Roger Marí

Gabriele Facciolo

Thibaud Ehret

Sat-NeRF: Learning Multi-View Satellite Photogrammetry With Transient Objects and Shadow Modeling Using RPC Cameras

We introduce the Satellite Neural Radiance Field (Sat-NeRF), a new end-to-end model for learning multi-view satellite photogrammetry in the wild. Sat-NeRF combines some of the latest trends in neural rendering with native satellite camera models, represented by rational polynomial coefficient (RPC) functions. The proposed method renders new views and infers surface models of similar quality to those obtained with traditional state-of-the-art stereo pipelines. Multi-date images exhibit significant changes in appearance, mainly due to varying shadows and transient objects (cars, vegetation). Robustness to these challenges is achieved by a shadow-aware irradiance model and uncertainty weighting to deal with transient phenomena that cannot be explained by the position of the sun. We evaluate Sat-NeRF using WorldView-3 images from different locations and stress the advantages of applying a bundle adjustment to the satellite camera models prior to training. This boosts the network performance and can optionally be used to extract additional cues for depth supervision.

Introduction

High-resolution satellite imagery is a valuable resource for countless economic activities, many of them based on knowledge of the geometry of the Earth's surface and its changes. This has triggered the development of a number of pipelines capable of highly accurate depth estimation from disparity using multiple satellite views [START_REF] Beyer | The Ames Stereo Pipeline: NASA's open source software for deriving and processing terrain data[END_REF][START_REF] Angelo | Dense multi-view stereo from satellite imagery[END_REF][START_REF] De Franchis | An automatic and modular stereo pipeline for pushbroom images[END_REF][START_REF] Facciolo | Automatic 3D reconstruction from multi-date satellite images[END_REF]19,[START_REF] Rupnik | 3D reconstruction from multi-view VHR-satellite images in MicMac[END_REF]45]. The output large-scale 3D models are usually represented using discrete point clouds or digital surface models (DSMs) of a certain resolution.

The latest works in 3D modeling from multiple views show that it is possible to achieve a superior representation of a 3D object or scene by learning it as a continuous function or field F [49]. Neural rendering methods learn F by integrating differentiable rendering techniques into a neural network. The tasks of novel view synthesis and 3D re- construction are then solved implicitly, as the network is trained to figure out which geometry and color radiances fit the camera projection mappings of the different views.

Neural radiance fields (NeRFs) have gained great popularity in the field of neural rendering [START_REF] Mildenhall | NeRF: Representing scenes as neural radiance fields for view synthesis[END_REF]49]. In this paper, we introduce a NeRF variant architecture that achieves state-of-the-art results in novel view synthesis and 3D reconstruction from high-resolution satellite imagery in the wild. We refer to our variant as Satellite NeRF or Sat-NeRF. The original NeRF approach is not adapted to satellite images, e.g. because of the specificities of the camera models, the large distance between the cameras and the scene or the appearance inconsistencies of multi-date collections [START_REF] Derksen | Shadow neural radiance fields for multi-view satellite photogrammetry[END_REF][START_REF] Martin-Brualla | NeRF in the wild: Neural radiance fields for unconstrained photo collections[END_REF]. Sat-NeRF addresses these challenges using some of the latest advances in NeRFs [START_REF] Deng | Depth-supervised NeRF: Fewer views and faster training for free[END_REF][START_REF] Derksen | Shadow neural radiance fields for multi-view satellite photogrammetry[END_REF][START_REF] Martin-Brualla | NeRF in the wild: Neural radiance fields for unconstrained photo collections[END_REF] and adapting well-known tools for satellite image processing. As a result, the model learns highly accurate 3D geometry, similar to that obtained with state-of-the-art stereo pipelines relying on handcrafted features (Figure 1).

Our contributions consist of:

-A NeRF variant that combines existing state-of-the-art methods to adapt to the satellite context. It is robust to the radiometric inconsistencies of multi-date satellite imagery, comprising shadows caused by a single nonstatic light source (the sun) and small transient objects (mainly trees or cars in open-air parkings). -A point sampling strategy adapted to satellite camera models. The rational polynomial camera (RPC) model [START_REF] Fraser | Sensor orientation via RPCs[END_REF][START_REF] Grodecki | IKONOS stereo feature extraction-RPC approach[END_REF] of each input image is directly used to cast rays in the object space. This RPC-based strategy provides independence to the satellite system and improves the results obtained with approximate pinhole cameras. -A study of the advantages of correcting RPC inconsistencies before training, e.g. by means of a bundle adjustment [31]. We show that eluding this step leads to a drop in the performance of the model. In addition, we detail how to reuse the sparse point cloud employed in the bundle adjustment to improve geometry learning.

We evaluate Sat-NeRF on different areas of interest covering 256 × 256 m each, using ∼10-20 RGB crops from multi-date WorldView-3 images for training [8,[START_REF] Bittner | Late or earlier information fusion from depth and spectral data? Large-scale digital surface model refinement by hybrid-cGAN[END_REF]. A lidar digital surface model (DSM) of resolution 0.5 m/pixel is used as ground truth model to assess the geometry. Sat-NeRF is compared to other NeRF variants [START_REF] Derksen | Shadow neural radiance fields for multi-view satellite photogrammetry[END_REF][START_REF] Mildenhall | NeRF: Representing scenes as neural radiance fields for view synthesis[END_REF] as well as a state-of-the-art traditional satellite stereo pipeline [START_REF] De Franchis | An automatic and modular stereo pipeline for pushbroom images[END_REF]. We also publish the code and data used for this article.

Related work

Current state-of-the-art 3D reconstruction pipelines for satellite images typically follow multi-view stereo approaches, which can outperform sophisticated true multiview software [18,[START_REF] Ozge | A comparison of stereo and multiview 3-D reconstruction using cross-sensor satellite imagery[END_REF]. Due to the complexity of the task, satellite stereo pipelines can still be improved in a number of aspects. Some of the most important limitations are:

-The 3D reconstruction usually follows the estimation of a dense disparity map using matching strategies derived from the semi-global matching algorithm [START_REF] Beyer | The Ames Stereo Pipeline: NASA's open source software for deriving and processing terrain data[END_REF][START_REF] Angelo | Dense multi-view stereo from satellite imagery[END_REF][START_REF] De Franchis | An automatic and modular stereo pipeline for pushbroom images[END_REF]19,24]. Therefore, human-crafted features and cost functions are at the core of the methodology. -The selection of suitable stereo pairs to estimate disparity is another major challenge. Criteria based on image metadata (e.g. acquisition dates, incidence angles, etc.) have proven to be useful, but do not guarantee the best choice [START_REF] Facciolo | Automatic 3D reconstruction from multi-date satellite images[END_REF]19,[START_REF] Han | State of the art in digital surface modelling from multi-view high-resolution satellite images[END_REF]. -The lack of consensus on how the geometry derived from multiple stereo pairs should be refined or aggregated. Local point-wise operations are common to merge altitude values derived from different pairs, e.g. median [START_REF] Angelo | Dense multi-view stereo from satellite imagery[END_REF]19,[START_REF] Marí | To bundle adjust or not: A comparison of relative geolocation correction strategies for satellite multiview stereo[END_REF] or k-medians [START_REF] Facciolo | Automatic 3D reconstruction from multi-date satellite images[END_REF]. However, recent work has shown that deep learning approaches can greatly improve the result, e.g. by exploiting geometric priors related to urban areas [6, 7, 28, 48].

-Very often, it is necessary to make adjustments or parameter tuning to handle different sources or types of satellite images [START_REF] Ozge | A comparison of stereo and multiview 3-D reconstruction using cross-sensor satellite imagery[END_REF]55].

Neural rendering represents an opportunity to find a natural solution to the previous issues, as it automatically learns the optimal features and operations adapted to each individual 3D scene. The main advantage of traditional pipelines is preserved as no explicit geometry supervision is required: the learning is self-supervised and based solely on the color of the input images. This is a key difference with respect to other state-of-the-art deep learning methods for DSM generation from satellite imagery [7, 17, 18, 48], which depend on ground truth geometry models.

Neural Radiance Fields

NeRF [START_REF] Mildenhall | NeRF: Representing scenes as neural radiance fields for view synthesis[END_REF] represents a static scene as a continuous volumetric function F, encoded by a fully-connected neural network. F predicts the emitted RGB color c = (r, g, b) and a non-negative scalar volume density σ at a 3D point x = (x, y, z) of the scene seen from a viewing direction d = (d x , d y , d z ), i.e.

F : (x, d) → (c, σ). (1)
Multi-view consistency is encouraged by restricting the network to predict the volume density σ based only on the spatial coordinates x, while allowing the color c to be predicted as a function of both x and the viewing direction d. The dependency of c on the viewing direction allows to recreate specular reflections caused by to static light sources. Given a set of input views and their camera poses, the training strategy is based on rendering the color of individual rays traced across the scene and projected onto the known pixels. Individual rays are chosen randomly, encouraging gradient flow at those ray intersections where the surface of the scene is susceptible of being located. Each ray r is defined by a point of origin o and a direction vector d. The color c(r) of a ray r(t) = o + td is computed as

c(r) = N i=1 T i α i c i .
(2)

The rendered color c(r) results from the weighted integration of the colors c i predicted at different points of the ray r, which is discretized into N 3D points x i between the near and far bounds of the scene, t n and t f . Each point x i in r is obtained as

x i = o + t i d, where t i ∈ [t n , t f ].
Following (2), the weight given to the color predicted for each point x i of r is defined by a transmittance factor T i representing the probability that light reaches the point without hitting any other particle, and an alpha compositing value α i encoding the opacity. Both T i and α i are set according to the volume density σ i predicted for x i :

α i = 1 -exp(-σ i δ i ); T i = i-1 j=1 (1 -α j ) , (3)
where δ i is the distance between two consecutive points along the ray, i.e. δ i = t i+1 -t i . Higher values of σ i will result in larger opacity α i , indicating that x i possibly belongs to a non-transparent surface. Occlusions are handled by the transmittance T i , equal to the cumulative product of the inverse opacity. Even if x i is given a large σ i , T i only allows it to contribute decisively to the rendered color if it is not preceded by previous opaque points in the ray. Given (3), the depth d(r) observed in the direction of a ray r can be rendered in a similar manner to (2) [START_REF] Deng | Depth-supervised NeRF: Fewer views and faster training for free[END_REF][START_REF] Barron | Mip-NeRF: A multiscale representation for anti-aliasing neural radiance fields[END_REF] as

d(r) = N i=1 T i α i t i . (4)
NeRF is optimized by minimizing the mean squared error (MSE) between the rendered color and the real color of the input images, at the positions where the rays project:

r∈R ∥c(r) -c GT (r)∥ 2 2 , (5)
where c GT (r) is the observed color of the pixel intersected by the ray r, and c(r) is the color predicted by the NeRF using (2). R is the set of rays in each input batch.

NeRF variants

NeRF assumes that the density, radiance and illumination of the target 3D scene is constant. This is a strong limitation, as these conditions are rarely encountered outside laboratory settings. Many variants have been proposed to address this problem. In this section we briefly review three models that inspired our work.

NeRF-W [START_REF] Martin-Brualla | NeRF in the wild: Neural radiance fields for unconstrained photo collections[END_REF] or NeRF in the Wild gains robustness to radiometric variation and transient objects by learning to separate transient phenomena from the static scene. An extra head of fully-connected layers is used to predict a transient color c τ and volume density σ τ for each input point, in addition to the usual c and σ. The transient outputs are linearly combined with the static ones to render the color of each ray. NeRF-W also uses the transient head to emit an uncertainty coefficient β, which measures the confidence of the network that a point belongs to a transient object. The value of β is used in the loss function to reduce the impact of transient/unreliable pixels in the learning process.

S-NeRF [START_REF] Derksen | Shadow neural radiance fields for multi-view satellite photogrammetry[END_REF] or Shadow NeRF is, to the best of our knowledge, the first attempt to apply NeRF for multi-view satellite photogrammetry. S-NeRF showed the benefits in geometry estimation of simultaneously exploiting the direction of solar rays to learn the amount of sunlight s that reaches each point x of the scene. The direction of solar rays is a common metadata of satellite images. Our work can be seen as an extension of S-NeRF that incorporates a modeling of transient objects similar to [START_REF] Martin-Brualla | NeRF in the wild: Neural radiance fields for unconstrained photo collections[END_REF] and a representation of the camera models more adapted to satellite data.

DS-NeRF [START_REF] Deng | Depth-supervised NeRF: Fewer views and faster training for free[END_REF] or Depth Supervised NeRF incorporates a depth supervision term to the loss function to accelerate the learning and reduce the amount of input images. The depth supervision term exploits a sparse set of 3D points that belong to the surface of the scene, which can be easily retrieved using structure-from-motion (SfM) pipelines. SfM is a common pre-processing step in NeRF frameworks, as it can estimate the camera poses needed to cast the input rays. A similar strategy to DS-NeRF is used in [START_REF] Barron | Mip-NeRF: A multiscale representation for anti-aliasing neural radiance fields[END_REF], which converts the sparse point clouds into dense depth priors.

Other recent NeRF variants are yet to be investigated in the context of satellite imagery. Some works are focused on achieving smoother scene representations or reducing the number of input views: e.g. DietNeRF [25] introduces an auxiliary semantic loss to maximize similarity between high-level features instead of RGB colors; Mip-NeRF [3] prevents blurring and aliasing in collections of images with different resolutions; PixelNeRF [54] describes a framework that is trained across multiple scenes and learns priors that can generalize to unseen scenes with few available images. Recent undergoing research is also progressing to extend NeRFs to dynamic scenes [40-42, 51], to gain efficiency and reduce the training time [23,36,52,[START_REF] Yu | PlenOctrees for real-time rendering of neural radiance fields[END_REF] or to handle complex illumination settings, under arbitrary, multiple light sources [START_REF] Bi | Neural reflectance fields for appearance acquisition[END_REF]47] or near-darkness conditions [START_REF] Mildenhall | NeRF in the dark: High dynamic range view synthesis from noisy raw images[END_REF].

Method

Sat-NeRF represents the scene as a static surface with an albedo color, i.e. the intrinsic color of static objects. The model learns to predict the geometry and the albedo color simultaneously with a set of additional outputs, which seek to explain the transient phenomena observed in the input images without inducing changes in the scene geometry.

We train the model following the ray casting strategy of NeRF (Section 2.1). Unlike the original NeRF (1), we assume a Lambertian surface and omit the color dependence on viewing angles. The inputs are x: 3-valued vector with the spatial coordinates of points located in the scene volume.

x is part of a ray r.

ω: 3-valued direction vector encoding the direction of solar rays. For each input image, ω is extracted from the azimuth and elevation angles (θ, ϕ) that indicate the position of the sun in the satellite image metadata. -t j : N (t) -valued embedding vector, learned as a function of the image index j. The objective of t j is to featurize the transient elements in the j-th view that cannot be explained by the position of the sun given by ω. We manually set N (t) = 4. The volumetric function of Sat-NeRF then writes F : (x, ω, t j ) → (σ, c a , s, a, β), where the outputs are σ: scalar encoding the volume density at location x.

c a : albedo RGB color, which depends exclusively on the geometry, i.e. the spatial coordinates x. -s: shadow-aware shading scalar, learned as a function of x and the solar rays direction vector ω. -a: ambient RGB color, independent of scene geometry, that defines a global hue bias according to the position of the sun given by ω. -β: uncertainty coefficient related to the probability that the color of x is explained by a transient object.

Shadow-aware irradiance model

This section describes how Sat-NeRF predicts the color c(r) of a ray r projected onto a certain pixel. We keep the rendering as in (2) and ( 4), with the transmittance and opacity factors as defined in (3), but adopt the shadow-aware irradiance model proposed in S-NeRF [START_REF] Derksen | Shadow neural radiance fields for multi-view satellite photogrammetry[END_REF] to compute the color c at each point x of a ray r:

c(x, ω, t j ) = c a (x)• s(x, ω)+(1-s(x, ω))•a(ω) , (6)
where c(x, ω, t j ) substitutes c i in (2). The shading scalar s(x, ω) takes values between 0 and 1 and is used to add shadows by darkening the albedo (Figure 2). Ideally, s ≈ 1 in those 3D points directly illuminated by the sun, whose color should be entirely explained by the albedo c a (x).

In addition, (6) attempts to capture the bluish hues of shadows [2, 30] by means of the ambient color a(ω), which contributes to the points where s takes values closer to 0. In practice, we find that the direction of the solar rays ω is narrowly related to the acquisition date (especially if the satellite passes at the same hours of the day), as shown in Figure 2. Thus, a(ω) ends up capturing ambient irradiance due to a mixture of phenomena, which is related to ω but also date-specific conditions like weather or seasonal changes.

As observed in S-NeRF [START_REF] Derksen | Shadow neural radiance fields for multi-view satellite photogrammetry[END_REF], the shading scalar s(x, ω) in (6) can produce unrealistic results for solar rays directions that are not seen in the training data. This can be minimized by adding a solar correction term to the loss:

L SC (R SC ) = r∈RSC NSC i=1 (T i -s i ) 2 + 1 - NSC i=1 T i α i s i , (7)
where R SC is a secondary batch of solar correction rays. Note that the rays in R SC follow the direction of solar rays ω, while the rays in R, in the main term of the loss (5), follow the viewing direction of the camera.

The solar correction term (7) uses the learned geometry, encoded by the transmittance T i and opacity α i (3), to further supervise the learning of the shadow-aware shading s(x, ω). The first part of (7) enforces that, for each ray r in R SC , the s i predicted at the i-th point should resemble T i , i.e. high values before reaching the visible surface, low values afterwards (both s i and T i take values between 0 and 1). The second part of (7) encourages that the integration of s over r reaches 1, since non-occluded and non-shadow areas have to be mostly explained by the albedo in (6).

Uncertainty weighting for transient objects

Similarly to NeRF-W [START_REF] Martin-Brualla | NeRF in the wild: Neural radiance fields for unconstrained photo collections[END_REF], we use the task-uncertainty learning approach introduced in [26] to gain robustness to transient objects by means of β. In our context, transient objects are punctual local changes across the input images that cannot be explained by the static surface or the available metadata, like the position of the sun. The irradiance model (6) does not handle transient objects explicitly. As a result, we observe that s and σ usually try to account for them, leading to wrong depth predictions, as shown in Fig- ure 4. Thanks to β, Sat-NeRF is given some margin to ignore the color inconsistencies caused by these objects.

The uncertainty prediction β weights the contribution of each ray to the MSE between rendered and known colors:

L RGB (R) = r∈R ∥c(r) -c GT (r)∥ 2 2 2β ′ (r) 2 + log β ′ (r)+η 2 , (8)
where β ′ (r) = β(r) + β min . In (8), we use β min = 0.05 and η = 3 to avoid negative values in the logarithm. The role of the logarithm in L RGB is to prevent β from converging to infinity to solve the problem. In this way the model is forced to find a compromise between the uncertainty coefficients β and the differences of colors.

The β(r) associated with a ray r is obtained by integrating the uncertainty predictions across the N points of r:

β(r) = N i=1 T i α i β(x i , t j ), (9)
where β(x i , t j ), is the uncertainty coefficient predicted at the i-th point of r. Sat-NeRF learns to predict the uncertainty β at each point of the scene based on its spatial coordinates x (some areas are more likely to exhibit transient objects, e.g. open-air parkings in Figure 2) and on the transient embedding vector t j of each input training image. Depending on each view, the areas typically affected by transient objects will arbitrarily differ to a greater or lesser extent with respect to the albedo. Note that the embedding vector t j is learned from the image index j during training. 1 We find that it is better to start using β after the second epoch, when the shadow-aware shading s is already well initialized. Otherwise the model may use β to overlook shadow areas instead of trying to explain them with s. Thus, we replace (8) with [START_REF] Bi | Neural reflectance fields for appearance acquisition[END_REF] in the first two epochs. 1 At test time, we use an arbitrary t j selected from the training set.

Figure 2. The shading scalar s related to the solar rays direction ω learns shadows and material roughness. We observe that ω is narrowly related to the acquisition date, causing the ambient color a associated with the low values of s (see ( 6)) to capture a mixture of phenomena, including seasonal changes reflected in the vegetation. The uncertainty prediction β does not affect shadows and concentrates on small color inconsistencies, mostly caused by cars in open-air parkings (green marks), large fans in rooftops (blue marks) or building edges.

Point sampling from satellite RPC models

Sat-NeRF casts rays directly using the RPC camera models of a set of satellite images. The RPC model is widely used for optical satellite imagery, as it allows to describe complex acquisition systems independently of satellitespecific physical modeling [START_REF] Akiki | Robust rational polynomial camera modelling for SAR and pushbroom imaging[END_REF][START_REF] Grodecki | IKONOS stereo feature extraction-RPC approach[END_REF]. Each RPC is defined by a projection function (to project 3D points onto image pixels) and its inverse, the localization function.

The use of RPCs in a NeRF framework represents an improvement with respect to previous work with satellite data. In S-NeRF [START_REF] Derksen | Shadow neural radiance fields for multi-view satellite photogrammetry[END_REF] the RPC model of each input view is replaced with a custom simplified pinhole camera matrix, which is the common representation used in NeRF for close-range imagery [START_REF] Mildenhall | NeRF: Representing scenes as neural radiance fields for view synthesis[END_REF]. The RPC-based sampling described here corresponds to a more general approach, which also leads to better results (see Section 4).

We denote the minimum and maximum altitudes of the scene as h min and h max , respectively. 2 The ray that crosses the scene and intersects the pixel p of the j-th image is modeled as a straight line between an initial and a final 3D point, i.e. x start and x end . These boundary points are obtained by localizing the pixel p at h min and h max , using the RPC localization function L j of the j-th image:

x start = L j (p, h max ) ECEF ; x end = L j (p, h min ) ECEF , (10)
where the subindex ECEF indicates that the 3D points returned by the localization function L j are converted to the Earth-centered, Earth-fixed coordinate system (or geocentric system), to work in a cartestian system of reference.

Given x start and x end , the origin o and direction vector d of the ray r(t) = o + td that intersects the pixel p of the 2 The altitude bounds [h min , hmax] can be selected in various ways, e.g. from a large-scale elevation model extracted from low-resolution data. j-th image are expressed as

o = x start ; d = x end -x start ∥x end -x start ∥ 2 . ( 11
)
The point of maximum altitude, x start , which is the closest to the camera, is taken as the origin o of the ray. The boundaries of the ray r(t) = o + td, i.e. [t min , t max ], are set as t min = 0 and t max = ∥x end -x start ∥ 2 . Since working with ECEF coordinates is impractical, due to large values used in the representation, we normalize all ray points in the interval [-1, 1] using an offset subtraction and scaling procedure similar to the one used in the RPC functions [START_REF] Grodecki | IKONOS stereo feature extraction-RPC approach[END_REF]. The set of 3D points resulting from localizing all pixels in the input images at h min and h max is used to compute the offset and scale in each spatial dimension.

RPC refinement for improved performance

Bundle adjustment approaches are a common good practice in remote sensing to correct inconsistencies between a collection of RPC models observing the same scene [START_REF] Grodecki | Block adjustment of highresolution satellite images described by rational polynomials[END_REF]31,[START_REF] Marí | To bundle adjust or not: A comparison of relative geolocation correction strategies for satellite multiview stereo[END_REF][START_REF] Ozge | Automatic geo-location correction of satellite imagery[END_REF]. In particular, bundle adjustment methods correct the RPCs by minimizing the reprojection error of a set of corresponding points seen across the images [50].

In absence of a prior RPC refinement, a 3D point projected with different raw RPC functions often falls on noncoincident image points, by a distance of up to tens of pixels [START_REF] Grodecki | Block adjustment of highresolution satellite images described by rational polynomials[END_REF]. This would cause a systematic loss of accuracy in any NeRF methodology for satellite imagery, because rays traced from corresponding pixels of different views would not intersect at an exact point in the object space. To prevent this situation, before training Sat-NeRF, we apply the bundle adjustment method described in [31], which performs a relative correction of the RPC models of all input images. The set of points used by the bundle adjustment is derived from correspondences of SIFT keypoints [29].

While the refined RPC models directly increase the accuracy of the point sampling strategy described in Section 3.3, a prior bundle adjustment can also improve the Sat-NeRF performance in other ways. DS-NeRF [START_REF] Deng | Depth-supervised NeRF: Fewer views and faster training for free[END_REF] discussed how the training of a NeRF can benefit from a sparse set of previously known 3D points, under the idea that such points can be easily produced using SfM pipelines. In the case of satellite imagery, the bundle adjustment produces an equivalent set of sparse 3D points derived from image features [10,31,39,45]. Based on this idea, we explore the benefits of adding the depth supervision term proposed in [START_REF] Deng | Depth-supervised NeRF: Fewer views and faster training for free[END_REF] to the loss of our Sat-NeRF model:

L DS (R DS ) = r∈RDS w(r) (d(r) -∥X(r) -o(r)∥ 2 ) 2 , (12)
where d(r) is the depth (4) predicted for a ray r, whose origin point is o(r). If r intersects X(r), a known 3D point, then ∥X(r) -o(r)∥ 2 is the target depth to be learned. R DS denotes a batch of rays that intersects known 3D points. Since the pixel coordinates associated with these 3D points are already provided by the bundle adjustment, all rays in R DS can be defined as explained in Section 3.3. Normalized coordinates between [-1, 1] are used in [START_REF] De Franchis | An automatic and modular stereo pipeline for pushbroom images[END_REF] to represent points in the object space, for consistency with Section 3.3.

Similarly to [START_REF] Deng | Depth-supervised NeRF: Fewer views and faster training for free[END_REF], we only use L DS in the initial 25% of training iterations. In our experience, this proportion is usually enough to gain accuracy in the learned geometry. Observe that the contribution of each depth supervision ray r in R DS is weighted by w(r) in [START_REF] De Franchis | An automatic and modular stereo pipeline for pushbroom images[END_REF], where w(r) is a scalar set according to the reprojection error of each point X(r) provided by the bundle adjustment.

Multi-task loss and network architecture

The main term of the Sat-NeRF loss function is the L RGB defined in (8), which is complemented by the solar correction term L SC (7) and the depth supervision term L DS [START_REF] De Franchis | An automatic and modular stereo pipeline for pushbroom images[END_REF]. The complete loss function can be expressed as

L = L RGB (R) + λ SC L SC (R SC ) + λ DS L DS (R DS ), (13)
where λ SC and λ DS are an arbitrary weight given to each secondary term. We empirically find λ SC = 0.1/3 and λ DS = 1000/3 to provide good results, to keep the secondary terms sufficiently relevant but below the magnitude of L RGB . For depth supervision, we used ∼2k-10k bundle adjustment points depending on the output of [31] for each area of interest. R, R SC and R DS have the same batch size.

The architecture of Sat-NeRF is shown in Figure 3. The main block of fully-connected layers, with h channels per layer, is dedicated to the prediction of the static properties of the scene: the volume density σ and the albedo color c a . A secondary head is added with fewer layers and half as many channels per layer to estimate the shading scalar s based on the direction of solar rays ω and the vector of h geometry-related features learned by the main block. Lastly, two single-layer heads are used to predict the uncertainty coefficient β and the ambient color a, from the transient embedding vector t j and ω, respectively. We employ SIREN layers with the initialization proposed in [START_REF] Vincent Sitzmann | Implicit neural representations with periodic activation functions[END_REF], as suggested in [START_REF] Derksen | Shadow neural radiance fields for multi-view satellite photogrammetry[END_REF]. The use of a softplus function to predict σ was crucial to achieve satisfactory results. The uncertainty β is also produced by a softplus [START_REF] Martin-Brualla | NeRF in the wild: Neural radiance fields for unconstrained photo collections[END_REF], which yields a smoother optimization problem compared to the usual ReLU [3]. The rest of the outputs result from sigmoid functions, since they are related to normalized RGB values and have to be in the interval [0, 1]. The value of h should be adjusted according to the resolution and the size of the observed area. In this work we set h = 512.

Evaluation

We evaluate Sat-NeRF on different areas of interest (AOI) of the 2019 IEEE GRSS Data Fusion Contest [8, 27], which provides 26 Maxar WorldView-3 images collected between 2014 and 2016 over the city of Jacksonville, Florida, US. From this data, we take as input a set of RGB crops of varying size, around 800×800 pixels, with a resolution of 0.3 m/pixel at nadir, covering 256×256 m for each AOI. The indices of the selected AOIs and the number of training and test images that were used are listed in Table 1.

In all conducted experiments we use a single NeRF model, trained with an Adam starting with a learning rate of 5e -4 , which is decreased at every epoch by a factor γ = 0.9 according to a step scheduler. The batch size is 1024 rays, and each ray r is discretized into 64 uniformly distributed 3D points. Training takes 300k iterations to converge, resulting in ∼10 h if a single batch of rays is used at each training iteration, or ∼20 h if a secondary term for solar correction or depth supervision is added to the loss (trained on a GPU with 16 GB RAM). We used bundle adjusted RPCs (Section 3.4) unless otherwise noted.

Ablation study

We evaluated the Sat-NeRF model starting from a simple NeRF and gradually adding new components. To this end, we propose three categories of experiments that are discussed below. Table 2 shows the quantitative results.

Category 1. Rows 0-3 are an ablation study dedicated to the irradiance model and the solar correction term described in Section 3.1. We verify that the S-NeRF irradiance model outperforms a basic NeRF and is strengthened by the solar correction term. Comparing our results with the ones reported in the original S-NeRF work [START_REF] Derksen | Shadow neural radiance fields for multi-view satellite photogrammetry[END_REF] reveals the impact of the proposed RPC-based point sampling and bundle adjustment detailed in Section 3, to which we attribute the difference between the metrics of row 0 and row 3.

Category 2. Rows 4-5 assess our Sat-NeRF model, which incorporates the uncertainty prediction of β and employs (8) as main term of the loss function. These rows show that the uncertainty modeling improves both the learned geometry and the novel view synthesis, as illustrated in Figure 4. Compared to the best S-NeRF results (row 3), Sat-NeRF (row 4) provides higher PSNR/SSIM and similar or even smaller altitude MAE without requiring a solar correction term. This insight could be exploited in settings that cannot afford additional training time to process a secondary batch of rays for solar correction. If we add the solar correction term (7) to the Sat-NeRF loss (8), the altitude MAE decreases even more: row 5 outperforms all the previous configurations across all AOIs.

Category 3. Rows 6-8 demonstrate the benefits of using a prior bundle adjustment to refine the RPC models of the satellite images (Section 3.4). The comparison of rows 6 and 5 reveals that the use of unrefined RPCs induces a performance drop: both PSNR/SSIM and altitude MAE are worse. Lastly, rows 7-8 add the depth supervision term [START_REF] De Franchis | An automatic and modular stereo pipeline for pushbroom images[END_REF] to the Sat-NeRF loss (8) with and without solar correction, to leverage the sparse point cloud provided by the bundle adjustment. This strategy proves to be beneficial to further improve the altitude MAE in some areas (214 and 260). In other areas (004 and 068), the sparse point cloud may contain outliers, especially in multi-date collections, leading to slightly worse MAE compared to a plain Sat-NeRF loss.

Comparison to traditional stereo pipelines

Sat-NeRF learns high quality 3D models, similar in accuracy to those obtained with satellite stereo pipelines relying on traditional algorithms for stereo matching [START_REF] Beyer | The Ames Stereo Pipeline: NASA's open source software for deriving and processing terrain data[END_REF][START_REF] De Franchis | An automatic and modular stereo pipeline for pushbroom images[END_REF]. In this work, we compare the DSMs produced by Sat-NeRF with a multi-view stereo DSM of the same area generated with S2P [START_REF] De Franchis | An automatic and modular stereo pipeline for pushbroom images[END_REF][START_REF] Facciolo | Automatic 3D reconstruction from multi-date satellite images[END_REF], the satellite stereo pipeline that won the 2016 IARPA Multi-View Stereo 3D Mapping Challenge [START_REF] Bosch | A multiple view stereo benchmark for satellite imagery[END_REF].

We follow the methodology described in [START_REF] Facciolo | Automatic 3D reconstruction from multi-date satellite images[END_REF] to produce the S2P DSMs. For each AOI, we manually select 10 stereo pairs for disparity estimation. The selection criterion prioritizes pairs with an angle between views of 5 to 45 degrees, with a maximum incidence angle of 40 degrees for each view. Within this set, we take the 10 pairs with closer acquisition dates and run S2P. The RPCs used by S2P were the same used to train Sat-NeRF, i.e. all RPCs are bundle adjusted using [31]. The 10 pairwise models are fused into a single DSM by taking the median altitude at each cell. To maximize the quality of the S2P DSMs we used the panchromatic product of the WorldView-3 images, instead of the RGB crops employed to train Sat-NeRF. Considering that the RGB images have a compressed dynamic (integer values in [0, 255]), i.e with less texture and more saturated areas, the Sat-NeRF DSMs are very encouraging compared to the state of the art with manual pair selection.

As shown in Figure 5 and 6, structures are more detailed in Sat-NeRF DSMs, but S2P provides more regular surfaces. Numerically, the global altitude MAE obtained with Sat-NeRF can be slightly better compared to the S2P DSMs ( For clarity, we provide an RGB view of the area from a near-nadir perspective. Water bodies are masked in the DSMs.

Conclusion

We introduced Sat-NeRF, a NeRF variant adapted for multi-date collections of multi-view satellite images. The geometry and appearance of permanent structures are simultaneously learned using a main backbone, while shadows and transient objects are learned by secondary heads.

The proposed method achieves state-of-the-art results in novel view synthesis and 3D modeling from satellite imagery. It also highlights the benefits of incorporating well-known techniques for satellite image processing into a NeRF framework. In particular, we show how to represent the input cameras using the RPC models characteristic of satellite images, instead of the pinhole cameras commonly used in NeRF for close-range imagery. We also demonstrate the advantages of applying a bundle adjustment step before training time, to improve reconstruction quality and, optionally, to provide additional cues for depth supervision.

Figure 1 .

 1 Figure 1. Images 1 and 2 exhibit color inconsistencies (e.g. shadows, cars, vegetation), hindering the direct use of NeRF. Sat-NeRF overcomes these problems and learns to render realistic views and underlying geometry. The digital surface model (DSM) derived from the network depth predictions is compared with a lidar equivalent, obtaining a mean absolute altitude error (MAE) similar to that of multi-view stereo (MVS) relying on handcrafted features.

Figure 3 .

 3 Figure3. Sat-NeRF network architecture, where x are the input spatial coordinates, ω is the direction of solar rays and tj is the learned transient embedding of image j. The model predicts the volume density σ, the components of the irradiance model (6), i.e. albedo color ca, shading scalar s, ambient color a, and an uncertainty coefficient β to weight the impact of transient objects.

Figure 4 .

 4 Figure 4. The uncertainty coefficient β learned by Sat-NeRF helps to improve the geometry learning with respect to S-NeRF [14]. In addition to shadows and textures, the shading scalar s of the irradiance model (6) usually attempts to account for transient objects (e.g. cars, marked in green). Sat-NeRF uses β to minimize the contribution of transient objects to the loss function (8), thus allowing the geometry and s to learn to ignore their color. In practice, we notice that s still retains some transient objects, but the learned geometry is much better, as shown in the corresponding DSM.

Figure 5 .

 5 Figure 5. Left to right: ground truth lidar DSM, Sat-NeRF DSM and S2P DSM [12]. The Sat-NeRF DSM corresponds to the model with lowest altitude MAE, in bold in Table2. Compared to S2P, structures are sharper and more detailed in Sat-NeRF DSMs (green arrows), which are also free of single-point outliers. However, Sat-NeRF produces more local irregularities: roofs and roads are less flat (red arrows). Certain roofs exhibit holes, that can be explained by their constant changes across the training sequence. The uncertainty coefficient β can only absorb occasional inconsistencies, which does not include roofs under construction (arrows 1-2) or roofs that are unusually free of parked cars (arrows 3-4). For clarity, we provide an RGB view of the area from a near-nadir perspective. Water bodies are masked in the DSMs.

Figure 6 .

 6 Figure 6. 3D visualization of the lidar, Sat-NeRF and S2P DSMs shown in Figure 5 (068). Compared to S2P, Sat-NeRF provides finer details and sharper edges but exhibits local irregularities.

Table 1 .

 1 Number of training and test images used for each area, and the altitude bounds of the scene considered in each case.

	Area index	004	068	214	260
	# train/test	9/2	17/2	21/3	15/2
	Alt. bounds [m]	[-24, 1]	[-27, 30]	[-29, 73]	[-30, 13]

Table 2 ,

 2 last row), which are affected by single-point outliers. Future work points to hybrid methods or the aggregation of contour-preserving regularization techniques.

			PSNR ↑			SSIM ↑			Altitude MAE [m] ↓
	Area index	004	068	214	260	004		214	260	004	068	214	260
	0. S-NeRF + SC [14]	-	-	-	-	0.344 0.459 0.384 0.416	4.418	3.644	4.829	7.173
	1. NeRF	17.93 10.26 15.26 14.95	0.559 0.536 0.736 0.443	3.327	2.591	2.691	3.257
	2. S-NeRF	25.87 24.20 24.51 21.52	0.864 0.900 0.939 0.829	1.830	1.496	3.687	3.245
	3. S-NeRF + SC	26.14 24.07 24.93 21.24	0.871 0.891 0.943 0.825	1.472	1.374	2.406	2.299
	4. Sat-NeRF	26.16 24.80 25.54 21.88	0.876 0.903 0.951 0.840	1.416	1.275	2.125	2.428
	5. Sat-NeRF + SC	26.67 25.07 25.50 21.78	0.884 0.908 0.950 0.842	1.288	1.249	2.009	1.864
	6. Sat-NeRF + SC (no BA)	21.55 22.87 24.53 20.96	0.571 0.874 0.942 0.816	1.577	1.392	2.176	1.875
	7. Sat-NeRF + DS	26.43 25.27 25.69 21.94	0.879 0.913 0.952 0.842	1.420	1.298	1.714	1.624
	8. Sat-NeRF + DS + SC	26.62 25.00 25.66 21.66	0.881 0.909 0.952 0.839	1.366	1.277	1.676	1.638
	S2P (10 pairs) [15]	-	-	-	-	-	-	-	-	1.370	1.174	1.811	1.640

Table 2 .

 2 Numerical results using the test images (unseen in training

Acknowledgements

This work was supported by a grant from Région Îlede-France. It was also partly financed by Office of Naval research grant N00014-17-1-2552, MENRT, and Kayrros.

This work was performed using HPC resources from GENCI-IDRIS (grants 2021-AD011012453 and 2022-AD011011801R1) and from the "Mésocentre" computing center of CentraleSupélec and ENS Paris-Saclay supported by CNRS and Région Île-de-France (http://mesocentre.centralesupelec.fr).


Artem Musiienko
email: artem.musiienko@helmholtz-berlin.de

Jakub Čížek

Hassan Elhadidy

Petr Praus

Kate Higgins

Bogdan Dryzhakov

Andriy Kanak

Franck Sureau

Jindřich Pipek

Eduard Belas

Eric Lukosi

Bin Hu

Mahshid Ahmadi
email: mahmadi3@utk.edu

Origin of Defects and Positron Annihilation in Hybrid and All-Inorganic Perovskites

Keywords: Perovskites, defects, charge transport, radiation sensors, positron annihilation

The emerging metal halide perovskites (MHP) have shown advanced charge transport properties suitable for application in solar cells, photodetectors and various other optoelectronic devices. The stability and uncontrolled defect structure are the main drawbacks limiting the efficient development of MHP devices. This study fills the gap in the understanding of recombination pathways in hybrid and all-inorganic perovskites. We resolved surface and bulk charge transport by combining current waveform time of flight (CW-ToF) and time-resolved photoluminescence (TRPL) spectroscopies. Further, by applying Raman and Infrared spectroscopies, we reveal the nature of the defects associated naturally with solvent trapping. We demonstrated that solvent-associated defects have a substitutional effect on the charge transport efficiency and stability of the MHP devices. Finally, we show the fundamental interaction of positron particles with perovskite vacancy defects and lattice. The positron annihilation is explained by using the ab-initio simulation of supercell. The results of this study suggest potential pathways for further improvement of hybrid and all-inorganic MHPs.

Lead halide perovskites with the formula APbX 3 , where A + is either cesium (Cs), methylammonium (MA) or formamidinium and X -is either chlorine (Cl), bromine (Br) or iodine (I -), have emerged as promising candidates for ionizing radiation detectors because of their high attenuation coefficient, tunable band-gap [START_REF] Wei | Dopant compensation in alloyed CH3NH3PbBr3-xClx perovskite single crystals for gammaray spectroscopy[END_REF][START_REF] Wei | Halide lead perovskites for ionizing radiation detection[END_REF][START_REF] Sutton | Bandgap-Tunable Cesium Lead Halide Perovskites with High Thermal Stability for Efficient Solar Cells[END_REF] , large mobility-lifetime (μτ) product [START_REF] Wei | Dopant compensation in alloyed CH3NH3PbBr3-xClx perovskite single crystals for gammaray spectroscopy[END_REF][START_REF] Wei | Halide lead perovskites for ionizing radiation detection[END_REF][START_REF] Yakunin | Detection of gamma photons using solution-grown single crystals of hybrid lead halide perovskites[END_REF] and low charge trap density and defect tolerance [START_REF] Wei | Halide lead perovskites for ionizing radiation detection[END_REF][START_REF] Shi | Low trapstate density and long carrier diffusion in organolead trihalide perovskite single crystals[END_REF][START_REF] Wen | Defect trapping states and charge carrier recombination in organic-inorganic halide perovskites[END_REF][START_REF] Zheng | Defect passivation in hybrid perovskite solar cells using quaternary ammonium halide anions and cations[END_REF][START_REF] Kang | High Defect Tolerance in Lead Halide Perovskite CsPbBr3[END_REF] . In particular, the all-inorganic cesium lead bromide (CsPbBr 3 ) perovskite is advantageous over its organic-inorganic perovskite counterparts due to its long-term chemical and structural stability [START_REF] Kang | High Defect Tolerance in Lead Halide Perovskite CsPbBr3[END_REF][START_REF] Dirin | Solution-Grown CsPbBr3 Perovskite Single Crystals for Photon Detection[END_REF][START_REF] Stoumpos | Crystal Growth of the Perovskite Semiconductor CsPbBr3: A New Material for High-Energy Radiation Detection[END_REF][START_REF] Ding | High Detectivity and Rapid Response in Perovskite CsPbBr3 Single-Crystal Photodetector[END_REF][START_REF] Rakita | Low-Temperature Solution-Grown CsPbBr3 Single Crystals and Their Characterization[END_REF] . Recently, more researchers have shown that CsPbBr 3 can achieve high-resolution detection of ionizing radiation [START_REF] He | High spectral resolution of gamma-rays at room temperature by perovskite CsPbBr3 single crystals[END_REF][START_REF] Feng | Low defects density CsPbBr3 single crystals grown by an additive assisted method for gamma-ray detection[END_REF] in addition to their excellent potential for application in tandem solar cells and LED technology [START_REF] Ahn | Trapped charge-driven degradation of perovskite solar cells[END_REF][START_REF] Duan | Hole-Boosted Cu(Cr,M)O2 Nanocrystals for All-Inorganic CsPbBr3 Perovskite Solar Cells[END_REF] .

Until recently, the growth of large-size, high-quality CsPbBr 3 single crystals required hightemperature growth from melts using the Bridgman melt-growth method [START_REF] He | High spectral resolution of gamma-rays at room temperature by perovskite CsPbBr3 single crystals[END_REF] . As this melt-growth method requires temperatures over 500 ˚C and multiple purification steps to purify the raw materials, this increases the cost of device fabrication to a level not lower than other radiation detection materials, such as CdZnTe [START_REF] Feng | Low defects density CsPbBr3 single crystals grown by an additive assisted method for gamma-ray detection[END_REF] . Therefore, the need for an inexpensive solution-based growth method is imperative. Over the past few years, the solution-based growth of CsPbBr 3 has been demonstrated [START_REF] Ding | High Detectivity and Rapid Response in Perovskite CsPbBr3 Single-Crystal Photodetector[END_REF][START_REF] Saidaminov | Inorganic Lead Halide Perovskite Single Crystals: Phase-Selective Low-Temperature Growth, Carrier Transport Properties, and Self-Powered Photodetection[END_REF] through the utilization of methods, such as the antisolvent vapor assisted [START_REF] Rakita | Low-Temperature Solution-Grown CsPbBr3 Single Crystals and Their Characterization[END_REF][START_REF] Zhang | Centimeter-Sized Inorganic Lead Halide Perovskite CsPbBr3 Crystals Grown by an Improved Solution Method[END_REF] and the inverse temperature crystallization methods [START_REF] Dirin | Solution-Grown CsPbBr3 Perovskite Single Crystals for Photon Detection[END_REF][START_REF] Wang | Low-Temperature Solution Growth and Characterization of Halogen (Cl, I)-Doped CsPbBr3 Crystals[END_REF][START_REF] Zhang | High-sensitivity X-ray detectors based on solution-grown caesium lead bromide single crystals[END_REF] . Despite progress in solution-based growth of lead halide perovskite single crystals, the crystallinity, dimensions, and charge transport properties of melt grown single crystals are superior.

It is well established that defects in semiconductor lattice have a critical effect on the performance of semiconductor devices. The crucial charge transport properties such as carrier mobility, lifetime, and recombination rate are greatly affected by the concentration of the defects [START_REF] Kang | High Defect Tolerance in Lead Halide Perovskite CsPbBr3[END_REF] . The effect of defects on charge transport properties of solution grown perovskites was broadly discussed in our recent studies [START_REF] Musiienko | Deciphering the effect of traps on electronic charge transport properties of methylammonium lead tribromide perovskite[END_REF][START_REF] Musiienko | Deep levels, charge transport and mixed conductivity in organometallic halide perovskites[END_REF] . Here, we explore the electrical structure, charge transport, and defects chemistry in solution and melt grown lead halide perovskite single crystals to reveal pathways to efficient and cheap perovskite semiconductor devices.

Positron annihilation spectroscopy (PAS) is commonly used to study defects and their chemical nature in various semiconductors [START_REF] Guin | Origin of the Order-Disorder Transition and the Associated Anomalous Change of Thermopower in AgBiS2 Nanocrystals: A Combined Experimental and Theoretical Study[END_REF][START_REF] Guin | Temperature Dependent Reversible p-n-p Type Conduction Switching with Colossal Change in Thermopower of Semiconducting AgCuS[END_REF][START_REF] Grebennikov | Application of positron annihilation and Raman spectroscopies to the study of perovskite type materials[END_REF][START_REF] Pansara | Defect structure transformation during substitution in quadruple perovskite CaCu3-xTi4-xFe2xO12 studied by positron annihilation spectroscopy[END_REF] . In conventional PAS, the material is bombarded with positrons. These positrons are then annihilated by the material's electrons, producing two 511 keV gamma rays. The positron annihilation rate is determined by the local electron density at the positron annihilation site [START_REF] Sundar | Positron annihilation spectroscopy in materials science[END_REF] . Because the electron density of a defect site is comparatively less than the electron density of the bulk material, the positron is trapped in a defect site for a longer time. Another positron annihilation technique, Coincidence Doppler Broadening (CDB) spectroscopy of the annihilation gamma rays, provides information about the electron momentum distribution in a material. Together, these techniques are incredibly powerful in characterizing the chemical nature of defects in a material.

Very few studies have used PAS to characterize the nature of defects in lead halide perovskites. Dhar et al. demonstrated the utilization of PAS in combination with Doppler broadening techniques to reveal the presence of MA defects in the MAPbI 3 crystal lattice [START_REF] Dhar | Investigation of Ion-Mediated Charge Transport in Methylammonium Lead Iodide Perovskite[END_REF] .

Particularly, single crystalline MAPbI 3 contains few defects than its polycrystalline counterpart but has an equal probability of developing both cationic and anionic (halide) vacancies, whereas the polycrystalline sample contains mainly cationic vacancies in its lattice. PAS has yet to be used to characterize the defects of other lead halide perovskites including CsPbBr 3 and MAPbBr 3 single crystals.

The free charge carrier losses at the surface, bulk, and interfaces are the main recombination pathways in semiconductor optoelectronic devices. Here, we first compare free charge recombination dynamics at the surface and in the bulk of hybrid and all-inorganic lead bromide perovskites using photoluminescence (PL) and time of flight current waveform (ToF-CW) spectroscopies. As can be seen in (Figure 1) there is a small deviation in the PL spectra of Bridgman grown CsPbBr 3 , solution grown CsPbBr 3 , and MAPbBr 3 single crystals with bandgap energies of 2.34 eV, 2.33 eV, and 2.31 eV, respectively. In addition, the excitation of the samples at 445 nm wavelength induces a bright luminescence emitted from the surface of these crystals, as shown in Figure 1 due to the presence of surface defects. Time-resolved photoluminescence (TRPL) measurements reveal nanosecond decay of PL signal in all studied samples (Figure 2). A faster and slower decay constants of around 40 ns and 140 ns is observed in MAPbBr 3 single crystal (Figure 2a). While the melt (Bridgman) and solution grown CsPbBr 3 crystals show decay constants of around 7 ns and 100 ns, and 2 ns and 37 ns, respectively.

To further explore charge carrier lifetime in these crystals, time of flight -current waveform (ToF-CW) spectroscopy was performed. The holes (generated by a short 1 µs laser pulse with a wavelength of 450 nm) drift in the bulk of lead halide perovskite crystals and induce ToF current waveform. Thus, the ToF CW reveals bulk recombination dynamics and lifetime of charge carriers. Recently we showed that charge trapping in the bulk defects is responsible for ToF CW decay [START_REF] Musiienko | Deciphering the effect of traps on electronic charge transport properties of methylammonium lead tribromide perovskite[END_REF] . Here, the ToF spectroscopy reveals microsecond dynamics of free holes drifting from anode to the cathode (Figure 2b) under an applied bias of 60 V. To estimate the lifetime of free holes in the bulk, we fit the ToF transients by an exponential fit. Interestingly, CsPbBr 3 crystal grown from melt shows holes lifetime of around τ CS =210 µs and MAPbBr 3 crystal shows holes lifetime of around τ CS =95 µs. While the CsPbBr 3 crystal grown from solution did not show a reliable ToF signal. Other studies also highlighted the critical effect of surface recombination on TRPL measurements [START_REF] Weiss | Bulk and surface recombination properties in thin film semiconductors with different surface treatments from time-resolved photoluminescence measurements[END_REF] . These results clearly explain the notable difference in charge decay dynamics shown by TRPL and ToF by different recombination rates at the surface and in the bulk of the lead halide perovskites.

Next, we utilized Raman spectroscopy and Fourier-transform infrared spectroscopy (FT-IR) to study the origin of defects and explain the superior transport properties of melt grown CsPbBr 3 . Firstly, the Raman spectra, specifically at low wavenumbers, of solution grown MAPbBr 3 , solution grown CsPbBr 3 , and melt grown CsPbBr 3 single crystals provide information regarding the vibrational modes of the metal-halide sublattice [START_REF] Cha | Formation and Encapsulation of All-Inorganic Lead Halide Perovskites at Room Temperature in Metal-Organic Frameworks[END_REF] , as shown in (Figure 3a). Regarding, The observed Raman peaks below 200 cm -1 in MAPbBr 3 are consistent with the literature and are attributed to lattice vibrations [START_REF] Wang | Structural and Photophysical Properties of Methylammonium Lead Tribromide (MAPbBr3) Single Crystals[END_REF] . Next, we note that the Raman spectra for both the solution-grown and melt grown CsPbBr 3 are similar. As shown in (Figure 3a) with arrows, the broad peaks around 73 cm -1 and 127 cm -1 , have previously been reported as the vibration mode of [PbBr 6 ] 4-octahedron and the motion of the Cs + cations [START_REF] Calistru | Identification of the symmetry of phonon modes in CsPbCl3 in phase IV by Raman and resonance-Raman scattering[END_REF][START_REF] Shibata | Exciton Localization and Enhancement of the Exciton-LO Phonon Interaction in a CsPbBr3 Single Crystal[END_REF] . Noticeably, we also see a shoulder peak at around 70 cm -1 for both the solution grown and melt-grown CsPbBr 3 . The peak has also been observed in the Raman spectrum of melt grown Cs 4 PbBr 6 single crystals [START_REF] Cha | Photoresponse of CsPbBr3 and Cs4PbBr6 Perovskite Single Crystals[END_REF] , indicating a possible second phase on the surface of these single crystals.

We proceeded further by obtaining the Raman spectra at higher wavenumbers to explore molecular modes, as shown in (Figure 3b). Overall, the Raman spectra for the solution grown MAPbBr 3 single crystal is consistent with the literature as the higher-frequency peaks being related to the different kinds of the MA + cation motions, such as the 918 cm -1 and 968 cm -1 being attributed to the CH 3 NH 3 + rocking and C-N stretching, respectively [START_REF] Xie | Organic-inorganic interactions of single crystalline organolead halide perovskites studied by Raman spectroscopy[END_REF][START_REF] Wang | Structural and Photophysical Properties of Methylammonium Lead Tribromide (MAPbBr3) Single Crystals[END_REF] . Next, we compare the Raman spectra for the solution grown CsPbBr 3 and melt grown CsPbBr 3 single crystals.

Noticeably, the spectra for the solution grown CsPbBr 3 single crystal contains well-defined peaks, whereas these peaks are absent for the melt grown CsPbBr 3 single crystal. We attribute these peaks shown in the solution-grown CsPbBr 3 single crystal to peaks commonly seen in the spectrum for dimethyl sulfoxide (DMSO) [START_REF] Batista | Selective DMSO-induced conformational changes in proteins from Raman optical activity[END_REF] , which is the solvent used for our growth process. In this Raman spectra peaks at 678 cm -1 and 713 cm -1 are attributed to C-S symmetric and antisymmetric stretches, and the peak at 1000 cm -1 is assigned to the S=O stretch. Further, individual peak assignments for both the solution grown MAPbBr 3 and solution grown CsPbBr 3 single crystals are shown in Table 1.

We utilize FT-IR characterization as shown in (Figure 3c) to further confirm solvent trapping within solution grown single crystals. Our measured FT-IR spectrum of the solutiongrown MAPbBr 3 single crystal is consistent with literature [START_REF] Mahapatra | Reducing ion migration in methylammonium lead tribromide single crystal via lead sulfate passivation[END_REF][START_REF] Xie | Understanding the Cubic Phase Stabilization and Crystallization Kinetics in Mixed Cations and Halides Perovskite Single Crystals[END_REF] . Prominent peaks are at 969 cm -1 , 1477 cm -1 , 1584 cm -1 , and 3165 cm -1 , which can be attributed to the C-N stretch, symmetric NH 3 + bend, asymmetric NH 3 + bend, and symmetric NH 3 + stretch, respectively. As seen with the Raman characterization, there are noticeably defined peaks, such as 951 cm -1 , 1028 cm -1 , and 1482 cm -1 , for the solution grown CsPbBr 3 single crystal that are not observed for the melt grown CsPbBr 3 single crystal. We attribute these peaks to the solvent trapped within the single crystal as most of these peaks correspond to peaks observed in the FT-IR spectrum for dimethyl sulfoxide (DMSO) as described in Table 1. In Figure 3c, we compare the FT-IR spectrum of DMSO to the solution grown CsPbBr 3 to further illustrate this point. Trapped solvent can form point defects and defect complexes responsible for the shorter carrier lifetime in solution grown single crystals. In addition, the trapped solvent can deteriorate the stability of the perovskite devices. For both the solution grown and melt grown CsPbBr 3 single crystals, we attribute a few visibly low intensity peaks at 1600 cm -1 and 3500 cm -1 to the environment, specifically water vibrational modes, as these measurements were performed in ambient conditions [START_REF] Zhang | Synthesis and single crystal growth of perovskite semiconductor CsPbBr 3[END_REF] . The modeled vacancies show larger ab-initio lifetimes than we observed in the positron annihilation experiment.

To explain positron decay, we further considered the annihilation of the positron in a free state, which is a lattice without defects. The calculated bulk positron lifetimes according to Eq. well. Hence, it can be concluded that CDB data are in agreement with results in Figure 3c that positrons in both samples (CsPbBr 3 and MAPbBr 3 ) are annihilated predominantly in the free state, i.e. not trapped at defects. Positron annihilates at the bulk of perovskites interacting with lattice but not with defects. Due to proven positron annihilation in the lattice of lead bromide perovskite crystals, we can estimate the maximum vacancy concentration to be around 10 15 cm -3 by considering a lower sensitivity limit of positron annihilation spectroscopy for negatively charged vacancies due to the dominance of delocalized bulk state and low positron trapping rate [START_REF] Gebauer | On the sensitivity limit of positron annihilation: detection of vacancies in as-grown silicon[END_REF] .

In summary, comparing the charge decay by TRPL and ToF, we can conclude that these two methods show different decay dynamics. Free carriers drifting in the ToF experiment are mainly affected by defects in the bulk of perovskites. Meanwhile, surface quality has a substantial effect on PL decay. TRPL does not reflect the lifetime of charge carriers in the bulk of perovskites. The dominant charge losses occur at the surface defects with a recombination lifetime of several to hundreds of nanoseconds. The bulk charge transport shows a much longer lifetime of hundreds of microseconds. Among studied hybrid and inorganic lead bromide perovskites, the CsPbBr 3 crystal grown from melt show a longer lifetime of charge carrier from bulk, 210 µs. We attribute the difference in charge transport properties between melt and solution grown crystals to solvent trapping within solution grown single crystals. DMSO peaks were detected by Raman and FT-IR spectroscopies in solution grown single crystals. By analyzing the PAS results, we conclude that positrons annihilate in the free state (i.e. not trapped at defects) in both all-inorganic and hybrid perovskite semiconductors grown by different methods. From a comparison of measured experimental positron lifetimes and Doppler broadening with ab-initio calculations, CsPbBr 3 grown from solution and melt and solution grown MAPbBr 3 show a similar mechanism of positron annihilation in the lattice. Thus, both inorganic and hybrid perovskites do not contain defects capable of positron trapping i.e. the concentration of vacancies is lower than 10 15 cm -3 .

This study clarifies the impact of growth conditions on the performance of hybrid and allinorganic radiation sensors. The comparison of the fundamental properties of single crystals grown from the melt and solution gives valuable information needed for further development of perovskite materials. The demonstrated defect nature and recombination channels reveal further pathways that can lead to more efficient, stable, and cheap perovskite devices.

Methods

Single Crystal Solution Growth

The MAPbBr 3 and CsPbBr 3 single crystals were grown using the inverse temperature crystallization method [START_REF] Saidaminov | High-quality bulk hybrid perovskite single crystals within minutes by inverse temperature crystallization[END_REF] . MABr (99.99% from Greatcell Solar Materials), CsBr (99.999% from SigmaAldrich), and PbBr 2 (99.999% from SigmaAldrich) were used without further purification.

For MAPbBr 3 , a molar ratio of 1.2 to 1 of MABr to PbBr 2 were dissolved in N,N-Dimethylformamide (99.8% anhydrous from SigmaAldrich) for a 1M solution. For CsPbBr 3 , a molar ratio of 1 to 2 of CsBr to PbBr 2 were dissolved in Dimethyl Sulfoxide (99.9% anhydrous from SigmaAldrich) for a 1M solution [START_REF] Saidaminov | Inorganic Lead Halide Perovskite Single Crystals: Phase-Selective Low-Temperature Growth, Carrier Transport Properties, and Self-Powered Photodetection[END_REF] . Both solutions were filtered using a 0.2 μm PTFE membrane syringe filter. Crystals precipitated during gradual heating at a rate of 5 ˚C/day from room temperature to ~70 ˚C for MAPbBr 3 and ~120 ˚C for CsPbBr 3 .

Single Crystal Bridgman technique

CsPbBr 3 crystals were grown by the Bridgman technique. CsBr (6N) and PbBr 2 (5N) (manufactured by Alfa Aesar) in equimolar amounts were loaded in a quartz ampoule and evacuated to 10 -5 mbar. The ampoule then was sealed and placed in a three-zone vertical furnace.

Synthesis of perovskite occurred by the fusion of starting materials. The ampoule with components was slowly heated to a temperature of 645C which exceeding the melting points of both reagents. At this temperature, an isothermal holding was carried out for several hours to achieve melt homogenization and complete the reaction between CsBr and PbBr 2 . Thereafter, the furnace was cooled down to room temperature for 20 hours. next step was growing a single crystal of CsPbBr 3 . The ampoule with the synthesized material was slowly heated up to 585 ° C and lowered with a rate of 3 mm/h into the "cold" zone of the furnace until the ingot crystallization was complete. After this, lowering was stopped and the grown crystal was slowly cooled down to room temperature for 30 hours.

Photoluminescent Spectroscopy

Photoluminescence (PL) and time-resolved photoluminescence measurements (TRPL) were performed using a Horiba Fluorolog3 spectrometer. A 405 nm CW laser was used for excitation during PL measurements. A 343 nm pulsed laser (280 fs, 200kHz) was used for TRPL measurements at the peak emission intensity.

Luminescence measurements

Confocal microspectrofluorimeter adapted for time-resolved fluorescence measurements by

Raman Spectroscopy and FT-IR Measurements

Raman spectroscopy was carried out on a HORIBA XploRA PLUS Raman system with a 785nm laser at 1mW through a confocal microscope with a 100x objective Multiple scans were averaged for each measurement between 600-2500cm -1 . FT-IR spectroscopy on uncrushed single crystals and dimethyl sulfoxide (DMSO) was performed on the Thermo-Scientific Nicolet iS50 FT-IR.

Positron annihilation spectroscopy

A [START_REF] Musiienko | Deep levels, charge transport and mixed conductivity in organometallic halide perovskites[END_REF] Na radioisotope with an activity of 1 MBq sealed between two Mylar foils with a thickness of 2 m.was used as a positron source. A digital spectrometer with a time resolution of 145 ps (FWHM of the resolution function) was used for positron lifetime measurements. Fitting of positron lifetime spectra was performed using the PLRF code [START_REF] Čížek | PLRF Code for Decomposition of Positron Lifetime Spectra[END_REF] . The source contribution to the LT spectra consisted of (i) a component with lifetime of 368 ps and relative intensity of 10% representing a contribution of positrons annihilated in the source spot and inside the Mylar foil, and (ii) a weak long lived component with lifetime of 1.5 ns and intensity of 1% originating from pick-off annihilation of ortho-positronium formed in the covering Mylar foil. The CDB studies were carried out using a digital spectrometer equipped with two high-purity Ge detectors.

The CDB spectrometer was characterized by an energy resolution of 0.9 keV at the annihilation line and a peak-to-background ratio higher than 10 5 . The results of the CDB measurements are presented in this paper as ratio curves related to a well-annealed pure Al (99.9999%) reference sample.

Positron annihilation ab-initio calculations

Positron annihilation related parameters (positron lifetimes and momentum distribution of annihilating electron-positron pairs) were calculated using the density functional theory (DFT).

Ab-initio calculations of positron parameters were performed within the so-called standard scheme [START_REF] Puska | Theory of positrons in solids and on solid surfaces[END_REF] . In this approximation, positron density is assumed to be everywhere vanishingly small

Next, to study the possible origin of point defects affecting free charges in single crystals, we study positron interaction we study positron interaction with bulk defects in melt grown CsPbBr 3 andFigure 3 d

 33 Figure 3 d-e). In this method, Positron implanted into the sample is thermalized within a few ps, i.e. its energy is quickly decreased down to thermal energy, kT. Thermalized positron is delocalized in the lattice as a modulated Blochas a modulated Bloch-like wave. Open volume defects, e.g. vacancies, represent potential wells for positron. Hence, if a for positron. Hence, if a sample contains open volume defects, the delocalized positron is trapped in these defects and its

S1 and (Figure 4 a

 4 and e) for a perfect lattice of CsPbBr 3 and MAPbBr 3 are around 355.1 and 333.4 ps, respectively. The bulk (lattice) positron lifetimes found from the experiment are close to the positron lifetimes measured in both crystals. The difference in positron lifetime (several tens of ps) in CsPbBr 3 and MAPbBr 3 crystals can be explained by the interaction of positrons with different lattices. Due to different cations (MA and Cs), the lattice has slightly different cation radii and lattice constant. As a result, positron trapping and positron lifetimes are slightly different for inorganic and hybrid lead halide perovskites, which was confirmed by ab-initio calculations.

  environment of defects. The momentum distribution of annihilating electron-positron can be calculated from the first principles and be directly compared with the experimental results. The present approach describes well the high momentum part (p > 10  10 -3 m 0 c) of the momentum distribution of annihilating pair with dominating contribution of core electrons which are localized in atomic shells retaining their atomic character. For the sake of comparison with the experiment, the calculated momentum distributions are presented as ratio curves with respect to Aluminium (Al), i.e. the calculated momentum distributions were divided by the momentum distribution calculated for a perfect Al crystal with an FCC structure and lattice parameter of a = 4.0495 Å.Coincidence Doppler broadening (CDB) ratio curves (related to pure Al reference) are plotted in (Figure4i). Calculated ratio curves for a perfect (defect-free) CsPbBr 3 and MAPbBr 3 lattice are shown in (Figure4 j). The ab-initio calculations of CDB curves are not accurate enough to allow for quantitative comparison with the experimental results. Nevertheless, there is clearly a good qualitative agreement between CDB curves calculated for perfect (defect-free) CsPbBr 3 and MAPbBr 3 lattice and experiment. A local maximum at p  8  10 -3 m 0 c followed by a minimum at p  13  10 -3 m 0 c and a broad peak located at higher momenta observed in the experimental curves are well reproduced in calculated curves. MAPbBr 3 exhibits a lower peak at p  8  10 -3 m 0 c and position of the local minimum and the following broad peak are shifted to slightly at higher momenta compared to CsPbBr 3 . This is reproduced in the calculated curves as

  using a phase-modulation principle with homodyne data acquisition was employed to obtain fluorescence spectra of Perovskite samples and to determine the fluorescence lifetimes. Laser diode module (Omicron LDM 442.50.A350, Germany) with sinusoidal intensity modulation (50 mW peak output, attenuated to one to tens of μW at the sample) was used for the excitation at 445 nm wavelength. Confocal epifluorescence upright microscope (Zeiss UMSP-80, Germany) with a 10× objective was used to collect the fluorescence emission signal from excited volume of the sample. It is focused on the entrance slit of the Jobin-Yvon HR640 spectrograph equipped with a 100 line/mm grating. The spectral detection window (375 nm wide) covered both the excitation wavelength (elastic scattering provided us a lifetime reference) and the perovskite emission spectrum. The fluorescence lifetime was calculated from the frequency-dependent phase shift and the intensity demodulation for modulation frequencies covering evenly 5 -165 MHz interval. Calculated phase differences and modulation ratios are used as an input table for GLOBAL fitting program (Laboratory for Fluorescence Dynamics, Irvine, USA) to calculate single fluorescence lifetime components and their percentage spectral representation.

Figure 1 Figure 2 a

 12 Figure 1 Photoluminescent spectra of CsPbBr 3 and MAPbBr 3 single crystals grown by melt (Bridgman) and solution methods. (The images on right show the surface emission after the excitation with 445 nm laser. More details can be found in SI Figure S1 and method section.

Figure 3

 3 Figure 3 Raman spectra for solution grown MAPbBr 3 , solution grown CsPbBr 3 , and melt grown CsPbBr 3 single crystals for a) low wavenumbers and b) high wavenumbers. Shown in c) is the FT-IR results for the same crystals. d) and e) Positron lifetime spectra measured in melt grown CsPbBr 3 and solution grown MAPbBr 3 respectively. The details on the fitting procedure can be found in the method section.

Table 1 .

 1 Individual peak assignments for both the solution grown MAPbBr 3 and solution grown CsPbBr 3 single crystals at high wavenumbers.

	Raman	Solution-grown	Solution-grown	FT-IR Peak	Solution-grown
	Shift	MAPbBr 3 Peak	CsPbBr 3 Peak	(cm -1 )	CsPbBr 3 Peak
		Assignment		Assignment	Assignment
	678			-		C-S Symmetric Stretch	1028	S=O
	713			-		C-S Asymmetric Stretch	1308	-CH 3 Symmetric
	887			-		-	1399	-CH 3 Asymmetric
	918	CH 3 NH 3	+ (rocking)	-	1432	-CH 3 Asymmetric
	968	C-N stretching		-	2909	-CH 3 Vibrations
	1000						S=O Stretch	2982	-CH 3 Vibrations
	1252	CH 3 -NH 3	+ rocking	-
	1345			-		C-H Symmetric
							Deformation
	1411			-		CH 3 Degenerate
							Deformation
	1426			-		-
	1477	Asymmetric NH 3	+	-
		bending
	1587	NH 3	+ twisting		-

Table 2

 2 Results of ab-initio calculations of positron lifetimes for a perfect lattice (bulk) and various types of vacancies. Calculated binding energy (E B ) of positron to vacancies is shown in the table as well.

	CsPbBr 3	 (ps)	E B (eV)	MAPbBr 3	 (ps)	E B (eV)
	Experimental PAS lifetime	349.6	-	Experimental PAS lifetime	343.2	-
	bulk	355.1	-	bulk	333.4	-
	Cs-vacancy	364.8	0.329	MA-vacancy	357.9	0.055
	Pb-vacancy	371.9	0.023	Pb-vacancy	347.0	0.010
	Br-vacancy	394.8	0.066	Br-vacancy	342.9	0.011
	Doppler broadening PAS method can bring additional information about the local chemical

Acknowledgment: K.H., B.D., E.L., B.H. and M.A. acknowledge financial support from US Department of Homeland Security (grant # 2016-DN-077-ARI01).

and not affecting the bulk electron structure. Modeling of point defects was performed using 320 atom-based supercells consisting of 4  4  4 perovskite units. Vacancies were created simply by removing corresponding atoms. Details of calculations of positron lifetimes are described in the Supplementary materials.

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Supplementary Materials

Positron annihilation ab-initio calculations

In the first step the Electron density n -(r) in the material was calculated without presence of positron. Subsequently, the effective potential for positron was constructed as

where ϕ(r) is the Coulomb potential produced by the charge distribution of electrons and nuclei and V corr is the electron-positron correlation potential in the limit of vanishing positron density [START_REF] Puska | Theory of positrons in solids and on solid surfaces[END_REF] .

In the second step the ground state of positron wave function 𝜓 + 𝒓 was calculated by solving of a single particle Schrödinger equation

) where E + is the positron ground state energy.

The positron lifetime is determined from the overlap of the electron density n -(r) and the positron density 𝑛 + 𝒓 = 𝜓 + 𝒓 2 using the expression

where r e is the classical electron radius and c is the speed of light. An enhancement of electron density at the site of positron due to positron-electron correlation was taken into account through the enhancement factor  which accounts for an enhancement of the electron density at the site of positron [START_REF] Puska | Theory of positrons in solids and on solid surfaces[END_REF] . . The enhancement factor  and the electron-positron correlation potential V corr were treated within the local density approximation (LDA) using the parametrization by Boronski and Nieminen [START_REF] Boroński | Electron-positron density-functional theory[END_REF] . Within this approach the enhancement factor  and the electron-positron correlation potential V corr are determined by the local electron density n -and its gradient n -at the positron annihilation site.

The momentum distribution of annihilating electron-positron pairs was calculated using the approach described in [START_REF] Alatalo | Theoretical and experimental study of positron annihilation with core electrons in solids[END_REF][START_REF] Kuriplach | Vacancies and vacancyoxygen complexes in silicon: Positron annihilation with core electrons[END_REF] . Electrons belonging to a shell of i-th atom described by the principal quantum number n and the angular quantum number l contribute to the momentum distribution by a contribution

where N i,nl is the number electrons in the shell n,l; B l is the spherical Bessel function and 𝑅 + 𝑖 , 𝑅 - 𝑖,𝑛𝑙 denote the radial part of positron and electron wave functions. The symbol  i,nl stands for the state-dependent positron enhancement factor [START_REF] Kuriplach | Vacancies and vacancyoxygen complexes in silicon: Positron annihilation with core electrons[END_REF] . The total momentum distribution of the annihilating electron-positron pairs was obtained by summing the partial contributions 𝜌 𝑖,𝑛𝑙 𝑝