both linear and angular
coefficients to compose the feature vector.

Finally, the multifractal spectrum is computed according to the description in
Section 2.2.4 and using $L=2,3,5,10,25,50,100,125,250$ as in Posadas et al.
(2001).

###  3.1 Motivation

Fractal dimension and other fractal measures in their strict sense, as in
Falconer (2004), are usually not suitable for the analysis of digital images.
To start with, digital images are discrete and as a consequence their
geometrical representation is countable. Fractal dimension of countable sets
is, by definition, zero Falconer (2004) and hence such images cannot be
identified using fractal dimension. One could construct a continuous model
based on the image, but that would introduce artificial data to the process.

Based on this context we develop here a statistical model to investigate how
measures like box counting or Bouligand-Minkowski dimension behave in the
analysis of digital images. For that purpose, LBP codes can be assumed to
follow a uniform distribution, which substantially simplifies the statistical
analysis. It is also worth to mention that the other measures explored here,
i.e., lacunarity and multifractal spectrum, can be employed in a similar
analysis, but the conclusions are expected to be similar, given that both also
share the same main objective of measuring the fractality of the image.

####  3.1.1 Box counting

For box counting, we can employ a model similar to that developed in Kenkel
(2013). Let us suppose that the probability of a white point in the binary
image (point of interest) is $p$ and we have a total of $N$ points. Let us
also suppose that in $\mathbb{R}^{1}$ the data is enclosed within the interval
$I=[0,1]$. The probability that a box with size $s$ (subinterval of $I$) does
not contain any point is

$p_{0}=(1-s)^{Np}.$ (16)

Correspondingly, the probability of the same box to contain at least one point
of interest is the complement of $p_{0}$:

$p_{1}=1-(1-s)^{Np}.$ (17)

The expected number of nonempty boxes used in the calculus of box counting
dimension is given by

$B(s)=[1-(1-s)^{Np}]/s.$ (18)

Likewise, in $\mathbb{R}^{2}$ we work on an interval $I^{2}=[0,1]\times[0,1]$
and have

$B(s)=[1-(1-s^{2})^{Np}]/s^{2}.$ (19)

The trick in Kenkel (2013) is that for a self-similar set with self-similar
dimension $d_{S}$ we would have

$B(s)=[1-(1-s^{d_{S}})^{Np}]/s^{d_{S}}.$ (20)

Box counting dimension is obtained by fitting a straight line to the curve
$\log s\times\log B(s)$ using least squares. Variable $s$ assumes value within
a predefined range (typically linear or exponential)
$s_{1},s_{2},\cdots,s_{n}$. To simplify the mathematical notation we define
$x_{i}=\log s_{i}$ and $y_{i}=\log B(s_{i})$. Least squares have a statistical
interpretation in which the straight line equation $y=\alpha x+\beta$ is
defined in terms of variances and covariances.

At first, we define the variances of $x$ and $y$ and the covariance between
$x$ and $y$:

$\begin{array}[]{ccc}\sigma_{x}^{2}=\frac{\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}}{n}&\sigma_{y}^{2}=\frac{\sum_{i=1}^{n}(y_{i}-\overline{y})^{2}}{n}&\mbox{cov}(x,y)=\frac{\sum_{i=1}^{n}(x_{i}-\overline{x})(y_{i}-\overline{y})}{n},\end{array}$
(21)

where $\overline{x}$ and $\overline{y}$ denote the mean of $x$ and $y$
respectively. Then it is well known from least squares theory that

$\alpha=\frac{\mbox{cov}(x,y)}{\sigma_{x}^{2}}\qquad\mbox{and}\qquad\beta=\overline{y}-\alpha\overline{x}.$
(22)

Such relations express how the measure (number of boxes) is statistically
associated to the scale $s$. Figure 2 shows $\alpha$ and $\beta$ for different
numbers of points $N_{p}$ and dimension $d_{S}$. For different similarity
dimensions, the box dimension ($\alpha$) and linear coefficient ($\beta$)
present different behaviors when plotted against the distribution of points
($Np$). In terms of image descriptors, this corresponds to a more complete
representation than the sole dimension or the probability $p$, even though
those important features continue to determine the shape of the curve.

|  |
---|---|---
|  |
$d_{S}=1.1$ | $d_{S}=1.5$ | $d_{S}=1.9$
Figure 2: $\alpha$ and $\beta$ in box counting for different numbers of points
$Np$ and dimension $d_{S}$.

###  3.2 Bouligand-Minkowski

A formal analysis of the dilation area of circles is significantly more
complicated than for box counting and the interested reader can have some idea
of that in da S. Oliveira et al. (2014). Nevertheless, for a statistical
study, we can reduce the problem to a one-dimensional binary “image” in
$\mathbb{R}^{1}$, more specifically over the $[0,1]$ interval. Now the
morphological dilation of Bouligand-Minkowski corresponds to placing bars with
length $s$ ($s<1$) centered at points (white points) randomly dropped over the
interval. The aim is to compute the expected length covered by at least one
bar.

As we need to take care of the boundaries of $[0,1]$, it turns out that for a
single bar, the probability $p_{1}$ of a point at the position $x$ being
covered can assume three possible values depending on the value of $x$:

$p_{1}=\left\\{\begin{array}[]{lll}x+\frac{s}{2}&\mbox{if}&s\in[0,\frac{r}{2}]\\\
s&\mbox{if}&x\in[\frac{s}{2},1-\frac{s}{2}]\\\
-x+\frac{s}{2}+1&\mbox{if}&x\in[1-\frac{s}{2},1].\\\ \end{array}\right.$ (23)

The probability of $x$ being uncovered is $1-p_{1}$. In a similar way to what
we did in box counting, we suppose that our “image” has $N$ points and the
probability of occurrence of a white point is $p$. Therefore the expected
number of white points is $Np$. The probability of $x$ being uncovered after
$n$ bars is randomly placed in $[0,1]$ is $(1-p_{1})^{Np}$. Hence the
probability of $x$ being covered by at least one bar is

$p_{n}=1-(1-p_{1})^{Np}.$ (24)

Finally the expected length of the covered region is given by

$\langle L\rangle=\int_{0}^{1}p_{n}dx.$ (25)

In practice we work with dilation radii much smaller than the size of the
image, which makes possible to disregard the effect of boundaries and focus on
the region $x\in[\frac{s}{2},1-\frac{s}{2}]$. This simplifies the integral to

$\langle L\rangle=1-(1-s)^{Np}.$ (26)

This expression can be rewritten as

$\langle L\rangle=1-\left(1-\frac{Nps}{Np}\right)^{Np},$ (27)

which for large $N$ allows for the use of the exponential limit:

$\lim\limits_{n\rightarrow\infty}\langle L\rangle=1-\mathrm{e}^{-Nps}.$ (28)

This limit confirms the power law relation naturally appearing in fractal-like
phenomenons. However, for our purposes, it is also worthwhile to investigate
(26), which determines the behavior of $\langle L\rangle$ for not so large
values of $N$.

Using a strategy similar to that in Kenkel (2013), in two dimensions we have
circles with radius $s$ and (26) is rewritten as

$\langle L\rangle=1-(1-\pi s^{2})^{Np}.$ (29)

For a self-similar structure with self-similar dimension $d_{S}$ we can resort
to the formula for the volume of the $n$-dimensional ball, yielding

$\langle
L\rangle=1-\left(1-\frac{\pi^{d_{S}/2}}{\Gamma\left(\frac{d_{S}}{2}+1\right)}s^{d_{S}}\right)^{Np},$
(30)

where $\Gamma$ is the Euler’s gamma function. $\alpha$ and $\beta$ are
provided by (22) as usual. Figure 3 shows $\alpha$ and $\beta$ for different
numbers of points $N_{p}$ and self-similar dimension $d_{S}$. As in Figure 2
we have different curves for each similarity dimension.

|  |
---|---|---
|  |
$d_{S}=1.1$ | $d_{S}=1.5$ | $d_{S}=1.9$
Figure 3: $\alpha$ and $\beta$ in Bouligand-Minkowski for different numbers of
points $Np$ and self-similar dimension $d_{S}$.

In general, all this analysis corroborates the interest in analyzing local
characteristics of the image like LBP codes by inspecting properties of the
$\log-\log$ dimension curve. Here we use the slope and linear coefficient of
such curve and verify that such potential theoretically predicted is also
confirmed in practical situations.

##  4 Experiments

The performance of the proposed descriptors was evaluated on three benchmark
databases widely used in the recent literature of texture classification
methods. We also applied the same method to a practical problem, namely the
identification of species of Brazilian plants using scanned images of leaves.

The first benchmark data set is KTHTIPS-2b Hayman et al. (2004), a database
comprising 4752 images uniformly divided into 11 material classes. An
important characteristic of this data is its focus on the material represented
in the image rather than on the instance of the photographed object. In each
material class the images can still be divided into 4 samples. Each sample
follows a particular scheme of scale, pose and illumination. The validation
protocol is the most typically employed in the literature, where 1 sample is
used for training and the remaining 3 samples are used for testing. The
accuracy (percentage of images correctly classified) and respective standard
deviation are obtained by averaging out the results for the 4 possible
combinations of training/testing.

The second database is UIUC Lazebnik et al. (2005), containing 1000 images
evenly divided into 25 texture categories. The images were collected under
non-controlled conditions and contain variation in albedo, perspective,
illumination and scale. For the validation split we randomly select 20 images
of each texture for training and the remaining 20 images for testing. This is
repeated 10 times to provide the average accuracy and deviation.

The third data set is UMD Xu et al. (2009). This is composed by a collection
of 1000 high-resolution images collected by a family camera without any
illumination control. The images are uncalibrated and unregistered and are
categorized into 25 classes, each one with 40 images. Each image has a
resolution of $1280\times 960$. The database is characterized by high variance
in viewpoint and scale, what makes its classification a more challenging task.

To reduce dimensionality and attenuate the effect of redundant information,
the proposed descriptors are subject to principal component analysis Jolliffe
(2002) before the classification. For the classification, we tested two
possibilities: support vector machines (SVM) Cortes and Vapnik (1995) with a
configuration similar to what is employed in Cimpoi et al. (2016), i.e.,
linear kernel, $C=1$ and $L^{2}$ normalization, and linear discriminant
analysis (LDA) Krzanowski (1988).

##  5 Results and Discussion

An initial test was accomplished to verify the performance of the proposed
method for the two compared classifiers, i.e., LDA and SVM. For this test we
employed box counting descriptors as those have more straightforward
interpretation and allow more unbiased evaluation of the classifier. Table 1
lists the percentage of images correctly classified in each compared database
as well as the respective standard deviation. LDA provided higher accuracies
in all tested data and, based on this, the remaining experiments are carried
out using this classification scheme.

Table 1: Classification accuracy using SVM and LDA classifier. Database | SVM | LDA
---|---|---
KTHTIPS-2b | 63.4$\pm$3.4 | 65.5$\pm$3.5
UIUC | 96.3$\pm$0.5 | 97.3$\pm$0.6
UMD | 98.0$\pm$0.7 | 98.6$\pm$0.6
1200Tex | 81.9$\pm$1.7 | 86.6$\pm$1.1

In Table 2 we analyze the performance of the individual fractal descriptors:
box counting (BC), Bouligand-Minkowski (BM), lacunarity (L), and multifractal
(MF). From that table it is possible to state that each fractal metric can be
more or less recommended depending on particularities of the data set. This
also suggests that combining different fractal features could provide even
better classification results.

Table 2: Classification accuracy of the individual fractal descriptors: BC (box counting), BM (Bouligand-Minkowski), L (lacunarity), and MF (multifractal). Database | BC | BM | L | MF
---|---|---|---|---
KTHTIPS-2b | 65.5$\pm$3.5 | 66.3$\pm$3.1 | 65.9$\pm$3.6 | 58.9$\pm$3.3
UIUC | 97.4$\pm$0.5 | 97.6$\pm$0.4 | 97.8$\pm$0.5 | 86.7$\pm$1.4
UMD | 98.6$\pm$0.6 | 99.0$\pm$0.6 | 99.1$\pm$0.4 | 97.2$\pm$0.6
1200Tex | 86.4$\pm$1.1 | 85.9$\pm$1.3 | 84.9$\pm$1.1 | 67.2$\pm$1.5

Table 3 shows the accuracy when some of the investigated fractal features are
combined into a single vector of descriptors. Other combinations were verified
but at the end those ones presented in Table 3 had the most competitive
performances. In general, some subtle improvement over the individual features
was obtained in most data sets when combined descriptors were used.

Table 3: Classification accuracy for some combinations of fractal descriptors.
Best results in each data set are in bold.

Database | BC+BM | BC+L | BC+MF | BM+L | BM+MF | L+MF
---|---|---|---|---|---|---
KTHTIPS-2b | 67.5$\pm$2.1 | 67.6$\pm$2.1 | 65.0$\pm$2.1 | 67.4$\pm$2.6 | 65.4$\pm$2.2 | 66.7$\pm$3.1
UIUC | 97.9$\pm$0.5 | 98.1$\pm$0.5 | 96.4$\pm$0.7 | 98.0$\pm$0.4 | 96.3$\pm$0.8 | 97.0$\pm$0.6
UMD | 98.9$\pm$0.6 | 99.1$\pm$0.5 | 99.1$\pm$0.5 | 99.1$\pm$0.4 | 99.3$\pm$0.2 | 99.1$\pm$0.3
1200Tex | 86.2$\pm$1.0 | 86.3$\pm$0.8 | 82.2$\pm$1.6 | 85.9$\pm$1.1 | 81.6$\pm$1.8 | 80.5$\pm$1.6

Database | BC+BM+L | BC+BM+L+MF
---|---|---
KTHTIPS-2b | 66.4$\pm$2.0 | 66.4$\pm$2.0
UIUC | 98.1$\pm$0.6 | 97.6$\pm$0.5
UMD | 99.0$\pm$0.5 | 99.2$\pm$0.3
1200Tex | 86.3$\pm$0.6 | 82.5$\pm$1.4

Table 4 lists the accuracy performance of the proposed descriptors in
KTHTIPS-2b, UMD and UIUC, compared with other results published in the
literature. Details concerning parameters and other implementation details for
each result can be obtained in the respective references. Here the fractal
features outperformed advanced approaches like SIFT $+$ VLAD or SIFT $+$ BoVW
in KTHTIPS-2b. Methods based on automatic (deep) learning like FC-CNN were
also outperformed in UIUC and UMD. These are canonical examples of what could
be defined as “textures in their strict sense” and the results confirmed the
potential of fractal-based methods to analyze such types of images. Finally,
MFS and PLS are examples of fractal-based approaches for texture recognition.
Both were also outperformed by the proposed descriptors in UIUC and UMD.

Table 4: Accuracy of the proposed descriptors compared with other texture descriptors in the literature. All the results except for the proposed method were obtained from the literature. A ‘-’ indicates that no result was published for that method on that database. A superscript 1 denotes a slightly different protocol in KTH-TIPS2b where 3 samples are used for training and 1 sample for testing. Method | KTH-TIPS2b | UIUC | UMD
---|---|---|---
VZ-MR8 Varma and Zisserman (2005) | 46.3 | 92.9 | -
LBP Ojala et al. (2002) | 50.5 | 88.4 | 96.1
VZ-Joint Varma and Zisserman (2009) | 53.3 | 78.4 | -
BSIF Kannala and Rahtu (2012) | 54.3 | 73.4 | 96.1
LBP-FH Ahonen et al. (2009) | 54.6 | - | -
CLBP Guo et al. (2010) | 57.3 | 95.7 | 98.6
SIFT+LLC Cimpoi et al. (2016) | 57.6 | 96.3 | 98.4
ELBP Liu et al. (2012) | 58.1 | - | -
SIFT + KCB Cimpoi et al. (2014) | 58.3 | 91.4 | 98.0
LBPriu2/VAR Ojala et al. (2002) | 58.51 | 84.4 | 95.9
SIFT + BoVW Cimpoi et al. (2014) | 58.4 | 96.1 | 98.1
PCANet (NNC) Chan et al. (2015) | 59.41 | 57.7 | 90.5
SIFT + VLAD Cimpoi et al. (2014) | 63.1 | 96.5 | 99.3
RandNet (NNC) Chan et al. (2015) | 60.71 | 56.6 | 90.9
SIFT+IFV Cimpoi et al. (2014) | 58.2 | 97.0 | 99.2
ScatNet (NNC) Bruna and Mallat (2013) | 63.71 | 88.6 | 93.4
DeCAF Cimpoi et al. (2014) | 70.7 | 94.2 | 96.4
FC-CNN VGGM Cimpoi et al. (2016) | 71.0 | 94.5 | 97.2
MFS Xu et al. (2009) | - | 92.7 | 93.9
PLS Quan et al. (2014) | - | 96.6 | 99.0
Proposed | 67.6 | 98.1 | 99.3

Figure 4 shows the confusion matrices for the benchmark databases. Box
counting descriptors were used to generate these pictures. Such
representations essentially confirm the accuracies in Table 2, but they also
provide information regarding the accuracy in each class, what opens
possibility for a more elaborate analysis on the classification outcomes. In
Figure 4 (a) we see KTHTIPS-2b with lower accuracy in classes 3 (“corduroy”)
and 5 (“cotton”). These are actually materials frequently confused as they are
types of clothing fabrics and possess similar texture patterns. UIUC, on the
other hand, yields a nearly perfect classification result, with some
significant misclassification only in class 8 - granite - (confused with 18 -
carpet). Despite being different materials, they are both characterized by a
granular appearance, which poses some difficulties for the automatic
discrimination.

|  |
---|---
(a) | (b)
|
---
(c)
Figure 4: Confusion matrices. (a) KTHTIPS-2b. (b) UIUC. (c) UMD.

###  5.1 Identification of Plant Species

Table 5 lists the accuracy of the fractal-LBP descriptors in 1200Tex database
Casanova et al. (2009), compared with a few state-of-the-art results published
in the literature. This is a set of images of plant leaves for 20 Brazilian
species collected in vivo. For each species 20 samples were collected, washed,
aligned with the vertical axis and photographed by a scanner. The image of
each sample was split into 3 non-overlapping windows with size $128\times
128$. Those windows were extracted from regions of the leaf presenting less
texture variance and were converted into gray scale images, resulting in a
database with 1200 images. From each species 30 images were randomly selected
for training and the remaining images for testing. This procedure was repeated
10 times, which allowed the computation of the average accuracy and respective
deviation (in Table 2 and Table 3).

Table 5: Accuracy of the fractal-LBP descriptors compared with other results in the identification of plant species. Method | Accuracy (%)
---|---
LBPV Guo et al. (2010) | 70.8
Network diffusion Gonçalves et al. (2016) | 75.8
FC-CNN VGGM Cimpoi et al. (2016) | 78.0
Gabor Casanova et al. (2009) | 84.0
FC-CNN VGGVD Cimpoi et al. (2016) | 84.2
Schroedinger Florindo and Bruno (2017) | 85.3
SIFT + BoVW Cimpoi et al. (2014) | 86.0
Proposed | 86.3

Figure 5 complements Table 5 by exhibiting the confusion matrix for the
proposed method (box counting features). Generally speaking, the accuracies in
all classes are high and the most critical situation occurs in class 8, which
is confused for example with class 6. These correspond to samples with similar
textures, especially with regards to the arrangement of nervures and leaf
microtexture, which are prominent elements for the process of distinguishing
among samples from different species.

Figure 5: Confusion matrix for the plant database.

Generally speaking, fractal descriptors still demonstrate competitiveness when
compared with several state-of-the-art approaches for texture recognition. It
is well known that characteristics intrinsic to the way that these materials
are formed in nature contribute to this relation to a significant extent. This
can be even more easily observed in data sets of “pure” textures, like UIUC
and UMD, as well as in practical problems where such types of images naturally
arise. This is the case in many biological applications and here is
illustrated by the foliar surface images. The results encourage more research
on this topic at the same time that it presents fractal descriptors as an
alternative that should be verified in practical problems, as they can achieve
competitive performance, for example, without requiring large amounts of
training data and they also provide more natural interpretation for the
obtained results as fractal sets have been classically associated with a
mathematical model of nature structures.

##  6 Conclusions

Here we developed a combination of local binary patterns with numerical
estimates of fractal dimension. More specifically, we compute the dimension of
the LBP codes thresholded at different levels to compose the image feature
vector.

The performance of our proposal was assessed in the classification of
benchmark databases typically used in the literature. We also employed such
descriptors in a practical problem with significant importance in botany and
related areas, namely, the identification of species of Brazilian plants. In
both cases our method obtained promising results, comparable to state-of-the-
art results recently published in the literature.

The results presented here suggest that the combination of fractal geometry
(and potentially other fractal measures) with a local encoding like LBP can be
rather useful to represent all the rich information conveyed by a texture
image.

## Conflict of interest

The authors declare that they have no conflict of interest.

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