Intriguing Property and Counterfactual Explanation of GAN for Remote Sensing Image Generation

Abstract

Generative adversarial networks (GANs) have achieved remarkable progress in the natural image field. However, when applying GANs in the remote sensing (RS) image generation task, an extraordinary phenomenon is observed: the GAN model is more sensitive to the amount of training data for RS image generation than for natural image generation (Fig. 1). In other words, the generation quality of RS images will change significantly with the number of training categories or samples per category. In this paper, we first analyze this phenomenon from two kinds of toy experiments and conclude that the amount of feature information contained in the GAN model decreases with reduced training data (Fig. 2). Then we establish a structural causal model (SCM) of the data generation process and interpret the generated data as the counterfactuals. Based on this SCM, we theoretically prove that the quality of generated images is positively correlated with the amount of feature information. This provides insights for enriching the feature information learned by the GAN model during training. Consequently, we propose two innovative adjustment schemes, namely uniformity regularization and entropy regularization, to increase the information learned by the GAN model at the distributional and sample levels, respectively. Extensive experiments on eight RS datasets and three natural datasets show the effectiveness and versatility of our methods. The source code is available at https://github.com/rootSue/Causal-RSGAN.

Appendices

A Proofs and Theoretical Analysis

1.1 A.1 Proof of Theorem 1

The global minimum of $\mathcal {L}_{\text{ AlignMax }}$ is achieved when the first term (distance between real and generated distributions) is minimized (i.e., equal to zero) and the second term (entropy) is maximized. Commonly, the uniform distribution on $(0, 1)^{d_c}$ gets the maximum entropy.

Step 1. First, we show that there exists a smooth function $\textbf{g}_{*}: \mathcal {X} \rightarrow (0, 1)^{d_c}$ which gets the global minimum of $\mathcal {L}_{\text{ AlignMax }}$. Consider the function $\textbf{f}^{-1}_{1:{d_c}}: \mathcal {X}\rightarrow \mathcal {C}$, i.e., the inverse of the true mixing $\textbf{f}$, restricted to its first $d_c$ dimensions. Further, we have $\textbf{f}^{-1}(x)_{1:{d_c}} = \textbf{c}$ by definition.

Then we construct a function $\textbf{k}:\mathcal {C} \rightarrow (0, 1)^{d_c}$ which maps $\textbf{c}$ to a uniform random variable on $(0, 1)^{d_c}$ using a recursive method known as the Darmois construction.

Specifically, we define

$$\begin{aligned} \begin{aligned} k_{i}(\textbf{c}):= & {} F_{i}\left( c_{i} \mid \textbf{c}_{1: i-1}\right)= & {} \mathbb {P}\left( C_{i} \le c_{i} \mid \textbf{c}_{1: i-1}\right) , \nonumber \end{aligned} \end{aligned}$$

where $i= 1, \ldots , d_{c}$, $F_i$ represents the conditional cumulative distribution function (CDF) of $c_i$ given $\textbf{c}_{1: i-1}$. By construction, $\textbf{k}(\textbf{c})$ is uniformly distributed on $(0, 1)^{d_c}$. Moreover, $\textbf{k}$ is smooth by the assumption that $p_\textbf{z}$ (and hence $p_\textbf{c}$) is a smooth density. Finally, we define

$$\begin{aligned} \begin{aligned} \textbf{g}^{*}:= & {} \textbf{k} \circ \textbf{f}_{1: d_{c}}^{-1}: \mathcal {X} \rightarrow (0,1)^{d_{c}} \nonumber \end{aligned} \end{aligned}$$

which is a smooth function since it is a composition of two smooth functions.

Next, we proof $\textbf{g}^{*}$ gets the global minimum of $\mathcal {L}_{\text{ AlignMax }}$. Using $\textbf{f}^{-1}(\textbf{x})_{1:{d_c}} = \textbf{c}$ and $\textbf{f}^{-1}(\widetilde{\textbf{x}})_{1:{d_c}} = \widetilde{\textbf{c}}$, we have

$$\begin{aligned} \begin{aligned}&\mathcal {L}_{\text{ AlignMax }}\left( \textbf{g}^{*}\right) \\ {}&=\mathbb {E}_{(\textbf{x}, \tilde{\textbf{x}}) \sim p_{(\textbf{x}, \tilde{\textbf{x}})}}\left[ \left\| \textbf{g}^{*}(\textbf{x})-\textbf{g}^{*}(\tilde{\textbf{x}})\right\| _{2}^{2}\right] -H\left( \textbf{g}^{*}(\textbf{x})\right) \\&=\mathbb {E}_{(\textbf{x}, \tilde{\textbf{x}}) \sim p_{(\textbf{x}, \tilde{\textbf{x}})}}\left[ \Vert \textbf{k}(\textbf{c})-\textbf{k}(\tilde{\textbf{c}})\Vert _{2}\right] -H(\textbf{k}(\textbf{c})) \\&=0 \nonumber \end{aligned} \end{aligned}$$

where in the last step we utilize the fact that $\textbf{c} = \widetilde{\textbf{c}}$ almost surely with respect to the ground truth generative process described in Fig. 4. As a result, the first term is zero. Moreover, since $\textbf{k}(\textbf{c})$ is uniformly distributed on $(0, 1)^{d_c}$ and the uniform distribution on the unit hypercube has zero entropy, the second term is also zero.

Next, let $\textbf{g}:\mathcal {X} \rightarrow (0, 1)^{d_c}$ be any smooth function which attains the global minimum of of $\mathcal {L}_{\text{ AlignMax }}$, i.e.,

$$\begin{aligned} \begin{aligned}&\mathcal {L}_{\textrm{AlignMax}}(\textbf{g})\\ {}&= \mathbb {E}_{(\textbf{x}, \tilde{\textbf{x}}) \sim p_{(\textbf{x}, \tilde{\textbf{x}})}}\left[ \Vert \textbf{g}(\textbf{x})-\textbf{g}(\tilde{\textbf{x}})\Vert _{2}^{2}\right] -H(\textbf{g}(\textbf{x}))&= 0 \nonumber \\ \end{aligned} \end{aligned}$$

Define $\textbf{h}: = \textbf{g} \circ \textbf{f}: \mathcal {Z} \rightarrow (0,1)^{d_{c}}$ which is smooth because both g and f are smooth. Given $\textbf{x}=\textbf{f}(\textbf{z})$, we can get

$$\begin{aligned} \begin{aligned} \mathbb {E}_{(\textbf{x}, \tilde{\textbf{x}}) \sim p_{(\textbf{x}, \tilde{\textbf{x}})}}\left[ \Vert \textbf{h}(\textbf{z})-\textbf{h}(\tilde{\textbf{z}})\Vert _{2}\right]&=0, \\ H(\textbf{h}(\textbf{z}))&=0 . \end{aligned} \end{aligned}$$

(11)

The first equation suggests that $\hat{\textbf{c}}=\textbf{h}(\textbf{z})_{1: d_{c}}= \textbf{h}(\tilde{\textbf{z}})_{1: d_{c}}$ must hold (almost surely with respect to p), and the second equation suggests that $\hat{\textbf{c}}=\textbf{h}(\textbf{z})$ must be uniformly distributed on $(0,1)^{d_c}$.

Step 2. Next, we show that $\textbf{h}(\textbf{z}) = \textbf{h}(\textbf{c}, \mathbf {\epsilon })$ can only depend on the true content $\textbf{c}$ and not on any of the noise variables $\mathbf {\epsilon }$.

Suppose for a contradiction that $\textbf{h}_{c}(\textbf{c}, \mathbf {\epsilon }):=\textbf{h}(\textbf{c}, \mathbf {\epsilon })_{1: d_{c}}=\textbf{h}(\textbf{z})_{1: d_{c}}$ depends on some component of the noise variable $\mathbf {\epsilon }$:

$$\begin{aligned} \begin{aligned} \exists l \in \left\{ 1, \ldots , d_{\epsilon }\right\} ,\left( \textbf{c}^{*}, \mathbf {\epsilon }^{*}\right) \in \mathcal {C} \times \mathcal {E}, \text{ s.t. } \frac{\partial \textbf{h}_{c}}{\partial {\epsilon }_{l}}\left( \textbf{c}^{*}, \mathbf {\epsilon }^{*}\right) \ne 0 \end{aligned} \end{aligned}$$

(12)

that is, we assume that the partial derivative of $\textbf{h}_c$ with respect to some noise variable ${\epsilon }_l$ is non-zero at some point $\textbf{z}^{*}=\left( \textbf{c}^{*}, \mathbf {\epsilon }^{*}\right) \in \mathcal {Z}=\mathcal {C} \times \mathcal {E}$.

Since $\textbf{h}$ is smooth, so is $\textbf{h}_c$. Therefore, $\textbf{h}_c$ has continuous (first) partial derivatives. By continuity of the partial derivative, $\frac{\partial \textbf{h}_{c}}{\partial {\epsilon }_{l}}$ must be non-zero in a neighbourhood of $(\textbf{c}^{*},\mathbf {\epsilon }^{*})$, i.e.

$$\begin{aligned}{} & {} \exists \eta >0 \text{ s.t. } {\epsilon }_{l} \mapsto \textbf{h}_{c}\left( \mathbf {\epsilon }^{*},\left( \mathbf {\epsilon }_{-l}^{*}, {\epsilon }_{l}\right) \right) \text{ is } \text{ strictly } \text{ monotonic } \text{ on } \nonumber \\{} & {} \quad \left( {\epsilon }_{l}^{*}-\eta , {\epsilon }_{l}^{*}+\eta \right) , \end{aligned}$$

(13)

where $\mathbf {\epsilon }_{-l} \in \mathcal {E}_{-l}$ denotes the vector of remaining noise variables except ${\epsilon }_l$.

Next, define the auxiliary function $\psi : \mathcal {C} \times \mathcal {E} \times \mathcal {E} \rightarrow \mathbb {R}_{\ge 0}$ as follows:

$$\begin{aligned} \begin{aligned} \psi (\textbf{c}, \mathbf {\epsilon }, \tilde{\mathbf {\epsilon }}):= & {} \left| \textbf{h}_{c}(\textbf{c}, \mathbf {\epsilon })-\textbf{h}_{c}(\textbf{c}, \tilde{\mathbf {\epsilon }})\right| \ge 0 \nonumber \end{aligned} \end{aligned}$$

To obtain a contradiction to the invariance condition under assumption Eq. (12), it remains to show that $\psi$ is strictly positive with probability greater than zero (w.r.t. p).

First, the strict monotonicity from Eq. (1) that suggests

$$\begin{aligned} \begin{aligned}&\psi \left( \textbf{c}^{*},\left( \mathbf {\epsilon }_{-l}^{*}, {\epsilon }_{l}\right) ,\left( \mathbf {\epsilon }_{-l}^{*}, \tilde{\epsilon }_{l}\right) \right) >0, \\&\quad \forall \left( {\epsilon }_{l}, \tilde{\epsilon }_{l}\right) \in \left( {\epsilon }_{l}^{*}, {\epsilon }_{l}^{*}+\eta \right) \times \left( {\epsilon }_{l}^{*}-\eta , {\epsilon }_{l}^{*}\right) \end{aligned} \end{aligned}$$

(14)

Note that in order to obtain the strict inequality in Eq. (14), it is important that ${\epsilon }_l$ and $\widetilde{\epsilon }_l$ take values in disjoint open subsets of the interval $({\epsilon }_{l}^{*}-\eta , {\epsilon }_{l}^{*}+\eta )$ from Eq. (1).

Since $\psi$ is a composition of continuous functions (absolute value of the difference of two continuous functions), it is also continuous. Considering the open set $\mathbb {R}_{>0}$, it is known that pre-images (or inverse images) of open sets under a continuous function are always open. For the continuous function $\psi$, this pre-image corresponds to an open set

$$\begin{aligned} \begin{aligned} \mathcal {U} \subseteq \mathcal {C} \times \mathcal {E} \times \mathcal {E} \nonumber \end{aligned} \end{aligned}$$

in the domain of $\psi$ on which $\psi$ is strictly positive.

Moreover, due to Eq. (14)

$$\begin{aligned} \begin{aligned} \left\{ \textbf{c}^{*}\right\} {\times }\left( \left\{ \mathbf {\epsilon }_{{-}l}^{*}\right\} {\times }\left( {\epsilon }_{l}^{*}, {\epsilon }_{l}^{*}{+}\eta \right) \right) {\times }\left( \left\{ \mathbf {\epsilon }_{{-}l}^{*}\right\} {\times }\left( {\epsilon }_{l}^{*}{-}\eta , {\epsilon }_{l}^{*}\right) \right) \subset \mathcal {U} \end{aligned} \end{aligned}$$

(15)

so $\mathcal {U}$ is non-empty.

Next, by assumption (iii), there exists at least one subset $A \subseteq {1,..., d_{\epsilon }}$ of changing noise variables such that linA and $p_A(A) > 0$; pick one such subset and call it A.

Then, also by assumption (iii), for any $\mathbf {\epsilon }_A \in \mathcal {E}_A$, there is an open subset $\mathcal {O}(\mathbf {\epsilon }_A) \subseteq \mathcal {E}_A$ containing $\mathbf {\epsilon }_A$, such that $p_{\tilde{\mathbf {\epsilon }}_{A} \mid \mathbf {\epsilon }_{A}}\left( \cdot \mid \mathbf {\epsilon }_{A}\right) >0$ within $\mathcal {O}\left( \mathbf {\epsilon }_{A}\right)$.

Define the following space

$$\begin{aligned} \begin{aligned} \mathcal {R}_{A}:= & {} \left\{ \left( \mathbf {\epsilon }_{A}, \tilde{\mathbf {\epsilon }}_{A}\right) : \mathbf {\epsilon }_{A} \in \mathcal {E}_{A}, \tilde{\mathbf {\epsilon }}_{A} \in \mathcal {O}\left( \mathbf {\epsilon }_{A}\right) \right\} \nonumber \end{aligned} \end{aligned}$$

and, recalling that $A^{\textrm{c}}=\left\{ 1, \ldots , d_{\epsilon }\right\} \backslash A$ denotes the complement of A, define

$$\begin{aligned} \begin{aligned} \mathcal {R}:=\mathcal {C} \times \mathcal {E}_{A^{\textrm{c}}} \times \mathcal {R}_{A} \nonumber \end{aligned} \end{aligned}$$

which is a topological subspace of $\mathcal {C} \times \mathcal {E} \times \mathcal {E}$.

By assumptions (ii) and (iii), $p_\textbf{z}$ is smooth and fully supported, and $p_{\tilde{\mathbf {\epsilon }}_{A} \mid \mathbf {\epsilon }_{A}}\left( \cdot \mid \mathbf {\epsilon }_{A}\right)$ is smooth and fully supported on $\mathcal {O}(\mathbf {\epsilon }_A)$ for any $\mathbf {\epsilon }_A \in \mathcal {E}_A$. Therefore, the measure $\mu _{\left( \textbf{c}, \mathbf {\epsilon }_{A^{\textrm{c}}}, \mathbf {\epsilon }_{A}, \tilde{\mathbf {\epsilon }}_{A}\right) \mid A}$ has fully supported, strictly-positive density on $\mathcal {R}$ w.r.t. a strictly positive measure on $\mathcal {R}$. In other words, $p_{z} \times p_{\tilde{\epsilon }_{A} \mid {\epsilon }_{A}}$ is fully supported (i.e., strictly positive) on $\mathcal {R}$.

Now consider the intersection $\mathcal {U}\cap \mathcal {R}$ of the open set $\mathcal {U}$ with the topological subspace $\mathcal {R}$. Since $\mathcal {U}$ is open, by definition of topological subspaces, the intersection $\mathcal {U}\cap \mathcal {R} \subseteq \mathcal {R}$ is open in $\mathcal {R}$ and thus has the same dimension as $\mathcal {R}$ if non-empty.

Moreover, since $\mathcal {O}\left( \mathbf {\epsilon }_{A}^{*}\right)$ is open containing $\mathbf {\epsilon }_{A}^{*}$, there exists $\eta ^{\prime }>0$ such that $\left\{ \mathbf {\epsilon }_{-l}^{*}\right\} \times \left( {\epsilon }_{l}^{*}-\eta ^{\prime }, {\epsilon }_{l}^{*}\right) \subset \mathcal {O}\left( \mathbf {\epsilon }_{A}^{*}\right)$. Thus, for $\eta ^{\prime \prime }=\min \left( \eta , \eta ^{\prime }\right) >0$,

$$\begin{aligned} \begin{aligned} \left\{ \textbf{c}^{*}\right\} \times \left\{ \mathbf {\epsilon }_{A^{c}}^{*}\right\} \times \left( \left\{ \mathbf {\epsilon }_{A \backslash \{l\}}^{*}\right\} \times \left( {\epsilon }_{l}^{*}, {\epsilon }_{l}^{*}+\eta \right) \right) \\ \times \left( \left\{ \mathbf {\epsilon }_{A \backslash \{l\}}^{*}\right\} \times \left( {\epsilon }_{l}^{*}-\eta ^{\prime \prime }, {\epsilon }_{l}^{*}\right) \right) \subset \mathcal {R} . \nonumber \end{aligned} \end{aligned}$$

In particular, this implies that

$$\begin{aligned} \begin{aligned} \left\{ \textbf{c}^{*}\right\} {\times }\left( \left\{ \mathbf {\epsilon }_{{-}l}^{*}\right\} {\times }\left( {\epsilon }_{l}^{*}, {\epsilon }_{l}^{*}{+}\eta \right) \right) {\times }\left( \left\{ \mathbf {\epsilon }_{{-}l}^{*}\right\} {\times }\left( {\epsilon }_{l}^{*}{-}\eta ^{\prime \prime }, {\epsilon }_{l}^{*}\right) \right) \subset \mathcal {R} \end{aligned} \end{aligned}$$

(16)

Now, since $\eta ^{\prime \prime } \le \eta$, the LHS of Eq. (16) is also in $\mathcal {U}$, so the intersection $\mathcal {U} \cap \mathcal {R}$ is non-empty.

In summary, the intersection $\mathcal {U} \cap \mathcal {R} \subseteq \mathcal {R}$:

• is non-empty since both $\mathcal {U}$ and $\mathcal {R}$ contain the LHS of Eq. (15);

• is an open subset of the topological subspace $\mathcal {R}$ of $\mathcal {C} \times \mathcal {E} \times \mathcal {E}$ since it is the intersection of an open set, $\mathcal {U}$, with $\mathcal {R}$;

• satisfies $\psi >0$ since this holds for all of $\mathcal {U}$;

• is fully supported w.r.t. the generative process since this holds for all of $\mathcal {R}$.

As a consequence,

$$\begin{aligned} \begin{aligned} \mathbb {P}(\psi (\textbf{c}, \mathbf {\epsilon }, \tilde{\mathbf {\epsilon }})>0 \mid A) \ge \mathbb {P}(\mathcal {U} \cap \mathcal {R})>0 \end{aligned} \end{aligned}$$

(17)

where $\mathbb {P}$ denotes probability w.r.t. the true generative process p. Since $p_A(A)>0$, this is a contradiction to the invariance in Eq. (11).

Hence, $\textbf{h}_c(\textbf{c},\mathbf {\epsilon })$ does not depend on any noise variable ${\epsilon }_l$. It is thus only a function of $\textbf{c}$, i.e., $\hat{\textbf{c}} = \textbf{h}_c(\textbf{c})$.

Step 3. Finally, we show that the mapping $\hat{\textbf{c}} = \textbf{h}(\textbf{c})$ is invertible. We use the following result from (Zimmermann et al., 2021).

Proposition 1

Let $\mathcal {M}$, $\mathcal {N}$ be simply connected and oriented $\mathcal {C}^{1}$ manifolds without boundaries and $h: \mathcal {M} \rightarrow \mathcal {N}$ be a differentiable map. Further, let the random variable $\textbf{z} \in \mathcal {M}$ be distributed according to $\textbf{z} \sim p(\textbf{z})$ for a regular density function p, i.e., $0<p<\infty$. If the pushforward $p_{\# h}(\textbf{z})$ of p through h is also a regular density, i.e., $0<p_{\# h}<\infty$, then h is a bijection.

We apply this result to the simply connected and oriented $\mathcal {C}^{1}$ manifolds without boundaries $\mathcal {M}=\mathcal {C}$ and $\mathcal {N}=(0,1)^{d_{c}}$, and the smooth (hence, differentiable) map $\textbf{h}: \mathcal {C} \rightarrow (0,1)^{d_{c}}$ which maps the random variable $\textbf{c}$ to a uniform random variable $\hat{\textbf{c}}$ (as established in Eq. 11).

Since both $p_{\textbf{c}}$ (by assumption) and the uniform distribution (the pushforward of $p_{\textbf{c}}$ through $\textbf{h}$) are regular densities in the sense of Proposition 1, we conclude that $\textbf{h}$ is a bijection, i.e., invertible.

We have shown that for any smooth $\textbf{g}: \mathcal {X} \rightarrow (0,1)^{d_{c}}$ which minimises $\mathcal {L}_{\text{ AlignMax }}$, we have that $\hat{\textbf{c}}=\textbf{g}(\textbf{x})=\textbf{h}(\textbf{c})$ for a smooth and invertible $\textbf{h}: \mathcal {C} \rightarrow (0,1)^{d_{c}}$, i.e., $\textbf{c}$ is block-identified by $\textbf{g}$.

1.2 A.2 Proof of Theorems 2 and 3

In this section, we present proofs for the Theorem 2 and 3 in main paper Sections 4.1. These two theorems illustrate the deep relations between the Gaussian kernel $K:S^d {\times } S^d\ {\rightarrow }\ \mathbb {R}$ and the uniform distribution on the unit hypersphere $S^d$. As we will show below, these properties directly follow well-known results on strictly positive definite kernels.

Definition 2

(Strict positive definiteness). A symmetric and lower semi-continuous kernel K on $A {\times } A$ (where A is infinite and compact) is called strictly positive definite if for every finite signed Borel measure $\mu$ supported on A whose energy

$$\begin{aligned} I_{K}[\mu ] \triangleq \int _{\mathcal {S}^{d}} \int _{\mathcal {S}^{d}} K(u, v) \textrm{d} \mu (v) \textrm{d} \mu (u) \end{aligned}$$

(18)

is well defined, we have $I_{K}[{\mu }]\ {\ge }\ 0$, where equality holds only if ${\mu }\ {\equiv }\ 0$ on the $\sigma$-algebra of Borel subsets of A.

Definition 3

Let M($S^d$) be the set of Borel probability measures on $S^d$.

Definition 4

(Uniform distribution on $S^d$). ${\sigma }^d$ denotes the normalized surface area measure on $S^d$.

According to Borodachov et al.(Borodachov et al., 2019), we can get the following two results.

Lemma 1

(Strict positive definiteness of $G_{\gamma }$). For ${\gamma }\ >\ 0$, the Gaussian kernel is $G_{\gamma }(x, y)\ {\triangleq }\ e^{-\gamma \Vert x-y\Vert _{2}^{2}}$ strictly positive definite on $S^d {\times } S^d$.

Lemma 2

(Strictly positive definite kernels on $S^d$). Consider kernel $K_{f}: \mathcal {S}^{d} \times \mathcal {S}^{d} \rightarrow (-\infty ,+\infty ]$ of the form:

$$\begin{aligned} K_{f}(u, v) \triangleq f\left( \Vert u-v\Vert _{2}^{2}\right) \end{aligned}$$

(19)

If $K_f$ is strictly positive definite on $S^d {\times } S^d$ and $I_{K_f}[{\sigma }^d]$ is finite, then ${\sigma }^d$ is the unique measure (on Borel subsets of $S^d$) in the solution of $\min _{\mu \in \mathcal {M}\left( \mathcal {S}^{d}\right) } I_{K_{f}}[\mu ]$, and the normalized counting measures associated with any $K_f$ -energy minimizing sequence of N-point configurations on $S^d$ converges weak* to ${\sigma }^d$. In particular, this conclusion holds whenever f has the property that $-f'(t)$ is strictly completely monotone on (0, 4] and $I_{K_f}[{\sigma }^d]$ is finite.

Based on Lemmas 1 and 2, we now can get two propositions.

Proposition 2

${\sigma }^d$ is the unique solution (on Borel subsets of $S_d$) of

$$\begin{aligned}{} & {} \min _{\mu \in \mathcal {M}\left( \mathcal {S}^{d}\right) } I_{G_{t}}[\mu ]\nonumber \\ {}{} & {} =\min _{\mu \in \mathcal {M}\left( \mathcal {S}^{d}\right) } \int _{\mathcal {S}^{d}} \int _{\mathcal {S}^{d}} G_{t}(u, v) \textrm{d} \mu (v) \textrm{d} \mu (u) \end{aligned}$$

(20)

Definition 5

($\text {Weak}^{*}$ convergence of measures). A sequence of Borel measures $\left\{ \mu _{n}\right\} _{n=1}^{\infty }$ in $\mathbb {R}^{p}$ converges $\text {weak}^{*}$ to a Borel measure $\mu$ if for all continuous function $f: \mathbb {R}^{p} \rightarrow \mathbb {R}$, we have

$$\begin{aligned} \lim _{n \rightarrow \infty } \int f(x) \textrm{d} \mu _{n}(x)=\int f(x) \textrm{d} \mu (x) \end{aligned}$$

(21)

Proposition 3

For each $N\ >\ 0$, the N point minimizer of the average pairwise potential is

$$\begin{aligned} \textbf{u}_{N}^{*}= & {} \underset{u_{1}, u_{2}, \ldots , u_{N} \in \mathcal {S}^{d}}{\arg \min } \sum _{1 \le i<j \le N} G_{t}\left( u_{i}, u_{j}\right) \end{aligned}$$

(22)

The normalized counting measures associated with the $\left\{ \textbf{u}_{N}^{*}\right\} _{N=1}^{\infty }$ sequence converge $\text {weak}^{*}$ to $\sigma _{d}$.

According to Propositions 2 and 3, the Theorems 2 and 3 can be naturally proven.

1.3 A.3 Information Theory

In Sect. 1, we mentioned that sparser features contain less information, and the uniform distribution gets the maximum information entropy. This conclusion is inspired from Theorems 2.6.4 and Theorem 8.2.2 in (Thomas & Joy, 2006).

Theorem 2.6.4 $H (X) \le log |X|$, where |X| denotes the number of elements in the range of X, with equality if and only X has a uniform distribution over X.

Theorem 8.2.2 The typical set $A_{\epsilon }^{(n)}$ has the following properties:

• ${\text {Pr}}\left( A_{\epsilon }^{(n)}\right) >1-\epsilon$ for n sufficiently large.

• ${\text {Vol}}\left( A_{\epsilon }^{(n)}\right) \le 2^{n(h(X)+\epsilon )}$ for all n.

• ${\text {Vol}}\left( A_{\epsilon }^{(n)}\right) \ge (1-\epsilon ) 2^{n(h(X)-\epsilon )}$ for n sufficiently large.”

Theorem 2.6.4 proves that the uniform distribution gets the maximum information entropy, and Theorem 8.2.2 proves that sparser features contain less information.

B Experiment Details

1.1 B.1 Datasets

Cloud Datset We collect three datasets: 38cloud (Mohajerani & Saeedi, 2019), Sentinel-2 Cloud Mask (Aybar et al., 2022) and SPCRAS (Hughes & Hayes, 2014). The 38cloud dataset consists of images collected from the Landsat 8 satellite. The size of images varies from 7000$\times$7000 to 8000$\times$8000 pixels. The spatial resolution of this dataset is 30 m. The Sentinel-2 Cloud Mask dataset consists of images collected from the Sentinel-2 satellite. Each image is 1022$\times$1022 pixels in size. The spatial resolution of this dataset is 20 m. The SPCRAS dataset consists of images collected from the Landsat 8 satellite. Each image is 1000$\times$1000 pixels in size. The spatial resolution of this dataset is 30 m. These datasets are filtered, cropped and downsampled to construct a new dataset: ’Cloud’. The Cloud dataset has 1386 images in total, and each image has 128$\times$128 pixels and kilometer resolution.

USGS Datset: We collect the USGS image dataset of SIRI-WHU (Zhao et al., 2015). The dataset was acquired from United States Geological Survey (USGS) covering Montgomery, Ohio, USA. It has four classes: farms, forests, parking lot and residential area. It consists of one image of 10000$\times$9000 pixels in size. The spatial resolution of this image is 2 feet. We first crop image patches of 64$\times$64 pixels from the USGS image. Then, these patches are upsampled to 256$\times$256 pixels. In this way, we construct a new dataset which has 21,840 images in total, and almost sub-centimeter resolution.

HIT-UAV Dataset: The HIT-UAV dataset (Suo et al., 2023) comprises 2898 infrared thermal images extracted from 43,470 frames in hundreds of videos captured by Unmanned Aerial Vehicles (UAVs) in various scenarios. Each image is 256$\times$256 pixels in size, and only has one channel.

SAR Dataset: We collect HRSID-JPG (Wang et al., 2019) and SAR-Ship datasets (Wei et al., 2020a). The HRSID-JPG dataset comprises 5604 high-resolution SAR images and 16,951 ship instances. The SAR-Ship-Dataset has 43,819 ship slices. We filter these datasets to remove images with too much background. Then we constuct a new Radar dataset which has 9407 images in total, and each image is 256$\times$256 pixels in size and only has one channel.

Thermal Dataset: We conduct experiments on thermal natural image dataset for object classification (Ashfaq et al., 2021). The images were captured using Seek Thermal and FLIR. There are three classes: cat, car and man. This dataset was filtered, center-cropped and resized. We finally get a new dataset which has 5543 images in total, and each image is 256$\times$256 pixels in size.

1.2 B.2 Implementation Details

For the infrared, thermal, and Radar images, their image shapes are similar to the natural images, so we do not need to adjust the network architecture. During our experiments, we made some adjustments to the hyperparameters of the loss function as shown in Table 9.

Table 9 The choice of hyperparameters ${\lambda }_G$, ${\lambda }_D$, ${\delta }_G$ and ${\delta }_D$

Fig. 9

The FID curves on the a NWPU-RESISC45 dataset and b PatternNet dataset

C Visual Results

Fig. 10

The RS images generated by BigGAN+ADA(first from the left), BigGAN+ADA+Lecam(second from the left), StyleGAN2+ADA(middle), StyleGAN2+ADA+Lecam(second from the right) and our method(first from the right) trained on NWPU dataset

Fig. 11

The RS images generated by BigGAN+ADA(first from the left), BigGAN+ADA+Lecam(second from the left), StyleGAN2+ADA(middle), StyleGAN2+ADA+Lecam(second from the right) and our method(first from the right) trained on PN dataset

Fig. 12

The RS images generated by BigGAN+ADA(first from the left), BigGAN+ADA+Lecam(second from the left), StyleGAN2+ADA(middle), StyleGAN2+ADA+Lecam(second from the right) and our method(first from the right) trained on AID dataset

Fig. 13

a Cloud dataset. b Images generated by StyleGAN2+ADA. (c) Images generated by our method

Fig. 14

a USGS dataset. b Images generated by StyleGAN2+ADA. (c) Images generated by our method

Fig. 15

a HIT-UAV dataset. b Images generated by StyleGAN2+ADA. (c) Images generated by our method

Fig. 16

a SAR dataset. b Images generated by StyleGAN2+ADA. c Images generated by our method

Fig. 17

a Thermal dataset. b Images generated by StyleGAN2+ADA. c Images generated by our method

Fig. 18

The images generated by StyleGAN2+ADA(left), and our method(right) trained on FFHQ256 dataset

Fig. 19

The images generated by StyleGAN2+ADA(left), and our method(right) trained on LSUN-cat dataset

The generated images on NWPU, PN and AID datasets are visulized in the Figs. 10, 11 and 12. Overall, the images generated by the BigGAN gets the worst quality. Compared with BigGAN and StyleGAN2, the images generated by our method have great diversity and rich content. The generated images on Cloud, USGS, HIT-UAV, SAR and Thermal datasets are visualized in the Figs. 13, 14, 15, 16 and 17. Our method are effective on RS images of different scales and different modalities. The generated images on FFHQ256(10k) and LSUN-cat(30k) datasets are visualized in the Figs. 18 and 19. Our method still outperforms StyleGAN2 on these datasets.