$\mu_{3}\in\Lambda_{3}^{*}/\Lambda_{3}$ by the simplification. Using the
equations in (104), we can write

$\displaystyle{\boldsymbol{Q}}_{1}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}8k_{12}+4k_{22}+\mu_{11}-\mu_{31}\\\
8k_{11}+4k_{21}-2k_{22}+\mu_{12}+\mu_{31}-\mu_{32}\end{pmatrix},$ (202)
$\displaystyle{}{\boldsymbol{Q}}_{2}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}8k_{22}+4k_{12}+\mu_{21}-\mu_{31}\\\
8k_{21}+4k_{11}-2k_{12}+\mu_{22}+\mu_{31}-\mu_{32}\end{pmatrix}.$

We restrict to $\mu=0=\mu_{1}+\mu_{2}+\mu_{3}$, such that

$\displaystyle{}{\boldsymbol{Q}}_{1}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}8k_{12}+4k_{22}+2\mu_{11}+\mu_{21}\\\
8k_{11}+4k_{21}-2k_{22}+2\mu_{12}+\mu_{22}-\mu_{11}-\mu_{21}\end{pmatrix},\quad{\boldsymbol{\mu}}_{1}=\begin{pmatrix}2\mu_{11}+\mu_{21}\\\
{}2\mu_{12}+\mu_{22}-\mu_{11}-\mu_{21}\end{pmatrix},$
$\displaystyle{\boldsymbol{Q}}_{2}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}8k_{22}+4k_{12}+2\mu_{21}+\mu_{11}\\\
8k_{21}+4k_{11}-2k_{12}+2\mu_{22}+\mu_{12}-\mu_{11}-\mu_{21}\end{pmatrix},\quad{\boldsymbol{\mu}}_{2}=\begin{pmatrix}2\mu_{21}+\mu_{11}\\\
{}2\mu_{22}+\mu_{12}-\mu_{11}-\mu_{21}\end{pmatrix}.$

### List of $\Phi_{\boldsymbol{\mu}}$

We next present explicit $q$-expansions of $\Phi_{\boldsymbol{\mu}}$ for a
various choices of ${\boldsymbol{\mu}}$. All the convergence conditions of the
types $C_{i}^{2}C_{i+1}^{2}-(C_{i}.C_{i+1})^{2}>0$ and
$C_{i}^{2}(C_{i-1}.C_{i+1})-(C_{i}.C_{i-1}).(C_{i}.C_{i+1})<0$ are satisfied
in this example along with $C_{j}^{2}>0$. As in explained in Section 4.1,
${\boldsymbol{\mu}}$ is fully determined by specifying
$(\mu_{1},\mu_{2},\mu_{3})$ and $\mu=\mu_{1}+\mu_{2}+\mu_{3}$. In the
following example, we fix $\mu=\mu_{1}=0$ and $\mu_{3}=-\mu_{2}$, and list
$\Phi_{\boldsymbol{\mu}}$ for 16 different values of $\mu_{2}$. Out of this
set, 10 give rise to different $q$-series. The overall sign of the
$q$-expansion is given by
$(-1)^{(a+b+c)}=(-1)^{\mu_{11}-\mu_{21}}=(-1)^{\mu_{21}}$. Furthermore, the
growth of the coefficients is moderate, $|c(n)|\leq C\,n^{\alpha}$ for some
positive constants $\alpha$ and $C$, which is expected for a holomorphic
(mock) modular form of weight 2.

It can be shown that in most of the sectors we study ($\mu_{1}=(0,0)$) there
is no contribution to the holomorphic part when the arguments of $E_{2}$ are
both zero. When $\mu_{2}=(1,j)$ or $\mu_{2}=(3,j)$ $b$ is odd while $a,c$ are
both odd or both even. Under these conditions at least two of
$(a+b-c),\;(a-b+c),\;(c+b-a)$ cannot vanish. Only in the conjugacy classes
$\mu_{2}=(0,0),(0,2),(2,3)$ we find integral $k_{11},k_{12},k_{21},k_{22}$ to
satisfy at least two of $(a+b-c),\;(a-b+c),\;(c+b-a)$ to vanish. The
contributions $A_{\ell}$ are given by

$\begin{split}A_{1}&=\frac{2}{\pi}\,\arctan(-5/\sqrt{11}),\\\
A_{2}&=A_{3}=\frac{2}{\pi}\,\arctan(-1/\sqrt{8}),\end{split}$ (203)

which are all irrational numbers.

For $\mu_{2}=(0,0)$, we have:

$\displaystyle{}\Phi_{\boldsymbol{\mu}}$ $\displaystyle=$ $\displaystyle
2q^{6}+4q^{20}+6q^{24}+4q^{30}+4q^{44}+4q^{50}+8q^{52}+2q^{54}+12q^{56}+4q^{60}$
$\displaystyle+$ $\displaystyle
4q^{64}+4q^{68}+4q^{70}+12q^{80}+2q^{88}+8q^{90}+8q^{92}+8q^{94}+14q^{96}+16q^{100}$
$\displaystyle+\frac{(A_{1}+A_{2}+A_{3}+1)}{2}+(A_{1}+A_{2}+A_{3}+3)(q^{8}+q^{32}+q^{72})+\dots$

For $\mu_{2}=(0,1)$, we have:

$\displaystyle{}\Phi_{\boldsymbol{\mu}}$ $\displaystyle=$ $\displaystyle
q^{5}\left(1+q+2q^{12}+2q^{15}+2q^{17}+2q^{20}+q^{22}+2q^{23}+q^{27}+3q^{28}+q^{30}+2q^{35}+2q^{38}\right.$
$\displaystyle+$
$\displaystyle\left.3q^{40}+2q^{42}+4q^{45}+3q^{46}+3q^{49}+4q^{51}+2q^{52}+2q^{53}+3q^{54}+2q^{55}+2q^{57}+3q^{60}\right.$
$\displaystyle+$
$\displaystyle\left.2q^{61}+2q^{63}+q^{64}+2q^{65}+5q^{69}+6q^{70}+2q^{71}+2q^{73}+3q^{75}+2q^{78}+2q^{79}+2q^{81}\right.$
$\displaystyle+$
$\displaystyle\left.q^{82}+2q^{83}+6q^{84}+2q^{86}+3q^{87}+4q^{88}+5q^{90}+4q^{91}+2q^{92}+6q^{93}+2q^{94}+2q^{95}\right)+\dots$

For $\mu_{2}=(0,2)$, we have:

$\displaystyle{}\Phi_{{\boldsymbol{\mu}}}$ $\displaystyle=$ $\displaystyle
2q^{14}+2q^{20}+2q^{24}+2q^{26}+2q^{32}+6q^{36}+4q^{40}+4q^{42}+2q^{44}+4q^{48}$
$\displaystyle+$ $\displaystyle
10q^{56}+6q^{60}+4q^{62}++4q^{68}+4q^{70}+8q^{72}+8q^{76}+10q^{80}+4q^{82}+4q^{86}$
$\displaystyle+$ $\displaystyle
8q^{88}+4q^{92}+4q^{96}+6q^{98}+8q^{100}+(A_{1}+1)(q^{50}+q^{54}+q^{62}+q^{74}+q^{90})+\dots$

For $\mu_{2}=(1,0)$, we have:

$\displaystyle{}\Phi_{{\boldsymbol{\mu}}}$ $\displaystyle=$
$\displaystyle-2q^{29/4}(1+q^{4}+q^{12}+q^{17}+q^{18}+q^{19}+2q^{21}+q^{25}+q^{31}+q^{37}+2q^{40}+q^{42}+q^{43}$
$\displaystyle+$ $\displaystyle
2q^{45}+q^{48}+2q^{49}+3q^{50}+q^{52}+q^{53}+2q^{54}+q^{55}+q^{56}+q^{57}+2q^{60}$
$\displaystyle+$ $\displaystyle
q^{66}+2^{71}+2q^{74}+q^{75}+q^{76}+3q^{79}+q^{80}+2q^{81}+q^{82}+3q^{84}+q^{85}+q^{86}$
$\displaystyle+$ $\displaystyle 3q^{87}+2q^{90}+3q^{91}+3q^{92})+\dots$

For $\mu_{2}=(1,1)$, we have:

$\displaystyle{}\Phi_{{\boldsymbol{\mu}}}$ $\displaystyle=$
$\displaystyle-2q^{83/4}\left(1+q^{3}+q^{5}+q^{8}+q^{9}+q^{10}+q^{13}+q^{16}+q^{22}+q^{23}+q^{24}+q^{26}+3q^{31}\right.$
$\displaystyle+$
$\displaystyle\left.2q^{33}+q^{34}+q^{35}+q^{36}+2q^{38}+q^{39}+q^{40}+q^{41}+2q^{42}+q^{44}+q^{45}+q^{46}+q^{49}+2q^{51}\right.$
$\displaystyle+$
$\displaystyle\left.2q^{54}+q^{57}+q^{58}+2q^{59}+2q^{60}+2q^{61}+q^{68}+q^{69}+q^{70}+2q^{71}+4q^{72}+3q^{72}+q^{74}\right.$
$\displaystyle+$
$\displaystyle\left.+2q^{75}+q^{76}+q^{77}+3q^{78}+q^{79}\right)+\dots$

For $\mu_{2}=(1,2)$, we have:

$\displaystyle{}\Phi_{{\boldsymbol{\mu}}}$ $\displaystyle=$
$\displaystyle-2q^{45/4}(1+2q^{8}+q^{15}+q^{18}+q^{21}+q^{23}+2q^{27}+q^{28}+q^{29}+q^{32}+2q^{35}+q^{39}+q^{41}$
$\displaystyle+$ $\displaystyle
2q^{43}+q^{44}+q^{46}+2q^{48}+q^{50}+q^{52}+q^{54}+q^{56}+3q^{57}+q^{58}+2q^{62}+3q^{63}+3q^{65}$
$\displaystyle+$ $\displaystyle
q^{66}+q^{68}+q^{70}+2q^{72}+q^{73}+2q^{74}+q^{75}+3q^{80}+q^{82}+2q^{85}+q^{86}+q^{87}+q^{88})+\dots$

For $\mu_{2}=(1,3)$, we have:

$\displaystyle{}\Phi_{{\boldsymbol{\mu}}}$ $\displaystyle=$
$\displaystyle-2q^{15/4}(1+q+q^{10}+q^{13}+q^{17}+q^{18}+q^{19}+q^{21}+q^{22}+2q^{32}+q^{35}+q^{39}$
$\displaystyle+$ $\displaystyle
2q^{40}+q^{42}+q^{45}+q^{47}+q^{50}+2q^{51}+q^{53}+3q^{54}+q^{56}+2q^{59}+q^{63}+q^{65}+2q^{66}$
$\displaystyle+$ $\displaystyle
2q^{68}+q^{71}+q^{72}+q^{74}+q^{75}+q^{76}+3q^{77}+q^{79}+2q^{80}+2q^{83}+q^{84}+2q^{85}+q^{86}$
$\displaystyle+$ $\displaystyle
q^{87}+q^{88}+2q^{90}+2q^{92}+3q^{93}+2q^{95})+\dots$

For $\mu_{2}=(2,0)$, we have:

$\displaystyle{}\Phi_{{\boldsymbol{\mu}}}$ $\displaystyle=$ $\displaystyle
2(q^{9}+q^{15}+q^{19}+2q^{27}+2q^{29}+q^{33}+q^{35}+q^{43}+4q^{45}+4q^{53}+2q^{55}+q^{57}+q^{59}$
$\displaystyle+$ $\displaystyle
2q^{61}+2q^{63}+3q^{65}+q^{67}+2q^{71}+q^{75}+4q^{77}+2q^{81}+3q^{83}+2q^{87}+q^{89}+5q^{91}$
$\displaystyle+$ $\displaystyle q^{93}+6q^{95}+q^{97}+2q^{99})+\dots$

For $\mu_{2}=(2,1)$, we have:

$\displaystyle{}\Phi_{{\boldsymbol{\mu}}}$ $\displaystyle=$ $\displaystyle
2q^{25}\left(2+2q+q^{2}+q^{4}+2q^{15}+q^{16}+2q^{18}+q^{22}+2q^{25}+2q^{29}+4q^{32}+3q^{34}+4q^{37}\right.$
$\displaystyle+$
$\displaystyle\left.2q^{39}+q^{40}+2q^{41}+q^{44}+q^{46}+2q^{51}+2q^{55}+q^{56}+2q^{59}+4q^{61}+3q^{62}+2q^{65}+2q^{66}\right.$
$\displaystyle+$
$\displaystyle\left.2q^{68}+4q^{70}+2q^{73}+2q^{74}+2q^{75}\right)+\dots$

For $\mu_{2}=(2,3)$, we have:

$\displaystyle{}\Phi_{{\boldsymbol{\mu}}}$ $\displaystyle=$ $\displaystyle
2q^{5}\left(1+q^{6}+q^{9}+q^{15}+q^{19}+2q^{20}+q^{27}+q^{28}+q^{33}+2q^{34}+q^{37}+q^{40}+q^{42}+2q^{47}+3q^{51}\right.$
$\displaystyle+$
$\displaystyle\left.q^{54}+2q^{55}+2q^{56}+q^{60}+q^{63}+2q^{64}+q^{66}+q^{69}+2q^{70}+4q^{71}+2q^{73}+q^{75}+2q^{78}+3q^{81}\right.$
$\displaystyle+$
$\displaystyle\left.2q^{84}+2q^{85}+2q^{88}+4q^{90}+3q^{91}+2q^{92}+3q^{94}\right)+(A_{3}+1)\;q^{2}(1+q^{16}+q^{48}+q^{96})+\dots$

### List of $\Psi_{{\boldsymbol{\mu}}}$

Similarly to the $\Phi_{{\boldsymbol{\mu}}}$, we list the $q$-expansion of
$\Psi_{\boldsymbol{\mu}}$ for a various choices of ${\boldsymbol{\mu}}$. In
these examples, we keep $\mu=\mu_{1}=0$ and $\mu_{3}=-\mu_{2}$, and list
$\Psi_{\boldsymbol{\mu}}$ for 16 different values of $\mu_{2}$. Out of these
16 only 10 give rise to different $q$-series. The factor
$(-1)^{(a+b+c)}=(-1)^{\mu_{11}-\mu_{21}}=(-1)^{\mu_{21}}$ fixes the overall
sign of the whole series. Comparison of the $\Phi_{{\boldsymbol{\mu}}}$ and
$\Psi_{{\boldsymbol{\mu}}}$ demonstrates that for small powers of $q$, scaling
solutions can be present kinematically, but that the number of bound states
vanishes quantum mechanically. We again observe a moderate, polynomial growth
of the coefficients, as expected for a (mock) modular form of weight 4.

For $\mu_{2}=(0,0)$, we have:

$\displaystyle{}\Psi_{{\boldsymbol{\mu}}}$ $\displaystyle=$ $\displaystyle
16q^{30}\left(1+2q^{22}+4q^{34}+q^{40}+2q^{60}+8q^{62}+2q^{64}+4q^{70}\right)+\dots$

For $\mu_{2}=(0,1)$, we have:

$\displaystyle{}\Psi_{{\boldsymbol{\mu}}}$ $\displaystyle=$ $\displaystyle
2q^{17}\left(1+q^{5}+q^{8}+q^{11}+q^{16}+q^{23}+q^{26}+q^{28}+4q^{30}+q^{33}+q^{34}+9q^{37}+5q^{39}+q^{40}\right.$
$\displaystyle+$
$\displaystyle\left.q^{41}+q^{42}+4q^{43}+4q^{45}+4q^{48}+q^{49}+9q^{51}+q^{53}+q^{57}+6q^{58}+q^{59}+q^{61}+q^{63}+9q^{66}\right.$
$\displaystyle+$
$\displaystyle\left.4q^{67}+16q^{69}+q^{71}+5q^{72}+16q^{74}+q^{75}+13q^{76}+26q^{78}+q^{79}+q^{80}+13q^{81}+q^{82}+q^{83}\right)$
$\displaystyle+\dots$

For $\mu_{2}=(0,2)$, we have:

$\displaystyle{}\Psi_{{\boldsymbol{\mu}}}$ $\displaystyle=$ $\displaystyle
16q^{36}(1+q^{4}+q^{12}+2q^{20}+q^{24}+q^{32}+q^{34}+2q^{36}+2q^{40}+q^{44}+q^{46}+2q^{52}+4q^{56}$
$\displaystyle+$ $\displaystyle 4q^{62}+2q^{64})+\dots$

For $\mu_{2}=(1,0)$, we have:

$\displaystyle{}\Psi_{{\boldsymbol{\mu}}}$ $\displaystyle=$
$\displaystyle-\frac{q^{29/4}}{2}\left(1+q^{4}+q^{12}+q^{17}+q^{18}+q^{19}+10q^{21}+9q^{25}+q^{31}+q^{37}+10q^{40}+q^{42}+q^{43}\right.$
$\displaystyle+$
$\displaystyle\left.34q^{45}+q^{48}+2q^{49}+27q^{50}+q^{52}+q^{53}+50q^{54}+q^{55}+q^{56}+q^{57}+58q^{60}+9q^{66}+2q^{71}\right.$
$\displaystyle+$
$\displaystyle\left.q^{74}+9q^{75}+q^{78}+51q^{79}+25q^{80}+10q^{81}+q^{82}+107q^{84}+25q^{85}+49q^{86}+43q^{87}+26q^{90}\right.$
$\displaystyle+$ $\displaystyle\left.107q^{91}+51q^{92}\right)+\dots$

For $\mu_{2}=(1,1)$, we have:

$\displaystyle{}\Psi_{{\boldsymbol{\mu}}}$ $\displaystyle=$
$\displaystyle-\frac{q^{83/4}}{2}\left(9+9q^{3}+q^{5}+q^{8}+q^{9}+q^{10}+9q^{13}+q^{16}+9q^{22}+q^{23}+9q^{24}+q^{26}+19q^{31}\right.$
$\displaystyle+$
$\displaystyle\left.34q^{33}+9q^{34}+25q^{35}+9q^{36}+26q^{38}+9q^{39}+25q^{40}+q^{41}+10q^{42}+25q^{44}+9q^{45}\right.$
$\displaystyle+$
$\displaystyle\left.25q^{46}+q^{49}+2q^{51}+10q^{54}+9q^{57}+9q^{58}+2q^{59}+18q^{60}+34q^{61}+q^{68}+9q^{69}+49q^{70}\right.$
$\displaystyle+$
$\displaystyle\left.74q^{71}+140q^{72}+131q^{73}+q^{74}+18q^{75}+q^{76}+49q^{77}+43q^{78}+25q^{79}\right)+\dots$

For $\mu_{2}=(1,2)$, we have:

$\displaystyle{}\Psi_{{\boldsymbol{\mu}}}$ $\displaystyle=$
$\displaystyle-\frac{q^{45/4}}{2}\left(1+2q^{8}+q^{15}+q^{18}+9q^{21}+9q^{23}+2q^{27}+q^{28}+9q^{29}+q^{32}+18q^{35}+9q^{39}\right.$
$\displaystyle+$
$\displaystyle\left.9q^{41}+26q^{43}+9q^{44}+25q^{46}+10q^{47}+9q^{50}+q^{52}+9q^{54}+9q^{56}+19q^{57}+25q^{58}+10q^{62}\right.$
$\displaystyle+$
$\displaystyle\left.75q^{64}+43q^{65}+9q^{66}+9q^{68}+q^{70}+10q^{72}+q^{73}+18q^{74}+q^{75}+131q^{80}+49q^{82}\right.$
$\displaystyle+$
$\displaystyle\left.50q^{85}+44q^{86}+49q^{87}+25q^{88}\right)+\dots$

For $\mu_{2}=(1,3)$, we have:

$\displaystyle{}\Psi_{{\boldsymbol{\mu}}}$ $\displaystyle=$
$\displaystyle-\frac{q^{15/4}}{2}\left(1+q+q^{10}+q^{13}+q^{17}+q^{18}+q^{19}+9q^{21}+q^{22}+10q^{32}+q^{35}+25q^{39}+2q^{40}\right.$
$\displaystyle+$
$\displaystyle\left.q^{42}+q^{45}+q^{47}+9q^{50}+50q^{51}+q^{53}+27q^{54}+9q^{56}+10q^{59}+q^{63}+9q^{65}+10q^{66}\right.$
$\displaystyle+$
$\displaystyle\left.50q^{68}+q^{71}+25q^{72}+q^{74}+9q^{75}+q^{76}+83q^{77}+q^{79}+10q^{80}+10q^{83}+25q^{84}\right.$
$\displaystyle+$
$\displaystyle\left.26q^{85}+49q^{86}+q^{87}+q^{88}+26q^{90}+10q^{92}+139q^{93}+34q^{95}\right)+\dots$

For $\mu_{2}=(2,0)$, we have:

$\displaystyle{}\Psi_{{\boldsymbol{\mu}}}$ $\displaystyle=$ $\displaystyle
2q^{9}(1+q^{6}+q^{10}+2q^{18}+2q^{20}+q^{24}+q^{26}+q^{34}+4q^{36}+12q^{44}+2q^{46}+q^{48}+9q^{50}+2q^{52}$
$\displaystyle+$ $\displaystyle
10q^{54}+3q^{56}+2q^{58}+10q^{62}+q^{66}+4q^{68}+18q^{72}+11q^{74}+2q^{78}+q^{80}+29q^{82}+q^{84}$
$\displaystyle+$ $\displaystyle 46q^{86}+q^{88}+10q^{90})+\dots$

For $\mu_{2}=(2,1)$, we have:

$\displaystyle{}\Psi_{{\boldsymbol{\mu}}}$ $\displaystyle=$ $\displaystyle
4q^{25}\left(1+q+q^{15}+q^{18}+4q^{25}+4q^{29}+5q^{32}+4q^{34}+8q^{37}+q^{39}+q^{41}+q^{51}+4q^{55}\right.$
$\displaystyle+$
$\displaystyle\left.4q^{59}+5q^{61}+4q^{62}+4q^{65}+q^{66}+16q^{68}+16q^{70}+9q^{73}+9q^{74}+4q^{75}\right)+\dots$

For $\mu_{2}=(2,3)$, we have:

$\displaystyle{}\Psi_{{\boldsymbol{\mu}}}$ $\displaystyle=$ $\displaystyle
4\left(q^{25}+q^{39}+q^{52}+9q^{56}+q^{61}+4q^{69}+q^{75}+17q^{76}+4q^{83}+9q^{89}+q^{90}+q^{93}\right.$
$\displaystyle+$
$\displaystyle\left.+5q^{95}+q^{96}+4q^{97}+25q^{99}\right)+\dots$

###  6.2 Charge configuration 2: $P_{1}=(1,0),P_{2}=P_{3}=(0,1)$

In this example the total magnetic charge is $P=(1,2)$ split into three
different centers as $P_{1}=(1,0),P_{2}=P_{3}=(0,1)$. The charge $P_{1}$
corresponds to the K3 fiber. The CFT central charge (183) of $P_{2,3}$ is
given above, while we have for $P_{1}$ and $P$,

$c_{L}(P_{1})=24,\qquad c_{L}(P)=176.$ (204)

In the $j$-th center, the quadratic form $D_{j}=d_{abc}P^{c}_{j}$ is given by

$\displaystyle D_{2}=D_{3}=\begin{pmatrix}0&4\\\ 4&2\end{pmatrix},\quad
D_{1}=\begin{pmatrix}0&0\\\ 0&4\end{pmatrix}.$ (205)

Note that $D_{1}$ is a singular matrix, and we only keep the direction with
the non-vanishing eigenvalue. The partition functions for $P_{1}$ can be
determined using Noether-Lefshetz theory [19, 21, 76]. On the other hand, the
modular properties are in this case very restrictive, and determine the
partition function with a little further inpute. First, the lowest exponent of
$h_{P,\mu}$ is $-c_{L}/24=-1$. Therefore, each of $h_{P,\mu}$ has a term with
a negative coefficient, and the vector space of weakly holomorphic vector-
valued modular forms is 3-dimensional. We can fix the coefficients following
[13, 77]: the coefficient for $\mu=0$ is the Euler number of the linear system
of the divisor, which is $c_{2}\cdot P_{1}/12=2$. Thus the linear system is a
$\mathbb{P}^{1}$. For $\mu=1$ and 2, we need to require that the divisor
contains a given curve class $\beta\in H^{2}(X,\mathbb{Z})$, and turn on a
world volume flux through this curve. This gives two complex constraints,
however since the linear system is a $\mathbb{P}^{1}$, there are generically
no solutions. As a result, the coefficients of the potential polar terms in
$h_{P_{1},1}$ and $h_{P_{1},2}$ vanish. This fixes $h_{P_{1},\mu}$ to be the
weight -3/2 modular form,

$h_{P_{1},\mu}(\tau)=\frac{2\,f_{P_{1},\mu}(\tau)}{\eta(\tau)^{24}},$ (206)

with

$\begin{split}f_{P_{1},\mu}(\tau)&=\frac{11}{18}E_{4}(\tau)\,E_{6}(\tau)\,\Theta_{\mu}(\tau)\\\
&\quad-6\,E_{4}(\tau)^{2}\,D_{1/2}\Theta_{\mu}(\tau)\\\
&\quad+16\,E_{6}(\tau)\,D_{5/2}D_{1/2}\Theta_{\mu}(\tau),\end{split}$ (207)

with $\Theta_{\mu}(\tau)=\sum_{n\in 4\mathbb{Z}+\mu}q^{n^{2}/8}$, $D_{k}$ the
modular derivative acting on weight $k$ modular forms, $D_{k}=\frac{1}{2\pi
i}\frac{d}{d\tau}-\frac{k}{12}E_{2}(\tau)$, and $E_{\ell}$ the Eisenstein
series of weight $\ell$. The $f_{P_{1},\mu}$ also appeared in the context of
Gromov-Witten theory [78, 76]. Finally since $P_{1}^{3}=0$, $h_{nP_{1},\mu}$
is also expected to be a weakly holomorphic modular form (rather than a mock
modular form), and given in terms of Hecke transformations of (206) [40, 21].

Returning to our original problem of $\Phi_{\boldsymbol{\mu}}$ and
$\Psi_{\boldsymbol{\mu}}$, note that we have for the products of quadratic
forms $D_{1}D_{3}^{-1}=\left(\begin{array}[]{cc}0&0\\\ 1&0\\\
\end{array}\right)$, $D_{1}D_{3}^{-1}D_{1}=\begin{pmatrix}0&0\\\
0&0\end{pmatrix}$, $D_{2}D_{3}^{-1}D_{1}=D_{1}D_{3}^{-1}D_{2}=D_{1}$ which
produces the quadratic form (98),

$\displaystyle\underline{D}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}[c]0&0&0&0\\\ 0&4&0&4\\\ 0&0&0&8\\\
0&4&8&4\end{pmatrix}.$ (208)

Excluding the null direction the determinant of $\underline{D}=-256$. The
quadratic form on the lattice of electric charge lattice is given by the
pseudo-inverse of $\underline{D}$ as follows:

$\displaystyle\underline{D}^{-1}=\left(\begin{array}[]{cccc}0&0&0&0\\\
0&\frac{1}{4}&-\frac{1}{8}&0\\\ 0&-\frac{1}{8}&0&\frac{1}{8}\\\
0&0&\frac{1}{8}&0\\\ \end{array}\right).$ (213)

Electric charge vectors $Q_{j}\in\Lambda_{j}^{*}$ have the following form

$Q_{1}=\begin{pmatrix}0\\\ q_{12}\end{pmatrix},\quad
Q_{2}=\begin{pmatrix}q_{21}\\\ q_{22}\end{pmatrix}.$

There are 64 conjugacy classes of the form

$\displaystyle q_{12}$ $\displaystyle=$ $\displaystyle 4k_{12}+\mu_{12},$
(214) $\displaystyle{}q_{21}$ $\displaystyle=$ $\displaystyle
4k_{22}+\mu_{21},\quad q_{22}=4k_{21}+2k_{22}+\mu_{22},$

where $\mu_{ij}\in\\{0,1,2,3\\}$, $k_{ij}\in\mathbb{Z}$. These are $N_{q}=64$
(100) conjugacy classes for fixed $\mu$.

Using equations in (104), we get

$\displaystyle{\boldsymbol{Q}}_{1}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}0\\\
4k_{12}+4k_{22}+\mu_{12}-\mu_{31}\end{pmatrix},$ (215)
$\displaystyle{}{\boldsymbol{Q}}_{2}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}8k_{22}+\mu_{21}-\mu_{31}\\\
8k_{21}+4k_{22}+4k_{12}+\mu_{22}-\mu_{32}\end{pmatrix}.$

When $\mu_{1}+\mu_{2}+\mu_{3}=0$, we have,

$\displaystyle{}{\boldsymbol{Q}}_{1}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}0\\\
4k_{12}+4k_{22}+\mu_{12}+\mu_{21}\end{pmatrix},\quad{\boldsymbol{\mu}}_{1}=\begin{pmatrix}0\\\
{}\mu_{12}+\mu_{21}\end{pmatrix},$ $\displaystyle{\boldsymbol{Q}}_{2}$
$\displaystyle=$ $\displaystyle\begin{pmatrix}8k_{22}+2\mu_{21}\\\
8k_{21}+4k_{22}+4k_{12}+2\mu_{22}+\mu_{12}\end{pmatrix},\quad{\boldsymbol{\mu}}_{2}=\begin{pmatrix}2\mu_{21}\\\
{}2\mu_{22}+\mu_{12}\end{pmatrix}.$

### List of $\Phi_{{\boldsymbol{\mu}}}$

We list $\Phi_{\boldsymbol{\mu}}$ for a various choices of
${\boldsymbol{\mu}}$. For this choice of charges, we find that some of the
inequalities for convergence are sometimes saturated. This however do not give
rise to divergences in the $q$ series. As in explained in Section 4.1,
${\boldsymbol{\mu}}$ is fully determined by specifying
$(\mu_{1},\mu_{2},\mu_{3})$ and $\mu=\mu_{1}+\mu_{2}+\mu_{3}$. The overall
sign of the $q$-expansion is given by $(-1)^{a+b+c}=(-1)^{\mu_{12}}$. The
contributions $A_{\ell}$ are for this example

$A_{1}=A_{2}=A_{3}=\frac{2}{\pi}\,\arctan(-3/4),$ (216)

which is an irrational number.

For $\mu_{1}=\mu_{2}=0$, we have:

$\displaystyle{}\Phi_{{\boldsymbol{\mu}}}$ $\displaystyle=$ $\displaystyle
4q^{6}+4q^{24}+12q^{30}+6q^{40}+4q^{54}+12q^{64}+12q^{70}+12q^{78}+12q^{88}+4q^{96}$
$\displaystyle+\frac{(A_{1}+A_{2}+A_{3}+1)}{2}+(A_{1}+A_{2}+A_{3}+3)(q^{8}+q^{32}+q^{72})+\dots$

For $\mu_{1}=(0,0),\;\mu_{2}=(1,0)$, we have:

$\displaystyle{}\Phi_{{\boldsymbol{\mu}}}$ $\displaystyle=$ $\displaystyle
2q^{33/8}\left(1+q^{4}+q^{7}+q^{16}+2q^{21}+q^{22}+q^{24}+q^{30}+3q^{31}+q^{34}+q^{41}+q^{42}+q^{44}\right.$
$\displaystyle+$
$\displaystyle\left.2q^{53}+q^{54}+q^{56}+2q^{58}+q^{60}+2q^{66}+4q^{67}+q^{72}+2q^{74}+q^{76}+2q^{77}+2q^{81}\right.$
$\displaystyle+$
$\displaystyle\left.q^{82}+q^{84}+q^{91}+q^{92}+2q^{93}\right)+\frac{(A_{3}+1)}{2}\;q^{1/8}(1+q^{6}+q^{10}+q^{28}+q^{36}+q^{66}+q^{78})$

For $\mu_{1}=(0,0),\;\mu_{2}=(2,0)$, we have:

$\displaystyle{}\Phi_{{\boldsymbol{\mu}}}$ $\displaystyle=$ $\displaystyle
2\left(\frac{1}{2}q^{5/2}+q^{21/2}+q^{29/2}+q^{33/2}+q^{41/2}+\frac{3}{2}q^{45/2}+q^{61/2}+q^{65/2}+q^{77/2}+2q^{81/2}+q^{85/2}\right.$
$\displaystyle+$
$\displaystyle\left.q^{89/2}+3q^{101/2}+2q^{105/2}+q^{109/2}+q^{117/2}+q^{121/2}+\frac{1}{2}q^{125/2}+2q^{129/2}+q^{133/2}+q^{145/2}\right.$
$\displaystyle+$
$\displaystyle\left.q^{149/2}+q^{153/2}+2q^{157/2}+q^{161/2}+q^{165/2}+2q^{173/2}+q^{181/2}+4q^{185/2}+2q^{189/2}\right)$
$\displaystyle+q^{1/2}\;\frac{(A_{3}+1)}{2}(1+q^{4}+q^{12}+q^{24}+q^{40})+\dots$

For $\mu_{1}=(0,0),\;\mu_{2}=(3,0)$, we have:

$\displaystyle{}\Phi_{{\boldsymbol{\mu}}}$ $\displaystyle=$ $\displaystyle
2q^{105/8}\left(2+q^{5}+q^{6}+q^{13}+q^{14}+2q^{24}+q^{30}+2q^{31}+2q^{33}+q^{34}+q^{38}+q^{40}+q^{43}+2q^{45}\right.$
$\displaystyle+$
$\displaystyle\left.q^{46}+q^{52}+3q^{55}+2q^{60}+q^{70}+2q^{71}+2q^{73}+q^{74}+q^{76}+2q^{78}+2q^{82}+q^{84}+q^{86}\right)$
$\displaystyle+\frac{(A_{3}+1)}{2}q^{9/8}(1+q^{2}+q^{14}+q^{20}+q^{44}+q^{54}+q^{90})+\dots$

For $\mu_{1}=(0,0),\;\mu_{2}=(0,1)$, we have:

$\displaystyle{}\Phi_{{\boldsymbol{\mu}}}$ $\displaystyle=$ $\displaystyle
2\left(q^{4}+q^{8}+q^{12}+q^{20}+q^{24}+2q^{26}+q^{28}+2q^{34}+2q^{36}+q^{40}+2q^{46}+q^{48}+q^{56}+2q^{58}\right.$
$\displaystyle+$
$\displaystyle\left.2q^{60}+2q^{64}+4q^{70}+2q^{72}+2q^{76}+q^{80}+2q^{82}+2q^{84}+2q^{86}+q^{88}+2q^{96}+2q^{98}+q^{100}\right)+\dots$

For $\mu_{1}=(0,0),\;\mu_{2}=(0,2)$, we have:

$\displaystyle{}\Phi_{{\boldsymbol{\mu}}}$ $\displaystyle=$ $\displaystyle
2\left(q^{10}+2q^{16}+2q^{22}+2q^{32}+2q^{38}+2q^{42}+2q^{48}+4q^{52}+2q^{58}+2q^{62}+2q^{66}+4q^{76}\right.$
$\displaystyle+$
$\displaystyle\left.2q^{80}+2q^{82}+3q^{90}+4q^{94}+2q^{96}\right)+q^{2}\;(A_{3}+1)(1+q^{16}+q^{48}+q^{96})+\dots$

For $\mu_{1}=(0,1),\;\mu_{2}=(1,0)$, we have:

$\displaystyle{}\Phi_{{\boldsymbol{\mu}}}$ $\displaystyle=$
$\displaystyle-2q^{19/4}\left(1+q^{2}+q^{6}+q^{17}+2q^{21}+q^{23}+q^{24}+q^{26}+q^{27}+q^{29}+q^{33}+q^{34}+q^{38}+q^{39}\right.$
$\displaystyle+$
$\displaystyle\left.q^{46}+q^{52}+q^{54}+q^{56}+q^{57}+q^{58}+q^{61}+2q^{62}+2q^{64}+q^{66}+2q^{68}+q^{69}+q^{71}+q^{72}\right.$
$\displaystyle+$
$\displaystyle\left.q^{75}+q^{76}+q^{77}+q^{78}+q^{81}+q^{82}+q^{86}+q^{87}+q^{88}+q^{89}+q^{90}+q^{95}\right)+\dots$

For $\mu_{1}=(0,1),\;\mu_{2}=(2,0)$, we have:

$\displaystyle{}\Phi_{{\boldsymbol{\mu}}}$ $\displaystyle=$
$\displaystyle-2q^{71/8}\left(1+q^{5}+q^{9}+q^{12}+q^{16}+q^{21}+q^{25}+q^{26}+q^{28}+q^{30}+q^{35}+q^{36}+q^{39}+q^{41}+q^{43}\right.$
$\displaystyle+$
$\displaystyle\left.q^{45}+q^{47}+q^{48}+q^{50}+2q^{54}+q^{58}+q^{60}+q^{63}+q^{65}+q^{67}+q^{68}+q^{69}+q^{71}+q^{72}+q^{75}\right.$
$\displaystyle+$
$\displaystyle\left.q^{76}+q^{78}+q^{80}+q^{81}+q^{82}+q^{86}+q^{88}+q^{89}+q^{90}\right)+\dots$

For $\mu_{1}=(0,0),\;\mu_{2}=(1,1)$, we have:

$\displaystyle{}\Phi_{{\boldsymbol{\mu}}}$ $\displaystyle=$ $\displaystyle
q^{45/8}\left(3+2q^{4}+q^{10}+4q^{18}+2q^{22}+2q^{23}+4q^{25}+2q^{26}+2q^{30}+4q^{35}+2q^{36}+2q^{43}+4q^{48}\right.$
$\displaystyle+$
$\displaystyle\left.2q^{56}+4q^{57}+4q^{59}+4q^{60}+2q^{62}+4q^{66}+2q^{68}+q^{70}+4q^{71}+2q^{72}+4q^{73}+2q^{74}\right.$
$\displaystyle+$
$\displaystyle\left.4q^{82}+8q^{85}+4q^{90}+2q^{92}\right)+\dots$

For $\mu_{1}=(0,0),\;\mu_{2}=(1,3)$, we have:

$\displaystyle{}\Phi_{{\boldsymbol{\mu}}}$ $\displaystyle=$ $\displaystyle
q^{5/8}\left(1+2q^{2}+2q^{10}+2q^{13}+2q^{14}+2q^{16}+2q^{20}+4q^{21}+q^{30}+2q^{32}+4q^{39}+2q^{40}+4q^{42}\right.$
$\displaystyle+$
$\displaystyle\left.2q^{44}+3q^{50}+8q^{51}+2q^{54}+4q^{56}+4q^{63}+2q^{64}+2q^{66}+2q^{73}+4q^{77}+2q^{78}\right.$
$\displaystyle+$
$\displaystyle\left.2q^{80}+4q^{84}+2q^{86}+2q^{87}+2q^{90}+4q^{92}+8q^{93}\right)+\dots$

For $\mu_{1}=(0,0),\;\mu_{2}=(2,1)$, we have:

$\displaystyle{}\Phi_{{\boldsymbol{\mu}}}$ $\displaystyle=$ $\displaystyle
4q^{15/2}\left(1+q^{12}+q^{20}+q^{24}+2q^{28}+q^{40}+2q^{48}+q^{52}+q^{60}+3q^{64}+2q^{72}+2q^{76}\right.$
$\displaystyle\left.+q^{80}+2q^{92}\right)+\dots$

For $\mu_{1}=(0,0),\;\mu_{2}=(2,3)$, we have:

$\displaystyle{}\Phi_{{\boldsymbol{\mu}}}$ $\displaystyle=$ $\displaystyle
4q^{3/2}\left(1+q^{12}+3q^{16}+q^{36}+3q^{44}+3q^{48}+3q^{56}+q^{72}+3q^{84}+3q^{92}+3q^{96}\right)+\dots$

For $\mu_{1}=(0,1),\;\mu_{2}=(0,0)$, we have:

$\displaystyle{}\Phi_{{\boldsymbol{\mu}}}$ $\displaystyle=$
$\displaystyle-2q^{39/8}\left(1+q^{2}+q^{3}+q^{17}+q^{21}+2q^{22}+q^{24}+q^{26}+q^{27}+2q^{28}+2q^{33}+q^{39}+q^{46}+q^{52}\right.$
$\displaystyle+$
$\displaystyle\left.2q^{55}+q^{57}+2q^{60}+q^{61}+2q^{63}+q^{64}+q^{66}+q^{67}+q^{68}+4q^{70}+2q^{76}+q^{78}+2q^{79}+2q^{81}\right.$
$\displaystyle+$ $\displaystyle\left.q^{87}+2q^{89}+q^{95}\right)+\dots$

For $\mu_{1}=(0,1),\;\mu_{2}=(1,1)$, we have:

$\displaystyle{}\Phi_{{\boldsymbol{\mu}}}$ $\displaystyle=$
$\displaystyle-2q^{45/4}\left(1+q^{3}+q^{6}+q^{10}+q^{14}+q^{18}+q^{23}+2q^{28}+q^{31}+q^{33}+q^{34}+q^{36}+2q^{39}+q^{42}\right.$
$\displaystyle+$
$\displaystyle\left.q^{45}+q^{46}+q^{49}+q^{51}+q^{53}+q^{55}+q^{58}+q^{59}+q^{62}+q^{65}+q^{67}+q^{70}+2q^{72}+q^{75}+q^{77}\right.$
$\displaystyle+$
$\displaystyle\left.q^{78}+q^{80}+q^{81}+q^{82}+q^{83}+2q^{85}+2q^{88}\right)\dots$

For $\mu_{1}=(0,1),\;\mu_{2}=(1,2)$, we have:

$\displaystyle{}\Phi_{{\boldsymbol{\mu}}}$ $\displaystyle=$
$\displaystyle-2q^{47/4}\left(1+q^{3}+q^{5}+q^{9}+q^{13}+q^{17}+q^{23}+q^{28}+q^{29}+q^{31}+q^{32}+q^{35}+q^{36}+q^{37}+q^{38}\right.$
$\displaystyle+$
$\displaystyle\left.q^{42}+2q^{44}+q^{49}+q^{50}+q^{52}+q^{53}+2q^{58}+q^{60}+q^{65}+q^{68}+q^{70}+q^{71}+q^{74}+q^{76}\right.$
$\displaystyle+$
$\displaystyle\left.q^{77}+q^{78}+q^{79}+q^{80}+q^{83}+q^{84}+3q^{85}+q^{87}\right)+\dots$

For $\mu_{1}=(0,1),\;\mu_{2}=(1,3)$, we have:

$\displaystyle{}\Phi_{{\boldsymbol{\mu}}}$ $\displaystyle=$
$\displaystyle-2q^{13/4}\left(1+q^{6}+q^{10}+q^{15}+q^{19}+q^{20}+q^{21}+q^{27}+q^{30}+q^{33}+q^{34}+q^{35}+q^{39}+q^{42}+q^{45}\right.$
$\displaystyle+$
$\displaystyle\left.q^{46}+q^{50}+q^{51}+q^{52}+q^{54}+q^{56}+q^{58}+q^{60}+q^{63}+q^{64}+q^{65}+q^{70}+q^{71}+q^{72}+q^{74}\right.$
$\displaystyle+$
$\displaystyle\left.q^{75}+q^{76}+q^{78}+q^{80}+q^{81}+q^{84}+q^{85}+q^{86}+q^{87}+q^{93}+q^{94}+q^{95}+q^{96}\right)+\dots$

For $\mu_{1}=(0,1),\;\mu_{2}=(2,2)$, we have:

$\displaystyle{}\Phi_{{\boldsymbol{\mu}}}$ $\displaystyle=$
$\displaystyle-2q^{15/8}\left(1+q^{10}+q^{13}+q^{15}+q^{16}+2q^{18}+q^{27}+q^{33}+q^{38}+2q^{40}+q^{42}+q^{45}+3q^{47}\right.$
$\displaystyle+$
$\displaystyle\left.q^{48}+2q^{52}+2q^{53}+2q^{58}+q^{60}+q^{68}+2q^{69}+q^{75}+2q^{79}+q^{81}+3q^{86}+2q^{88}+q^{90}\right.$
$\displaystyle+$
$\displaystyle\left.q^{93}+q^{94}+q^{95}+3q^{96}+2q^{97}\right)+\dots$

For $\mu_{1}=(0,2),\;\mu_{2}=(0,1)$, we have:

$\displaystyle{}\Phi_{{\boldsymbol{\mu}}}$ $\displaystyle=$ $\displaystyle
2q^{11/2}\left(1+q^{4}+q^{10}+2q^{18}+q^{22}+q^{24}+2q^{26}+q^{30}+q^{34}+2q^{36}+q^{42}+q^{44}\right.$
$\displaystyle+$
$\displaystyle\left.2q^{48}+2q^{56}+2q^{58}+3q^{60}+q^{64}+2q^{68}+q^{70}+3q^{72}+2q^{74}+q^{80}+3q^{84}+2q^{86}\right.$
$\displaystyle+$ $\displaystyle\left.3q^{90}+q^{94}\right)+\dots$

For $\mu_{1}=(0,2),\;\mu_{2}=(1,0)$, we have:

$\displaystyle{}\Phi_{{\boldsymbol{\mu}}}$ $\displaystyle=$ $\displaystyle
2q^{41/8}\left(1+2q^{5}+q^{10}+2q^{11}+4q^{18}+q^{21}+q^{25}+2q^{26}+2q^{27}+2q^{32}+2q^{35}+2q^{36}\right.$
$\displaystyle+$
$\displaystyle\left.2q^{37}+q^{41}+2q^{44}+3q^{45}+4q^{48}+2q^{55}+2q^{56}+2q^{57}+q^{58}+2q^{59}+2q^{60}+2q^{61}\right.$
$\displaystyle+$
$\displaystyle\left.q^{66}+2q^{68}+3q^{70}+2q^{71}+2q^{73}+2q^{74}+4q^{75}+2q^{81}+2q^{83}+4q^{86}+4q^{87}+q^{88}\right.$
$\displaystyle+$ $\displaystyle\left.4q^{90}+2q^{93}\right)+\dots$

For $\mu_{1}=(0,2),\;\mu_{2}=(1,1)$, we have:

$\displaystyle{}\Phi_{{\boldsymbol{\mu}}}$ $\displaystyle=$ $\displaystyle
q^{29/8}\left(1+q^{4}+3q^{9}+q^{15}+2q^{16}+q^{19}+q^{22}+2q^{23}+2q^{24}+q^{29}+q^{30}+4q^{33}+2q^{38}\right.$
$\displaystyle+$
$\displaystyle\left.q^{39}+q^{40}+2q^{42}+2q^{43}+2q^{44}+q^{49}+2q^{51}+q^{52}+3q^{54}+2q^{55}+2q^{56}+q^{60}+2q^{62}\right.$
$\displaystyle+$
$\displaystyle\left.3q^{65}+4q^{66}+4q^{69}+q^{70}+q^{72}+2q^{78}+3q^{79}+2q^{80}+2q^{82}+4q^{83}+2q^{84}+q^{85}+q^{87}\right.$
$\displaystyle+$
$\displaystyle\left.2q^{91}+2q^{93}+q^{94}+4q^{95}+2q^{96}\right)+\dots$

For $\mu_{1}=(0,2),\;\mu_{2}=(2,0)$, we have:

$\displaystyle{}\Phi_{{\boldsymbol{\mu}}}$ $\displaystyle=$ $\displaystyle
4\left(q^{13}+q^{19}+q^{27}+q^{37}+q^{43}+q^{45}+q^{47}+q^{53}+q^{57}+q^{59}+q^{67}+q^{69}+q^{73}+q^{83}\right.$
$\displaystyle+$
$\displaystyle\left.q^{85}+q^{87}+q^{89}+q^{93}+q^{97}+q^{99}\right)+\dots$

### List of $\Psi_{{\boldsymbol{\mu}}}$

Similarly as the $\Phi_{{\boldsymbol{\mu}}}$ $q$-series we list the $q$-series
of $\Psi_{\boldsymbol{\mu}}$ for a various choices of ${\boldsymbol{\mu}}$. In
this example $(-1)^{a+b+c}=(-1)^{\mu_{12}}$. For $\mu_{1}=\mu_{2}=(0,0)$, we
have:

$\displaystyle{}\Psi_{{\boldsymbol{\mu}}}$ $\displaystyle=$ $\displaystyle
16(q^{6}+4q^{24}+3q^{30}+9q^{54}+12q^{64}+3q^{70}+3q^{78}+16q^{96})+\dots$

For $\mu_{1}=(0,0)\;\mu_{2}=(1,0)$, we have:

$\displaystyle{}\Psi_{{\boldsymbol{\mu}}}$ $\displaystyle=$ $\displaystyle
2q^{33/8}\left(1+4q^{4}+9q^{16}+2q^{21}+4q^{22}+16q^{24}+4q^{30}+8q^{31}+q^{42}+25q^{44}+18q^{53}+16q^{54}\right.$
$\displaystyle+$
$\displaystyle\left.36q^{56}+2q^{58}+4q^{60}+17q^{66}+40q^{67}+4q^{72}+8q^{74}+8q^{81}+9q^{82}+49q^{84}+2q^{93}\right)+\dots$

For $\mu_{1}=(0,0)\;\mu_{2}=(2,0)$, we have:

$\displaystyle{}\Psi_{{\boldsymbol{\mu}}}$ $\displaystyle=$ $\displaystyle
8q^{21/2}\left(1+q^{6}+q^{12}+4q^{22}+q^{28}+2q^{30}+4q^{32}+2q^{40}+5q^{42}+q^{48}+2q^{54}+9q^{56}+\right.$
$\displaystyle+$
$\displaystyle\left.4q^{66}+8q^{68}+9q^{70}+q^{72}+2q^{76}+10q^{82}+13q^{84}\right)+\dots$

For $\mu_{1}=(0,0)\;\mu_{2}=(3,0)$, we have:

$\displaystyle{}\Psi_{{\boldsymbol{\mu}}}$ $\displaystyle=$ $\displaystyle
2q^{105/8}\left(5+4q^{6}+q^{14}+25q^{24}+4q^{30}+2q^{31}+8q^{33}+16q^{34}+4q^{40}+8q^{45}+9q^{46}+2q^{55}\right.$
$\displaystyle+$
$\displaystyle\left.61q^{60}+16q^{70}+18q^{71}+32q^{73}+36q^{74}+4q^{76}+2q^{78}+8q^{82}+16q^{84}+q^{86}\right)+\dots$

For $\mu_{1}=(0,0)\;\mu_{2}=(0,1)$, we have:

$\displaystyle{}\Psi_{{\boldsymbol{\mu}}}$ $\displaystyle=$ $\displaystyle
2q^{4}\left(1+q^{4}+q^{8}+9q^{16}+q^{20}+2q^{22}+9q^{24}+2q^{30}+10q^{32}+q^{36}+2q^{42}+25q^{44}\right.$
$\displaystyle+$
$\displaystyle\left.9q^{52}+18q^{54}+26q^{56}+2q^{60}+20q^{66}+34q^{68}+2q^{72}+9q^{76}+2q{78}+2q^{80}+18q^{82}\right.$
$\displaystyle+$
$\displaystyle\left.49q^{84}+2q^{92}+2q^{94}+25q^{96}\right)+\dots$

For $\mu_{1}=(0,0)\;\mu_{2}=(0,2)$, we have:

$\displaystyle{}\Psi_{{\boldsymbol{\mu}}}$ $\displaystyle=$ $\displaystyle
16\left(q^{16}+q^{32}+4q^{42}+q^{48}+2q^{52}+4q^{66}+2q^{76}+9q^{80}+4q^{90}+8q^{94}+q^{96}\right)+\dots$

For $\mu_{1}=(0,0)\;\mu_{2}=(1,1)$, we have:

$\displaystyle{}\Psi_{{\boldsymbol{\mu}}}$ $\displaystyle=$ $\displaystyle
2q^{45/8}\left(1+q^{4}+13q^{18}+q^{22}+2q^{25}+9q^{26}+q^{30}+2q^{32}+4q^{36}+41q^{48}+9q^{56}+8q^{57}\right.$
$\displaystyle+$
$\displaystyle\left.18q^{59}+26q^{60}+2q^{66}+9q^{68}+2q^{71}+q^{72}+18q^{73}+16q^{74}+2q^{82}+10q^{85}+85q^{90}\right)+\dots$

For $\mu_{1}=(0,0)\;\mu_{2}=(1,3)$, we have:

$\displaystyle{}\Psi_{{\boldsymbol{\mu}}}$ $\displaystyle=$ $\displaystyle
2q^{21/8}\left(1+4q^{8}+q^{12}+9q^{14}+q^{18}+2q^{19}+16q^{30}+8q^{37}+9q^{38}+25q^{40}+q^{42}+9q^{48}\right.$
$\displaystyle+$
$\displaystyle\left.20q^{49}+q^{52}+2q^{54}+2q^{61}+4q^{62}+36q^{64}+32q^{75}+25q^{76}+49q^{78}+8q^{82}+9q^{84}+q^{88}\right.$
$\displaystyle\left.+29q^{90}+68q^{91}\right)\dots$

For $\mu_{1}=(0,0)\;\mu_{2}=(2,1)$, we have:

$\displaystyle{}\Psi_{{\boldsymbol{\mu}}}$ $\displaystyle=$ $\displaystyle
4q^{15/2}\left(1+q^{12}+9q^{20}+q^{24}+2q^{28}+9q^{40}+2q^{48}+25q^{52}+9q^{60}+19q^{64}+2q^{72}+2q^{76}\right.$
$\displaystyle+$ $\displaystyle\left.25q^{80}+18q^{92}\right)+\dots$

For $\mu_{1}=(0,0)\;\mu_{2}=(2,3)$, we have:

$\displaystyle{}\Psi_{{\boldsymbol{\mu}}}$ $\displaystyle=$ $\displaystyle
4q^{3/2}\left(1+9q^{12}+3q^{16}+25q^{36}+27q^{44}+3q^{48}+3q^{56}+49q^{72}+75q^{84}+27q^{92}+3q^{96}\right)+\dots$

For $\mu_{1}=(0,2)\;\mu_{2}=(2,0)$, we have:

$\displaystyle{}\Psi_{{\boldsymbol{\mu}}}$ $\displaystyle=$ $\displaystyle
4\left(q^{13}+q^{19}+q^{27}+9q^{37}+q^{43}+q^{45}+9q^{47}+q^{53}+q^{57}+9q^{59}+q^{67}+q^{69}\right.$
$\displaystyle+$
$\displaystyle\left.25q^{73}+9q^{83}+9q^{85}+25q^{87}+q^{89}+q^{93}+9q^{97}+q^{99}\right)+\dots$

For $\mu_{1}=(0,1)\;\mu_{2}=(1,0)$, we have:

$\displaystyle{}\Psi_{{\boldsymbol{\mu}}}$ $\displaystyle=$
$\displaystyle-\frac{q^{19/4}}{2}\left(1+q^{2}+q^{6}+25q^{17}+26q^{21}+q^{23}+9q^{24}+q^{26}+q^{27}+25q^{29}+q^{33}+9q^{34}+q^{38}\right.$
$\displaystyle+$
$\displaystyle\left.9q^{39}+81q^{46}+81q^{52}+25q^{54}+25q^{56}+49q^{57}+q^{58}+25q^{61}+26q^{62}+90q^{64}+q^{66}+2q^{68}\right.$
$\displaystyle+$
$\displaystyle\left.49q^{71}+25q^{72}+q^{75}+q^{76}+25q^{77}+49q^{78}+9q^{81}+9q^{82}+q^{86}+169q^{87}+9q^{88}+q^{89}\right.$
$\displaystyle\left.+9q^{90}+169q^{95}\right)+\dots$

For $\mu_{1}=(0,1)\;\mu_{2}=(2,0)$, we have:

$\displaystyle{}\Psi_{{\boldsymbol{\mu}}}$ $\displaystyle=$
$\displaystyle-\frac{q^{71/8}}{2}\left(1+q^{5}+9q^{9}+q^{12}+9q^{16}+25q^{21}+9q^{25}+q^{26}+q^{28}+25q^{30}+q^{35}+49q^{36}\right.$
$\displaystyle+$
$\displaystyle\left.q^{39}+25q^{41}+9q^{43}+9q^{45}+49q^{47}+q^{48}+q^{50}+90q^{54}+9q^{58}+49q^{60}+25q^{63}\right.$
$\displaystyle+$
$\displaystyle\left.25q^{65}+81q^{67}+q^{68}+9q^{69}+9q^{71}+q^{72}+121q^{75}+25q^{76}+q^{78}+25q^{80}\right.$
$\displaystyle+$
$\displaystyle\left.q^{81}+81q^{82}+49q^{86}+49q^{88}+q^{89}+121q^{90}\right)+\dots$

For $\mu_{1}=(0,1)\;\mu_{2}=(0,0)$, we have:

$\displaystyle{}\Psi_{{\boldsymbol{\mu}}}$ $\displaystyle=$
$\displaystyle-\frac{q^{39/8}}{2}\left(1+9q^{2}+q^{3}+25q^{17}+49q^{21}+2q^{22}+25q^{24}+9q^{26}+q^{27}+18q^{28}+2q^{33}+9q^{39}\right.$
$\displaystyle+$
$\displaystyle\left.81q^{46}121q^{52}+50q^{55}+81q^{57}+2q^{60}+49q^{61}+98q^{63}+25q^{64}+9q^{66}+q^{67}+q^{68}\right.$
$\displaystyle+$
$\displaystyle\left.68q^{70}+18q^{76}+49q^{78}+2q^{79}+2q^{81}+169q^{87}+18q^{89}+225q^{95}\right)+\dots$

For $\mu_{1}=(0,1)\;\mu_{2}=(1,1)$, we have:

$\displaystyle{}\Psi_{{\boldsymbol{\mu}}}$ $\displaystyle=$
$\displaystyle-\frac{q^{45/4}}{2}\left(1+q^{3}+q^{6}+9q^{10}+9q^{14}+9q^{18}+25q^{23}+26q^{28}+q^{31}+25q^{33}+q^{34}+q^{36}\right.$
$\displaystyle+$
$\displaystyle\left.50q^{39}+q^{42}+49q^{45}+9q^{46}+9q^{49}+49q^{51}+9q^{53}+9q^{55}+81q^{58}+9q^{59}+9q^{62}+81q^{65}\right.$
$\displaystyle+$
$\displaystyle\left.25q^{67}+25q^{70}+82q^{72}+25q^{75}+25q^{77}+q^{78}+121q^{80}+q^{81}+25q^{82}+q^{83}+26q^{85}+122q^{88}\right)$
$\displaystyle+\dots$

For $\mu_{1}=(0,1)\;\mu_{2}=(1,2)$, we have:

$\displaystyle{}\Psi_{{\boldsymbol{\mu}}}$ $\displaystyle=$
$\displaystyle-\frac{q^{47/4}}{2}\left(9+9q^{3}+q^{5}+9q^{9}+q^{13}+q^{17}+49q^{23}+49q^{28}+9q^{29}+9q^{31}+25q^{32}+9q^{35}\right.$
$\displaystyle+$
$\displaystyle\left.9q^{36}+q^{37}+49q^{38}+q^{42}+34q^{44}+9q^{49}+25q^{50}+q^{52}+q^{53}+122q^{58}+q^{60}+121q^{65}+49q^{68}\right.$
$\displaystyle+$
$\displaystyle\left.49q^{70}+81q^{71}+9q^{74}+49q^{76}+49q^{77}+9q^{78}+121q^{79}+25q^{80}+9q^{83}+9q^{84}+25q^{85}+81q^{87}\right)$
$\displaystyle+\dots$

For $\mu_{1}=(0,1)\;\mu_{2}=(1,3)$, we have:

$\displaystyle{}\Psi_{{\boldsymbol{\mu}}}$ $\displaystyle=$
$\displaystyle-\frac{q^{13/4}}{2}\left(1+9q^{6}+q^{10}+25q^{15}+q^{19}+q^{20}+9q^{21}+49q^{27}+q^{30}+9q^{33}+9q^{34}+25q^{35}\right.$
$\displaystyle+$
$\displaystyle\left.q^{39}+81q^{42}+q^{45}+9q^{46}+25q^{50}+25q^{51}+49q^{52}+q^{54}+q^{56}+9q^{58}+121q^{60}\right.$
$\displaystyle+$
$\displaystyle\left.q^{63}+9q^{64}+25q^{65}+49q^{70}+49q^{71}+81q^{72}+q^{74}+q^{75}+9q^{76}+9q^{78}+25q^{80}\right.$
$\displaystyle+$
$\displaystyle\left.169q^{81}+q^{84}+9q^{85}+25q^{86}+49q^{87}+81q^{93}+81q^{94}+121q^{95}+q^{96}\right)+\dots$

For $\mu_{1}=(0,1)\;\mu_{2}=(2,2)$, we have:

$\displaystyle{}\Psi_{{\boldsymbol{\mu}}}$ $\displaystyle=$
$\displaystyle-\frac{q^{15/8}}{2}\left(1+9q^{10}+25q^{13}+9q^{15}+q^{16}+2q^{18}+q^{27}+49q^{33}+81q^{38}+18q^{40}+49q^{42}+25q^{45}\right.$
$\displaystyle+$
$\displaystyle\left.59q^{47}+q^{48}+2q^{52}+18q^{53}+2q^{58}+25q^{60}+121q^{68}+2q^{69}+169q^{75}+98q^{79}+121q^{81}\right.$
$\displaystyle+$
$\displaystyle\left.99q^{86}+162q^{88}+49q^{90}+25q^{93}+9q^{94}+9q^{95}+99q^{96}+50q^{97}\right)+\dots$

For $\mu_{1}=(0,2)\;\mu_{2}=(0,1)$, we have:

$\displaystyle{}\Psi_{{\boldsymbol{\mu}}}$ $\displaystyle=$ $\displaystyle
16q^{47/2}\left(1+q^{8}+q^{18}+4q^{30}+q^{38}+q^{40}+4q^{42}+q^{50}+q^{54}+4q^{56}+q^{66}+q^{68}+9q^{72}\right)+\dots$

For $\mu_{1}=(0,2)\;\mu_{2}=(1,0)$, we have:

$\displaystyle{}\Psi_{{\boldsymbol{\mu}}}$ $\displaystyle=$ $\displaystyle
2q^{81/8}\left(1+q^{6}+5q^{13}+q^{21}+9q^{22}+q^{27}+q^{30}+4q^{31}+9q^{32}+q^{39}+q^{40}+25q^{43}+4q^{50}\right.$
$\displaystyle+$
$\displaystyle\left.q^{51}+q^{52}+4q^{54}+9q^{55}+25q^{56}+q^{63}+9q^{65}+q^{66}+9q^{68}+16q^{69}+26q^{70}+q^{76}\right.$
$\displaystyle+$
$\displaystyle\left.4q^{78}+10q^{81}+13q^{82}+61q^{85}+q^{88}\right)+\dots$

For $\mu_{1}=(0,2)\;\mu_{2}=(1,1)$, we have:

$\displaystyle{}\Psi_{{\boldsymbol{\mu}}}$ $\displaystyle=$ $\displaystyle
2q^{101/8}\left(1+4q^{7}+q^{14}+4q^{15}+13q^{24}+q^{29}+q^{33}+4q^{34}+16q^{35}+4q^{42}+4q^{45}+9q^{46}\right.$
$\displaystyle+$
$\displaystyle\left.16q^{47}+q^{53}+4q^{56}+5q^{57}+41q^{60}+9q^{69}+4q^{70}+4q^{71}+9q^{73}+17q^{74}\right.$
$\displaystyle+$
$\displaystyle\left.36q^{75}+q^{82}+4q^{84}+17q^{86}+4q^{87}\right)+\dots$

Acknowledgement: We thank Sergey Alexandrov and Boris Pioline for useful
discussions and comments on the draft. The authors are supported by Laureate
Award 15175 “Modularity in Quantum Field Theory and Gravity” of the Irish
Research Council.

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# On training deep networks for satellite image super-resolution

###### Abstract

The capabilities of super-resolution reconstruction (SRR)—techniques for
enhancing image spatial resolution—have been recently improved significantly
by the use of deep convolutional neural networks. Commonly, such networks are
learned using huge training sets composed of original images alongside their
low-resolution counterparts, obtained with bicubic downsampling. In this
paper, we investigate how the SRR performance is influenced by the way such
low-resolution training data are obtained, which has not been explored up to
date. Our extensive experimental study indicates that the training data
characteristics have a large impact on the reconstruction accuracy, and the
widely-adopted approach is not the most effective for dealing with satellite
images. Overall, we argue that developing better training data preparation
routines may be pivotal in making SRR suitable for real-world applications.

Index Terms—  Super-resolution reconstruction, deep learning, convolutional
neural networks, satellite imaging

##  1 Introduction

_Super-resolution reconstruction_ (SRR) is aimed at generating a high-
resolution (HR) image from a low-resolution (LR) observation (a single image
or multiple images) [10]. SRR is a deeply explored research topic of
considerable practical potential, as developing effective SRR techniques may
allow for overcoming the spatial resolution limitations of the imaging
sensors, which is a common problem in remote sensing.

###  1.1 Related work

Existing single-image SRR methods can be categorized into: (i) frequency-
domain techniques [2], (ii) reconstruction-based methods which exploit prior
knowledge on the object appearance [11], and (iii) algorithms that learn the
mapping between LR and HR [12]. Recently, we have witnessed a breakthrough in
the learning-based single-image SRR, attributed to the use of deep
_convolutional neural networks_ (CNNs). Deep learning SRR originates from
sparse coding [12], aimed at creating a dictionary of LR patches, associated
with their HR counterparts. The reconstruction consists in exploiting that
dictionary for converting each LR patch from the input image into HR.

_Super-resolution CNN_ (SRCNN) [3], followed by its faster version (FSRCNN)
[4], was proposed for learning the LR-to-HR mapping from a number of LR–HR
image pairs. Despite relatively simple architecture, SRCNN outperforms the
state-of-the-art example-based methods. In [8], SRCNN was trained with
Sentinel-2 images, which according to the authors improved its capacities of
enhancing satellite data. Certain limitations of SRCNN were addressed with a
_very deep super-resolution_ network [6], trained relying on fast residual
learning. The domain expertise was exploited in a _sparse coding network_ [9],
achieving high training speed and model compactness. Recently, _generative
adversarial networks_ are being actively explored for SRR [7]. They are
composed of a _generator_ (ResNet in [7]), trained to perform SRR, and a
_discriminator_ which tries to distinguish the ResNet reconstruction outcomes
from real HR images.

###  1.2 Contribution

Deep CNNs for SRR are trained from a dataset of corresponding LR–HR patches.
As deep networks commonly require huge amounts of training data, LR images are
obtained by subjecting the original HR images to a degradation procedure based
on an assumed imaging model. In most works [3, 7, 8], bicubic downsampling is
applied to transform HR into LR, and in some cases [4], the _training set_
($\bm{T}$) is additionally augmented with translation, rotation, and scaling.
However, it has not been analyzed whether and how $\bm{T}$ (including the
degradation procedure) influences the reconstruction accuracy.

In this paper, our contribution consists in addressing the aforementioned
research gap. We investigate the influence of $\bm{T}$, used for training a
CNN, on the reconstruction performance. We trained two different CNNs (Section
2) with natural images from the DIV2K single-image SRR benchmark, and with
Sentinel-2 images. The trained CNNs are tested in two settings: for
reconstructing artificially-degraded satellite images (original images are
treated as reference HR data), as well as in a real-world scenario—for
original Sentinel-2 images, matched with SPOT and Digital Globe WorldView-4
images of the same region. The results of our extensive experiments (reported
in Section 3) indicate that the degradation procedure used for creating
$\bm{T}$ plays a pivotal role here. Not only does it have a larger impact on
the SRR performance than the domain of images exploited for training (natural
vs. satellite), but it is also more important than the choice of the CNN
architecture.

##  2 Deep learning for super-resolution

In this work, we exploit two CNNs of different complexity, namely: FSRCNN [4],
which is a relatively shallow CNN, and a much deeper residual network
(SRResNet [7]), to investigate their behavior in different training scenarios.

Figure 1 shows the architecture of FSRCNN [4]—the network is composed of five
major parts aimed at: (i)  _feature extraction_ , realized by the first
convolutional layer (denoted as Conv) with $n=56$ kernels of size $k=5\times
5$, (ii)  _shrinking_ , performed using $n=16$ kernels ($1\times 1$) to reduce
the number of features (from $56$ to $16$), (iii)  _non-linear mapping_ using
multiple ($m=4$) convolutional layers with $n=16$ kernels ($3\times 3$), (iv)
_expansion_ which inverses the shrinking and increases the dimensionality of
the feature vectors from $16$ back to $56$, and (v)  _deconvolution_ which
produces the reconstructed HR image. FSRCNN can be trained faster than SRCNN
and it offers real-time performance after training [3, 4].

LR Conv

$\bm{k5n56}$

Conv

$\bm{k1n16}$

Conv

$\bm{k3n16}$

$\bm{m}\enspace\times$Non-linear map. Conv

$\bm{k1n56}$

Deconv

$\bm{k1n1}$

HR

Fig. 1: FSRCNN architecture for SRR proposed in [4]. For each layer, we report
the number of filters $n$, and the size of a square filter $k$ (e.g., $k5n56$
means $56$ filters of size $5\times 5$).

The SRResNet [7] architecture (Fig. 2) benefits from the residual connections
between the layers [5]. The _residual blocks_ (RBs) are the groups of layers
stacked together with the input of the block added to the output of the final
layer contained in this block. In SRResNet, each block encompasses two
convolutional layers, each followed by a _batch normalization_ (BN) layer that
neutralizes the internal co-variate shift. The _upsampling blocks_ (UBs) allow
for image enlargement by _pixel shuffling_ (PS) layers that increase the
resolution of the features. The number of both RBs and UBs is variable—by
increasing the number of RBs, the network may model a better mapping, whereas
by changing the number of UBs, we may tune its scaling factor. However, by
adding more blocks, the architecture of the network becomes increasingly
complex, which makes it harder to train.

LR Conv

$\bm{k9n64}$

Conv

$\bm{k3n64}$

BN Conv

$\bm{k3n64}$

BN $+$ $\bm{N_{rb}\,\times\,}$RB Conv

$\bm{k3n64}$

BN $+$ Conv

$\bm{k3n256}$

PS $\bm{N_{ub}\,\times\,}$UB Conv

$\bm{k9n1}$

HR

Fig. 2: SRResNet architecture proposed in [7]. In this work, we used
$N_{rb}=16$ RBs and $N_{ub}=1$ UB.

##  3 Experimental study

We trained FSRCNN and SRResNet using natural images from the DIV2K
dataset111Available at https://data.vision.ee.ethz.ch/cvl/DIV2K, and
Sentinel-2 images. From these images, the patches were extracted randomly to
create $\bm{T}$ and _validation set_ ($\bm{V}$), as specified in Table 1. LR
images were obtained from HR ones using different downsampling techniques:
nearest neighbor (NN), bilinear, bicubic, and Lanczos. We also created a mixed
set—the downsampling technique was randomly selected for each image. For
Lanczos, we additionally applied Gaussian blur with $\sigma_{b}=0.7$
(Lanczos-B), Gaussian noise with $\sigma_{n}=0.01$ (Lanczos-N), and both blur
and noise with $\sigma_{n}=0.022$ (Lanczos-BN). Examples of patches in
$\bm{T}$ are shown in Fig. 3. We used Python with Keras to implement the CNNs.
The experiments were run on an Intel i9 4 GHz computer with 64 GB RAM, and two
RTX 2080 8 GB GPUs. We used ADAM optimizer with learning rate of $10^{-3}$.
The optimization stops, if after 50 epochs the accuracy over $\bm{V}$ does not
increase.

Table 1: Datasets used for training FSRCNN and SRResNet. Dataset |  No. of patches in $\bm{T}$ |  No. of patches in $\bm{V}$ | LR patch size | HR patch size
---|---|---|---|---
DIV2K | 12800 | 1600 |  112$\times$112 |  224$\times$224
Sentinel | 4825 | 535 |  112$\times$112 |  224$\times$224
|  |  |  |

(a)(a)(a)(a)(a)(b)(b)(b)(b)(b)(c)(c)(c)(c)(c)(d)(d)(d)(d)(d)(e)(e)(e)(e)(e)(f)(f)(f)(f)(f)(g)(g)(g)(g)(g)(h)(h)(h)(h)(h)

Fig. 3: A $64\times 64$ patch (a) degraded with different techniques: b) NN,
c) bilinear, d) bicubic, e) Lanczos, f) Lanczos-B, g) Lanczos-N, and h)
Lanczos-BN. Table 2: Reconstruction accuracy obtained for $\bm{\Psi}$ after
training FSRCNN and SRResNet using different $\bm{T}$’s (best scores for each
category are marked as bold). The scenarios commonly reported in the
literature are marked as gray.

$\bm{\Psi}\rightarrow$ | Artificially-degraded (AD) satellite images | Real satellite (RS) images
---|---|---
SRR method $\rightarrow$ |  FSRCNN [4] |  SRResNet [7] |  FSRCNN [4] |  SRResNet [7]
Downsampling of $\bm{T}$ $\downarrow$ | PSNR | SSIM | UIQI | VIF | KFS | PSNR | SSIM | UIQI | VIF | KFS | PSNR | SSIM | UIQI | VIF | KFS | PSNR | SSIM | UIQI | VIF | KFS
DIV2K | NN | 31.95 | 0.915 | 0.891 | 0.545 | 12.92 | 30.81 | 0.905 | 0.879 | 0.523 | 12.708 | 16.79 | 0.454 | 0.268 | 0.122 | 2.638 | 17.29 | 0.439 | 0.263 | 0.117 | 2.64
Bilinear | 22.26 | 0.679 | 0.652 | 0.351 | 7.669 | 22.03 | 0.67 | 0.643 | 0.35 | 7.747 | 16.83 | 0.454 | 0.292 | 0.125 | 2.768 | 16.77 | 0.457 | 0.292 | 0.124 | 2.762
Bicubic | 26.61 | 0.819 | 0.792 | 0.435 | 10.209 | 26.84 | 0.82 | 0.794 | 0.44 | 10.375 | 16.44 | 0.433 | 0.262 | 0.109 | 2.61 | 16.97 | 0.465 | 0.287 | 0.124 | 2.705
Lanczos | 28 | 0.844 | 0.818 | 0.454 | 11.093 | 28.57 | 0.854 | 0.829 | 0.466 | 11.364 | 16.9 | 0.459 | 0.283 | 0.126 | 2.664 | 16.62 | 0.47 | 0.274 | 0.117 | 2.661
Lanczos-B | 11.02 | 0.175 | 0.168 | 0.118 | 3.591 | 10.98 | 0.167 | 0.162 | 0.11 | 3.514 | 15.32 | 0.313 | 0.21 | 0.098 | 2.817 | 15.45 | 0.336 | 0.218 | 0.099 | 2.819
Lanczos-N | 28.6 | 0.866 | 0.836 | 0.473 | 11.429 | 28.14 | 0.869 | 0.836 | 0.47 | 11.256 | 16.91 | 0.456 | 0.271 | 0.124 | 2.624 | 17.74 | 0.479 | 0.271 | 0.122 | 2.634
Lanczos-BN | 19.97 | 0.676 | 0.624 | 0.337 | 5.94 | 18.35 | 0.583 | 0.543 | 0.299 | 5.451 | 16.49 | 0.434 | 0.257 | 0.117 | 2.659 | 16.47 | 0.46 | 0.263 | 0.118 | 2.689
Mixed | 30.16 | 0.885 | 0.858 | 0.481 | 11.504 | 28.5 | 0.856 | 0.829 | 0.456 | 11.729 | 16.84 | 0.453 | 0.289 | 0.126 | 2.717 | 16.32 | 0.476 | 0.29 | 0.126 | 2.737
Sentinel-2 | NN | 31.64 | 0.91 | 0.88 | 0.531 | 12.794 | 31.59 | 0.908 | 0.875 | 0.527 | 12.43 | 16.88 | 0.438 | 0.242 | 0.11 | 2.553 | 16.08 | 0.441 | 0.254 | 0.11 | 2.591
Bilinear | 23.01 | 0.701 | 0.676 | 0.358 | 7.467 | 23.01 | 0.669 | 0.632 | 0.308 | 6.378 | 17.12 | 0.491 | 0.292 | 0.124 | 2.772 | 16.9 | 0.507 | 0.279 | 0.109 | 2.682
Bicubic | 27.82 | 0.837 | 0.804 | 0.426 | 10.636 | 27.97 | 0.844 | 0.797 | 0.435 | 9.702 | 16.38 | 0.502 | 0.287 | 0.126 | 2.769 | 16.93 | 0.458 | 0.227 | 0.079 | 2.568
Lanczos | 28.41 | 0.85 | 0.823 | 0.459 | 11.04 | 26.18 | 0.833 | 0.807 | 0.445 | 11.04 | 16.93 | 0.49 | 0.285 | 0.126 | 2.686 | 15.52 | 0.482 | 0.254 | 0.105 | 2.593
Lanczos-B | 12.2 | 0.216 | 0.207 | 0.134 | 3.722 | 12.21 | 0.221 | 0.202 | 0.073 | 2.54 | 15.63 | 0.337 | 0.216 | 0.099 | 2.806 | 15.49 | 0.4 | 0.225 | 0.098 | 2.769
Lanczos-N | 28.67 | 0.865 | 0.839 | 0.474 | 11.348 | 28.68 | 0.868 | 0.842 | 0.48 | 11.557 | 16.88 | 0.487 | 0.275 | 0.127 | 2.652 | 17.35 | 0.528 | 0.265 | 0.122 | 2.664
Lanczos-BN | 20.7 | 0.702 | 0.663 | 0.342 | 6.014 | 18.79 | 0.6 | 0.553 | 0.281 | 5.203 | 16.53 | 0.455 | 0.269 | 0.12 | 2.718 | 17.02 | 0.515 | 0.261 | 0.114 | 2.71
Mixed | 28.23 | 0.847 | 0.817 | 0.431 | 10.576 | 20.83 | 0.843 | 0.805 | 0.45 | 9.555 | 16.99 | 0.461 | 0.291 | 0.126 | 2.778 | 13.47 | 0.46 | 0.237 | 0.084 | 2.563
|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |

After training, the nets were tested using two kinds of test sets
($\bm{\Psi}$): (i)  _artificially-degraded_ (AD) images—10 HR images of size
$500\times 500$ pixels, bicubically downsampled to $250\times 250$ pixels, and
(ii)  _real satellite_ (RS) images acquired at different resolution—we used
three Sentinel-2 scenes as LR, two of which are matched with SPOT images and
one is matched with Digital Globe WorldView-4 image. We evaluate the
reconstruction accuracy relying on _peak signal-to-noise ratio_ (PSNR),
_structural similarity index_ (SSIM), _visual information fidelity_ (VIF),
_universal image quality index_ (UIQI), and _keypoint features similarity_
(KFS) [1]. For all these metrics, higher values indicate higher similarity
between the reconstruction outcome and the reference image.

In Table 2, we show the reconstruction accuracy obtained with FSRCNN and
SRResNet trained using different $\bm{T}$’s. We highlight PSNR and SSIM scores
(in gray) for AD after training with bicubically downsampled $\bm{T}$, as this
is the scenario most often reported in the literature. From these scores,
SRResNet is slightly better than FSRCNN, and using satellite data for training
appears to be beneficial. However, from all the scores, it is clear that the
degradation procedure is more significant than both the type of images in
$\bm{T}$ and the network architecture. Actually, the nets trained with
$\bm{T}$’s based on NN perform in the best way, which can also be assessed
qualitatively from Fig. 4. If $\bm{T}$ is blurred (Lanczos-B), then the image
sharpening is too strong, resulting in many high-frequency artifacts. A
surprising outcome can be observed for SRResNet (Bicubic, Sentinel)—the
details in the sea area are lost after reconstruction, but the land area is
reliably restored.

For RS images, it is not clear from the reported metrics (Table 2), which
$\bm{T}$ is the best. The values are much lower than for AD, as the HR images
used for reference are acquired using a different sensor, so even if an image
is well reconstructed, it is substantially different from HR. From Fig. 5, it
can be seen that NN downsampling (best for AD) results in a blurry outcome.
Interestingly, Lanczos-B (very poor for AD), delivers better results in this
case (and it is consistently picked by the KFS metric—the similarity to HR in
the domain of the detected keypoints is the highest here). Similarly to AD,
severe artifacts in the sea area can be observed for SRResNet trained with
some $\bm{T}$’s (Mixed and NN, for Sentinel). In our opinion, visually most
plausible results are obtained using bilinear downsampling (for both Sentinel
and DIV2K), which is also reflected with the highest UIQI scores in Table 2.

High resolution High resolution High resolution High resolution High
resolution FSRCNN
(NN, DIV2K) FSRCNN
(NN, DIV2K) FSRCNN
(NN, DIV2K) FSRCNN
(NN, DIV2K) FSRCNN
(NN, DIV2K) FSRCNN
(NN, Sentinel) FSRCNN
(NN, Sentinel) FSRCNN
(NN, Sentinel) FSRCNN
(NN, Sentinel) FSRCNN
(NN, Sentinel) FSRCNN
(Bicubic, DIV2K) FSRCNN
(Bicubic, DIV2K) FSRCNN
(Bicubic, DIV2K) FSRCNN
(Bicubic, DIV2K) FSRCNN
(Bicubic, DIV2K) FSRCNN
(Bicubic, Sentinel) FSRCNN
(Bicubic, Sentinel) FSRCNN
(Bicubic, Sentinel) FSRCNN
(Bicubic, Sentinel) FSRCNN
(Bicubic, Sentinel) FSRCNN
(Lanczos-B, DIV2K) FSRCNN
(Lanczos-B, DIV2K) FSRCNN
(Lanczos-B, DIV2K) FSRCNN
(Lanczos-B, DIV2K) FSRCNN
(Lanczos-B, DIV2K) FSRCNN
(Lanczos-B, Sentinel) FSRCNN
(Lanczos-B, Sentinel) FSRCNN
(Lanczos-B, Sentinel) FSRCNN
(Lanczos-B, Sentinel) FSRCNN
(Lanczos-B, Sentinel) Low resolution Low resolution Low resolution Low
resolution Low resolution SRResNet
(NN, DIV2K) SRResNet
(NN, DIV2K) SRResNet
(NN, DIV2K) SRResNet
(NN, DIV2K) SRResNet
(NN, DIV2K) SRResNet
(NN, Sentinel) SRResNet
(NN, Sentinel) SRResNet
(NN, Sentinel) SRResNet
(NN, Sentinel) SRResNet
(NN, Sentinel) SRResNet
(Bicubic, DIV2K) SRResNet
(Bicubic, DIV2K) SRResNet
(Bicubic, DIV2K) SRResNet
(Bicubic, DIV2K) SRResNet
(Bicubic, DIV2K) SRResNet
(Bicubic, Sentinel) SRResNet
(Bicubic, Sentinel) SRResNet
(Bicubic, Sentinel) SRResNet
(Bicubic, Sentinel) SRResNet
(Bicubic, Sentinel) SRResNet
(Lanczos-B, DIV2K) SRResNet
(Lanczos-B, DIV2K) SRResNet
(Lanczos-B, DIV2K) SRResNet
(Lanczos-B, DIV2K) SRResNet
(Lanczos-B, DIV2K) SRResNet
(Lanczos-B, Sentinel) SRResNet
(Lanczos-B, Sentinel) SRResNet
(Lanczos-B, Sentinel) SRResNet
(Lanczos-B, Sentinel) SRResNet
(Lanczos-B, Sentinel)

Fig. 4: Examples of reconstructing an artificially degraded Digital Globe
WorldView-4 image ($250\times 250$ pixels, presenting Rio de Janeiro, Brasil)
using SRResNet and FSRCNN, trained with $\bm{T}$’s obtained relying on
different degradation procedures.

High resolution High resolution High resolution High resolution High
resolution FSRCNN
(Bilinear, Sentinel) FSRCNN
(Bilinear, Sentinel) FSRCNN
(Bilinear, Sentinel) FSRCNN
(Bilinear, Sentinel) FSRCNN
(Bilinear, Sentinel) FSRCNN
(Lanczos, DIV2K) FSRCNN
(Lanczos, DIV2K) FSRCNN
(Lanczos, DIV2K) FSRCNN
(Lanczos, DIV2K) FSRCNN
(Lanczos, DIV2K) FSRCNN
(Lanczos-BN, DIV2K) FSRCNN
(Lanczos-BN, DIV2K) FSRCNN
(Lanczos-BN, DIV2K) FSRCNN
(Lanczos-BN, DIV2K) FSRCNN
(Lanczos-BN, DIV2K) FSRCNN
(Bicubic, Sentinel) FSRCNN
(Bicubic, Sentinel) FSRCNN
(Bicubic, Sentinel) FSRCNN
(Bicubic, Sentinel) FSRCNN
(Bicubic, Sentinel) FSRCNN
(Mixed, Sentinel) FSRCNN
(Mixed, Sentinel) FSRCNN
(Mixed, Sentinel) FSRCNN
(Mixed, Sentinel) FSRCNN
(Mixed, Sentinel) FSRCNN
(NN, Sentinel) FSRCNN
(NN, Sentinel) FSRCNN
(NN, Sentinel) FSRCNN
(NN, Sentinel) FSRCNN
(NN, Sentinel) Low resolution Low resolution Low resolution Low resolution Low
resolution SRResNet
(Bilinear, DIV2K) SRResNet
(Bilinear, DIV2K) SRResNet
(Bilinear, DIV2K) SRResNet
(Bilinear, DIV2K) SRResNet
(Bilinear, DIV2K) SRResNet
(Lanczos-B, DIV2K) SRResNet
(Lanczos-B, DIV2K) SRResNet
(Lanczos-B, DIV2K) SRResNet
(Lanczos-B, DIV2K) SRResNet
(Lanczos-B, DIV2K) SRResNet
(Lanczos-N, Sentinel) SRResNet
(Lanczos-N, Sentinel) SRResNet
(Lanczos-N, Sentinel) SRResNet
(Lanczos-N, Sentinel) SRResNet
(Lanczos-N, Sentinel) SRResNet
(Bicubic, DIV2K) SRResNet
(Bicubic, DIV2K) SRResNet
(Bicubic, DIV2K) SRResNet
(Bicubic, DIV2K) SRResNet
(Bicubic, DIV2K) SRResNet
(Mixed, Sentinel) SRResNet
(Mixed, Sentinel) SRResNet
(Mixed, Sentinel) SRResNet
(Mixed, Sentinel) SRResNet
(Mixed, Sentinel) SRResNet
(NN, Sentinel) SRResNet
(NN, Sentinel) SRResNet
(NN, Sentinel) SRResNet
(NN, Sentinel) SRResNet
(NN, Sentinel)

Fig. 5: Examples of reconstructing a real Sentinel-2 image ($240\times 266$
pixels, presenting Bandar Abbas, Iran) using SRResNet and FSRCNN, trained with
different $\bm{T}$’s based on different degradation procedures. HR image
(SPOT) is given for reference.

##  4 Conclusions

In this paper, we reported our experimental study on preparing the data to
train deep CNNs for satellite image SRR. The results indicate that the
degradation procedure used to generate the training data has a tremendous
impact on the SRR performance, which is usually neglected in the literature.
Furthermore, it is worth noting that much deeper architecture of SRResNet does
not seem to outperform a relatively simple FSRCNN, when appropriate $\bm{T}$
is used.

Currently, we are exploring how to combine different degradation procedures,
including data augmentation techniques, to create training sets which better
reflect the actual imaging conditions. We expect that this will allow deep
CNNs to increase their performance for real satellite images.

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