Title: Abstract Rendering:  Certified Rendering Under 3D Semantic Uncertainty
Venue: NeurIPS 2025
Accepted: yes
Keywords: Abstract Rendering

Certifiable Classification

Gaussian Splat

Neural Radiance Field

alpha-beta-CROWN
Abstract: Rendering produces 2D images from 3D scene representations, yet how continuous variations in camera pose and scenes influence these images—and, consequently, downstream visual models—remains underexplored. We introduce **abstract rendering**, a framework that computes provable bounds on all images rendered under continuously varying camera poses and scenes. The resulting abstract image, expressed as a set of constraints over the image matrix, enables rigorous uncertainty propagation through downstream neural networks and thereby supports certification of model behavior under realistic 3D semantic perturbations, far beyond traditional pixel-level noise models. Our approach propagates camera pose uncertainty through each rendering step using efficient piecewise linear bounds, including custom abstractions for three rendering-specific operations—matrix inversion, sorting-based aggregation, and cumulative product summation—not supported by standard tools. Our implementation, ABSTRACTRENDER, targets two state-of-the-art photorealistic scene representations—3D Gaussian Splats and Neural Radiance Fields (NeRF)—and scales to complex scenes with up to 1M Gaussians. Our computed abstract images achieve up to 3% over-approximation error compared to sampling results (baseline). Through experiments on classification (ResNet), object detection (YOLO), and pose estimation (GATENet) tasks, we demonstrate that abstract rendering enables formal certification of downstream models under realistic 3D variations—an essential step toward safety-critical vision systems.

Abstract Rendering: Certified Rendering Under 3D Semantic Uncertainty

Abstract

Rendering produces 2D images from 3D scene representations, yet how continuous variations in camera pose and scenes influence these images-and, consequently, downstream visual models-remains underexplored. We introduce abstract rendering , a framework that computes provable bounds on all images rendered under continuously varying camera poses and scenes. The resulting abstract image, expressed as a set of constraints over the image matrix, enables rigorous uncertainty propagation through downstream neural networks and thereby supports certification of model behavior under realistic 3D semantic perturbations, far beyond traditional pixel-level noise models. Our approach propagates camera pose uncertainty through each rendering step using efficient piecewise linear bounds, including custom abstractions for three rendering-specific operations-matrix inversion, sorting-based aggregation, and cumulative product summation-not supported by standard tools. Our implementation, ABSTRACTRENDER, targets two state-of-the-art photorealistic scene representations-3D Gaussian Splats and Neural Radiance Fields (NeRF)-and scales to complex scenes with up to 1M Gaussians. Our computed abstract images achieve up to 3% over-approximation error compared to sampling results (baseline). Through experiments on classification (ResNet), object detection (YOLO), and pose estimation (GATENet) tasks, we demonstrate that abstract rendering enables formal certification of downstream models under realistic 3D variations-an essential step toward safety-critical vision systems.

1 Introduction

Rendering produces 2D images from 3D scenes and underpins visual computing. Two prominent neural scene representations are Gaussian Splats [1] and Neural Radiance Fields (NeRFs) [2]. Gaussian Splats represent a scene as a collection of 3D Gaussians whose colors are blended after projection onto the image plane, whereas NeRFs encode scenes as neural networks mapping 3D positions and viewing directions to color and density, rendered via ray casting and volumetric integration. These methods exemplify rasterization- and ray-casting-based paradigms that achieve photorealistic reconstruction and novel-view synthesis [3]. However, rigorous analysis of how variations in camera pose or scene geometry affect the rendered outputs-and consequently the predictions of downstream models such as classifiers, object detectors, and pose estimators-remains limited, leaving the robustness of vision systems under realistic 3D perturbations largely unexplored.

Although formal verification has been extensively developed for standalone neural networks [4, 5], verifying rendering pipelines poses a fundamentally different challenge. A recent work [6] considers a version of this problem for simpler mesh-based scenes, but a general treatment remains open. Formal verification of rendering seeks to compute the set of all possible images generated under continuous variations in camera pose and scene. When coupled with downstream perception models this enables guarantees like 'no misclassification occurs during camera panning within a specified range' or

'the relative pose error remains bounded within tolerance across viewpoints.' Such guarantees are crucial for safety-critical applications [7] including aircraft auto-landing [8] and formation-flight[9]. Verifying rendering is challenging because ray-casting and rasterization involve neural components, volumetric integration, and nonsmooth operations such as sorting. Consequently, reasoning about rendering requires composing verification techniques across heterogeneous computational layers, a capability absent from existing frameworks.

In this paper, we introduce Abstract Rendering , a method for computing the set of all possible images rendered from a continuously varying range of camera poses looking at a scene with a range variations. This set of images, referred to as an Abstract Image , is compactly represented using constraints on the image matrix, such as interval or linear bounds. This computed abstract image can then be passed through an existing neural network verification tool [4] to verify models for visual tasks. Our framework is illustrated in Figure 1, with representative results in Figure 4. For example, consider a ResNet classifier downstream of a camera rotating 360 ◦ around an airplane in a 3D scene; our method certifies the subrange of viewing angles over which the airplane is classified correctly.

Ourframework:AbstractRendering+DownstreamCertification

Dozer: [0.75,0.8]

Chair: [0.15,0.2]

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Truck: [0.25,0.5]

AbstractRendering

Abstractimage

DownstreamNetwork

ClassScores

(withuncertainty)

(boundpropagation)

(lowerandupperbound)

(boundpropagation)

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Our abstract rendering algorithm applies linear sets to represent and propagate uncertainties from camera pose and scene parameters, through the rendering pipeline. By decomposing the rendering process into composition of basic operations, we develop linear bound propagation rules that maintain tight output bounds when transforming linear input sets. This compositional approach ultimately produces an abstract image that preserves uncertainty information. To optimize the trade-off between bound precision and computational efficiency, we adopt CROWN [10, 11] for handling standard operations like division and matrix multiplication.

However, certain operators in the rendering process lack native support. To address this, we develop novel linear bound propagation techniques for three rendering-specific operations: matrix inversion, sorting-based aggregation, and cumulative product summation. For matrix inversion required in computing projected 2D Gaussian probability densities, our Taylor series-based approximation produces tighter linear bounds than standard adjugate methods. Sorting-based aggregation, critical for occlusion determination between Gaussians, is reformulated as an equivalent indicator-based process to mitigate over-approximation caused by independently bounding sorted outputs while ignoring index mutual exclusivity. Cumulative product summation, used in opacity/occlusion-weighted value aggregation, employs an ordered strategy that significantly improves bound tightness (particularly beyond 1,000 terms) at moderate computational expense.

In summary, our key contributions are as follows. (1) We introduce the first abstract rendering algorithm to compute abstract images for scenes represented by Gaussian Splats and NeRFs under semantic variations in the 3D world. (2) We develop a unified mathematical and software framework for abstracting rendering pipelines, with novel techniques for matrix inversion, sorting-based aggregation, and cumulative product summation, extending beyond what is supported by existing tools like CROWN. (3) We integrate abstract rendering with neural network verification tool CROWN, to certify visual tasks such as image classification, pose estimation, and object detection, enabling statements like: 'Across camera viewpoints or scene variations, the target model consistently classifies, estimates, or detects object within specified error bounds.'

2 Related Work

The only closely related work computes abstract images for triangular mesh scenes [6]. To handle camera-pose uncertainty, that method collects each pixel's RGB values into intervals over all possible viewpoints. In practice, this amounts to computing partial interval depth-images for each triangle and merging them using a depth-based union, resulting in an interval image that over-approximates all concrete images for that region of poses. This work considers only camera translations, not rotations. Our method works with Gaussian splatting or NeRF which involve more complex operations and are learnable.

Numerous methods exist in neural network verification [5, 12, 13, 14, 15, 16], including SMTbased approaches [17] and MIP-based techniques [18]. Among these, linear bound propagation has emerged as a prominent strategy, leveraging linear relaxations of nonlinear activations and propagating these bounds layer-wise to efficiently compute output guarantees [19, 11, 20]. Researchers have further advanced the field by introducing complex geometric abstractions such as polyhedra [21], zonotopes [22], and starsets [16, 14]. However, our work diverges conceptually from traditional neural network verification: rather than verifying neural networks themselves, we focus on verifying a broader algorithmic framework that embeds neural networks as components. Crucially, many operations encountered in this context, such as matrix inverse, sorting or cumulative product, are not handled by existing neural network verification tools, necessitating novel theoretical and technical innovations.

Adversarial robustness has been studied most deeply for pixel-space perturbations, where attacks [23, 24, 25, 26, 27] and certificates [11, 20, 14, 16, 28] are constrained to small l p balls around the original image. Follow-up work broadened the threat model to 2D semantic transforms [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], including global hue or brightness shifts, in-plane rotations, translations, elastic warps, but still treats the scene as a rigid 2D grid. Our method, instead, handles semantic, 3D perturbations that stem from changing the camera pose itself (e.g., orbiting or translating the camera around a fixed object), which jointly affect all pixels in a viewpoint-consistent manner.

Only a handful of papers explore this regime. Athalye [41] synthesizes 3D-printable adversarial objects that fool classifiers over many viewpoints, yet offer no formal guarantee of success across the entire pose distribution. Camera-Motion Smoothing (CMS) by Hu [42] provides the first certificates for small six-DoF camera perturbations via randomized smoothing, but it relies on dense point-cloud supervision and is restricted to narrow pose radii. In Hu's later work [43], he partitions pose space and smooths in the image domain to improve efficiency, yet still yields conservative bounds over only modest displacements. View-Invariant Adversarial Perturbations (VIAP) [44] craft attacks that remain effective across multiple viewpoints, but-as with Athalye et al.-provide no certification. Our approach is the first to deliver tight, end-to-end formal guarantees for standard vision networks across a full 360 ◦ camera sweep with realistic rendering.

3 Preliminaries: Linear Approximations and Rendering Algorithms

Abstract Rendering bounds uncertainty in rendered images by propagating linear bound through each operation within the rendering pipeline. In this section, we delineate the fundamental operations integral to rendering algorithms and demonstrate their amenability to tight linear over-approximations. Subsequently, we provide an overview of two prominent neural rendering techniques: 3D Gaussian Splatting and Neural Radiance Fields (NeRF).

3.1 Linear Over-approximability

Notations. For vectors or matrices x, y of the same shape, x < y denotes element-wise comparison. The Frobenius norm of a vector or matrix x is denoted by ∥ x ∥ . Boldface (e.g., pc ) indicates setvalued variables, while non-bold (e.g., pc ) denotes fixed values. A linear set (or polytope) is defined as { x ∈ R n | Ax ≤ b } , where Ax ≤ b specifies linear constraints. A piecewise linear relation R ⊆ R n × R m has the form { ( x, y ) | Ax + b ≤ y ≤ Ax + b, Ax ≤ b } , where A,A,b,b define linear bounds on y given x . A constant relation is a special case where A = A = 0 .

We consider a class of functions that can be tightly over-approximated by piecewise linear lower and upper bounds over any compact input domain.

Definition 1. A function f : S → R m is called linearly over-approximable if, for any compact X ⊆ S and ε > 0 , there exist piecewise linear maps ℓ X , u X : X → R m such that for all x ∈ X , ℓ X ( x ) ≤ f ( x ) ≤ u X ( x ) and | u X ( x ) -ℓ X ( x ) | < ε .

Proposition 1. Any continuous function is linearly over-approximable.

Although Proposition 1 implies piecewise linear bounds on continuous functions can be adjusted sufficiently tight by shrinking the neighborhood, finding accurate bounds over larger neighborhoods remains a significant challenge. Our implementation of abstract rendering employs CROWN [10] for computing such bounds. Except for Ind and Sort , all the basic operations listed in Table 1 are continuous and therefore linearly over-approximable. Moreover, since Ind is a piecewise constant function, it can be tightly bounded on each subdomain by partitioning at x = 0 , allowing it to be treated as a linearly over-approximable function. This is formalized in Corollary 1.

Corollary 1. All operations in Table 1, except for Sort , are linearly over-approximable.

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3.2 Rendering Algorithms: Gaussian Splat and NeRF

Scenes and Camera Model A 3D Gaussian scene ScG is defined as a finite collection of 3D Gaussians in world coordinate frame w indexed by I . Each 3D Gaussian is specified by its mean µ w , covariance Σ w , opacity o and RGB color c . For simplicity, we use RGB instead of spherical harmonics to represent 3D Gaussians' colors. A neural radiance field ScN consists of an opacity network F o mapping 3D points to opacity, a color network F c mapping 3D points and view directions to color, maximum sampling distance L , and sampling count N . A camera C is characterized by translation T ∈ R 3 , rotation R ∈ R 3 × 3 , focal lengths ( f x , f y ) ∈ R 2 , and principal point offsets ( c x , c y ) ∈ R 2 .

Gaussian Splat rendering Algorithm 1 takes a Gaussian scene ScG , a camera C and a pixel coordinate u , and outputs the rendered RGB value pc at pixel u . Rendering proceeds in four steps: (1) it transforms 3D Gaussians from world to camera coordinate ( µ c , Σ c ) (Line 2-3); (2) it projects 3D Gaussians onto image plane ( µ p , Σ p ) and computes its weighed probability density a at pixel u (Line 4-10); (3) it sorts the Gaussians according to the distance of their means to the image plane (Line 11-14); (4) it computes pc by aggregating the colors of all Gaussians weighted according to their relative positions (Line 16-18). The full image is rendered by applying GAUSSIANSPLAT at every pixel location.

NeRF rendering Algorithm 2 takes as input a neural radiance field ScN , a camera C , and a pixel coordinate u , and outputs the rendered RGB value pc at that pixel. The rendering proceeds in three steps: (1) it computes the normalized direction of the camera ray dir w corresponding to u and samples N points x along the ray within the maximum range L (Line 1-4); (2) it evaluates the opacity and

Algorithm 1 GAUSSIANSPLAT ( ScG , C , u ) 1: for all i ∈ I do 2: µ c [ i ] ← R × ( µ w [ i ] -T ) 3: Σ c [ i ] ← R × Σ w [ i ] 4: K [ i ] ← [ fx µ c [ i , 2] 0 cx µ c [ i , 2] 0 fy µ c [ i , 2] cy µ c [ i , 2] ] 5: J [ i ] ← [ fx µ c [ i , 2] 0 -fx · µ c [ i , 0] µ c 2 [ i , 2] 0 fy µ c [ i , 2] -fy · µ c [ i , 1] µ c 2 [ i , 2] ] 6: µ p [ i ] ← K × µ c [ i ] 7: Σ p [ i ] ← J [ i ] × Σ c [ i ] × J [ i ] ⊤ 8: Conic [ i ] ← Inv( Σ p [ i ]) 9: q [ i ] ← ( u -µ p [ i ]) ⊤ × Conic [ i ] × ( u -µ p [ i ]) 10: a [ i ] ← o [ i ] · Exp( -1 2 · q [ i ]) 11: d [ i ] ← µ c [ i , 2] 12: end for 13: as ← Sort( a , d ) 14: cs ← Sort( c , d ) 15: for all i ∈ I do 16: oc [ i ] ← ∏ i -1 j =1 (1 -as [ j ]) 17: end for 18: pc ← ∑ N i =1 ( oc [ i ] · as [ i ] · cs [ i ]) 19: return pc

Algorithm 2 NERF ( ScN , C , u )

Algorithm 3 MATRIXINV ( X , X ref , k )

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color of each sampled point using the networks F o and F c (Line 5-6); (3) it aggregates the colors along the ray by computing a weighted sum based on the opacity and depth ordering of the samples (Line 7-10).

Having introduced two representative rendering algorithms, we now present a formal definition of abstract rendering.

Abstract rendering problem. Our goal is to design an algorithm that for all pixel u ∈ WH , it takes as input a linear set of scenes Sc and cameras C , and outputs a linear set of pixel color pc such that GAUSSIANSPLAT ( Sc , C , u ) ∈ pc or NERF ( Sc , C , u ) ∈ pc ) for each Sc ∈ Sc and C ∈ C .

4 Methodology: Abstract Rendering

Our abstract rendering algorithm processes scene and camera inputs by incrementally building piecewise linear relations between intermediate variables and inputs through standard rendering steps, proceeding until pixel-input relationships are established. These relations enable linear bound computation for each pixel, yielding linear bound for image. Since most GAUSSIANSPLAT and NERF operations are linear, their relations compose directly. In this section, we highlight three renderingspecific nonlinear operations (matrix inverse, sorting-based aggregation and cumulative product summation) and introduce tailored techniques to tightly bound them with linear over-approximations.

4.1 Abstracting Matrix Inverse by Taylor Series

Bounding matrix inversion (Line 8 of GAUSSIANSPLAT) via the adjugate method 2 often introduces significant over-approximation due to uncertainty in the denominator. To address this, we propose a Taylor series-based algorithm, MATRIXINV (Algorithm 3), which reformulates matrix inversion using only addition, multiplication, and the inverse of a fixed reference matrix. Although the remainder term still involves division, its influence diminishes as the Taylor order increases.

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MATRIXINV takes a non-singular matrix X , a non-singular reference matrix X ref , and a Taylor order k as input, and returns lower and upper bounds lXinv and uXinv for X -1 . The algorithm first checks the convergence condition (Line 2), then computes the k -th order Taylor approximation and its remainder (Lines 3-4), and finally derives bounds via basic addition and subtraction (Lines 5-6).

Lemma 1. Given any non-singular matrix X , the output of Algorithm MATRIXINV ⟨ lXinv , uXinv ⟩ satisfies that: lXinv ≤ Inv( X ) ≤ uXinv .

Lemma 1 guarantees that MATRIXINV yields valid bounds for the Inv operation with fixed input. When extended to a linear set of input matrices, the linear bounds of lXinv and uXinv can be obtained by propagating linear relations line by line. Their union defines the overall linear bounds for the inversion. Example 1 illustrates the advantage of using MATRIXINV over the adjugate method in terms of bound tightness.

Example 1. Given the constant lower and upper bounds of input matrices,

we combine lower bound of lXinv and upper bound of uXinv as overall bounds for Inv operation, and take || uXinv -lXinv || as measurement for bound tightness. The adjugate method yields a bound width of 1 . 22 , while MATRIXINV reduces it to 0 . 70 , closely matching the empirical value of 0 . 66 .

4.2 Abstracting Sorting-based Summation by Indicator

The sorting step in Lines 13-14 of GAUSSIANSPLAT models occlusion: a front Gaussian occludes those behind, but not vice versa. Bounding sorted elements independently introduces large overapproximations in the final pixel color (Line 18), as it neglects the inherent ordering constraint. To mitigate this, we introduce VR-IND (Algorithm 4), which replaces sorting with pairwise indicatorbased occlusion modeling. This approach preserves the output while significantly improving bound tightness in linear relaxation.

Lemma 2. For any given inputs - effective opacities a , colors c and depth d , VR-IND outputs the same value as computation from Line 13 to Line 18 in GAUSSIANSPLAT .

Lemma 2 confirms that VR-IND produces the same output as Lines 13-18 in GAUSSIANSPLAT, capturing both occlusion (Line 2) and color aggregation (Line 4) without explicit sorting. Example 2 further demonstrates its advantage in reducing over-approximation.

Example 2. Consider a scene with three Gaussians (red, green, blue) and a camera at the origin facing upward, with each coordinate perturbed by ±0.3. As shown in Figure 2, the upper bound obtained via linear relaxation through VR-IND is significantly tighter than that derived from direct bound propagation through GAUSSIANSPLAT . An analytical example is given in Appendix A.

4.3 Abstract Summation of Cumulative Product

The summation of cumulative products arises in Lines 8-10 of NERF, where occlusion effects are computed and colors are aggregated to produce the final pixel color. Naively bounding each element of oc independently and then aggregating can lead to significant over-approximation, since the intermediate terms a are shared across multiple entries. To mitigate this, we propose a factorizationbased formulation, SUMCUMPROD (Algorithm 5), which computes each element of a only once in the final expression for pc . This approach preserves exactness for fixed inputs while enabling substantially tighter bounds during linear relaxation.

Lemma 3. For any given inputs - effective opacities a , colors c , SUMCUMPROD outputs the same value as computation from Line 8 to Line 10 in NERF .

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Lemma 3 confirms that, for fixed input, SUMCUMPROD exactly reproduces the original NERF output without explicitly computing occlusion for each sampled point. Example 3 shows that it produces much tighter bounds than the standard method. However, SUMCUMPROD has two limitations: (1) its recursive structure necessitates iterative bound propagation, thereby increasing computational cost; (2) it assumes a fixed order of a , restricting its applicability to Gaussian-represented scenes where the Gaussian depth order remains unchanged, such as in scenarios involving camera translation uncertainty.

Example 3. Consider a single-channel color c = [ 0 . 8 , 1 . 0 , 0 . 9 , 0 . 8 , 0 . 5 ] and effective opacity a = [ 0 . 2 , 0 . 5 , 0 . 6 , 0 . 8 , 0 . 1 ] with ± 0 . 1 perturbations on each a [ i ] . SUMCUMPROD yields a tight bound on output pixel color, [0 . 826 , 0 . 926] , versus a looser [0 . 780 , 1 . 029] from Lines 8-10 of NERF . Sampling 10,000 random perturbations gives [0 . 831 , 0 . 922] , closely matching the result from SUMCUMPROD .

4.4 Linear Approximability of ABSTRACTRENDER

The operations in MATRIXINV, VR-IND, and SUMCUMPROD fall within those listed in Table 1. Since sorting can be eliminated using VR-IND, and all remaining operations are linear overapproximable, we arrive at Theorem 1. It states that both GAUSSIANSPLAT and NERF can be rewritten using only linearly approximable operations, enabling computation of tight bounds via input domain partitioning.

Theorem 1. By replacing components of GAUSSIANSPLAT and NERF with MATRIXINV , VR-IND , and SUMCUMPROD , both rendering algorithms are fully linear over-approximable.

5 Experiments with Abstract Rendering

We implemented the abstract rendering algorithm ABSTRACTRENDER described in Section 4 with both GAUSSIANSPLAT and NERF. As mentioned earlier, the linear approximation of the continuous operations in Table 1 is implemented using CROWN [10].

Scene Description. We evaluate ABSTRACTRENDER on scenes of varying scales and complexities, represented using GAUSSIANSPLAT and NERF. The scenes include: Lego , Chair , and Drums [2], which are single-object scenes on empty backgrounds; PineTree [6], a synthetic boulevard scene with trees; Airport [8], a large-scale photorealistic airport environment; Garden [1], a real-world scene; and Airplane , Truck , and Car , containing objects corresponding to CIFAR-10 classes. GAUSSIANSPLAT reconstructions are generated using Splatfacto [45], while those of NERF are trained with the vanilla NERF algorithm [2], both implemented with standard Nerfstudio settings. Further details of these scenes are provided in the appendix.

Experimental Setup. For each scene, we evaluate ABSTRACTRENDER under varying camera poses and input perturbations. As a baseline, we construct empirical bound images by (1) sampling 50 images per input partition (the full perturbation range is divided into hundreds or thousands of partitions) and (2) computing pixel-wise lower and upper bounds across the samples. Since exhaustive enumeration is infeasible, these empirical bound images are under-approximations of the exact set of renderable images, whereas our framework provides sound over-approximations . The proximity of ABSTRACTRENDER results to the empirical bound images indicates the tightness of our computed bounds.

Metrics and Evaluation. To measure the tightness between the pixel-wise lower and upper bound images, we report Mean Pixel Gap ( MPG ) and Max Pixel Gap ( XPG ), defined as:

where pc j and pc j denote the pixel-wise upper and lower bounds, and WH represents the set of all pixel coordinates on the image plane. Smaller values of MPG or XPG indicate a smaller difference between the lower and upper bound images. Table 2 summarizes the results across different scene representations and perturbations, while Figure 3 visualizes the comparison between ABSTRACTRENDER results and empirical bounds for two scenes: Airplane and Lego .

AbstractRenderResults

EmpiricalResults

Airplane

Lego

From Table 2, we see that ABSTRACTRENDER is applicable to diverse scenes, including single-object ( Lego ), synthetic ( PineTree ), and large-scale realistic ( Airport , Garden ) scenes, with reasonable runtime at 200 × 200 resolution. The framework supports camera perturbations in both position (x, y, z) and orientation (roll, pitch, yaw). The metrics ( MPG and XPG ) computed by ABSTRACTRENDER (under GS and NeRF columns in Table 2) are larger than the corresponding empirical bounds, as expected, since our framework produces over-approximations, which inherently encompass a larger image space than the under-approximated empirical bounds. We further observe that bounds computed for NERF exhibit tighter margins than those for GAUSSIANSPLAT, as indicated by smaller MPG / XPG gaps relative to empirical samples. This is because NERF can leverage our linear bound

relaxation for cumulative product summation (Section 4.3), whereas GAUSSIANSPLAT cannot, due to variations in the depth of 3D Gaussians on the image plane under camera rotation uncertainty.

Overall, these experimental results confirm that our method produces sound over-approximations that are meaningfully tight for rendered images from both GAUSSIANSPLAT and NERF under camera pose uncertainty.

ABSTRACTRENDER for Certified Classification This experiment evaluates ABSTRACTRENDER for certifying neural classifier robustness against camera pose perturbations. We test whether a pretrained CIFAR-10 ResNet [46] maintains correct predictions as the camera orbits the target object azimuthally over 360 ◦ at fixed distance and elevation. The 360 ◦ range is partitioned into angular intervals, and ABSTRACTRENDER computes abstract images for each, which are propagated through the classifier via CROWN [10] for set-based certification. An interval is robust if the lower bound of the target label exceeds the upper bounds of all other classes. Verification coverage is the percentage of certifiably robust intervals. Table 3 shows results for Airplane , Car , and Truck in GAUSSIANSPLAT and NERF scenes. Figure 4 visualizes camera regions where the classifier is certifiably correct (green) and regions without such guarantees (red). Regions capturing more of the object or its distinctive features are more likely to be verified as robust, e.g., the lateral views of the airplane and car, and the front-left view of the truck.

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ABSTRACTRENDER for certified pose estimation This experiment evaluates ABSTRACTRENDER for certifying the robustness of neural pose estimators against camera pose perturbations. We assess a target pose estimator built upon GateNet [47] (fine-tuned on our airplane and truck datasets) by partitioning the camera's translational perturbation range into intervals. ABSTRACTRENDER generates abstract images capturing all photometric variations within each interval, which are propagated through the estimator via CROWN to verify whether positional errors between pose estimation and ground truth holds below a given error tolerance (20% divation from ground truth). Verification coverage is measured as the proportion of robust intervals across the workspace. Table 4 reports results

for Airplane and Truck in GAUSSIANSPLAT and NERF scenes, while Figure 5 visualizes camera regions where the estimator meets the error tolerance (green) versus regions without guarantees (red). Regions where the target object occupies a substantial portion of the rendered image are easier to certify-for example, distant regions in GAUSSIANSPLAT scenes and closer regions in NERF scenes, reflecting the different default camera-object distances in the two settings.

Obj

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Certified Object Detection

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ABSTRACTRENDER for certified object detection This experiment evaluates ABSTRACTRENDER for certifying neural object detectors under camera pose perturbations. We test a pretrained detector based on the YOLO2 architecture [48], with a backbone from the VNN-COMP 2023 benchmark 3 . The goal is to assess whether the detector can consistently produce bounding boxes that exceeds a predefined confidence threshold under camera translations. The camera's translational range is partitioned into small intervals. For each interval, ABSTRACTRENDER computes abstract images. These images are propagated through the YOLO2 detector. An interval is considered robust if the detector maintains a high-confidence bounding box across all images rendered from that interval. Table 4 reports results for Airplane and Truck in GAUSSIANSPLAT and NERF scenes, demonstrating ABSTRACTRENDER's capacity to certify object detection models.

6 Conclusions

Limitations. ABSTRACTRENDER currently has several limitations: First, like other verification approaches, it takes more time than sampling-based approaches. Second, the correctness of the analysis assumes the reconstructed scene accurately represents reality. Third, our framework computes bounds for each pixel's color independently, ignoring correlations between neighboring pixels.

Despite these limitations, ABSTRACTRENDER is the first framework for computing abstract images of scenes represented by Gaussian Splats and NeRF under camera pose or scene uncertainty. It is built using novel linear relational approximations of three rendering-specific operations. By integrating ABSTRACTRENDER with CROWN, we have enabled certification of visual tasks with respect to semantic variations in 3D environments. A promising direction for future work is to accommodate scenes inaccuracies and to certify visual control functions.

Acknowledgments and Disclosure of Funding

Chenxi Ji, Yangge Li, and Sayan Mitra are supported by a research grant from The Boeing Company and NSF (FMITF-2525287). Huan Zhang and Xiangru Zhong are supported in part by the AI2050 program at Schmidt Sciences (AI2050 Early Career Fellowship) and NSF (IIS SLES-2331967, CCF FMITF-2525287).

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A An analytical example illustrating the advantage of Algorithm VR-IND

Consider two purely white Gaussians (with color values set to 1 ) whose probability densities and depth to the target pixel are given by a = { 0 . 8 , 0 . 5 } and d = { x, x + 1 } , respectively, where x ∈ [ -1 , 1] . We aim to compute the worst-case bound of volumetric blending as follows:

we first determine the output range of pc via Naive interval approach. By plugging in range of x into the expression of d , we have d [0] = x ∈ [ -1 , 1] , and d [1] = x +1 ∈ [0 , 2] . Since the relative ordering between d [0] and d [1] cannot be inferred from their interval bound, we must consider both possible cases: one where ( as [0] , as [1]) = ( a [0] , a [1]) , and the other where the assignment is reversed. Thus, we have

Hence, the resulting interval bound for pc is:

Next, we determine the output range of pc via proposed algorithm VR-IND, in which the same computation is encoded symbolically to account for value-dependent assignments. The volumetric blending procedure is reformulated using indicator functions ( Ind ), making the dependencies on d explicit:

Given d [0] and d [1] , we can compute the values of indicator functions Ind in the above formulation.

Substituting these indicator values yields:

Hence, using VR-IND, we obtain a precise bound pc ∈ [0 . 9 , 0 . 9] , which is significantly tighter than the interval-based bound of [0 . 6 , 1 . 2] . Remark: in the real verification pipeline, VR-INDproduces a linear relaxation set instead of a fixed value. Here we use a simplified case for highlighting VR-IND's advantage in terms of bound tightness.

The naive interval method leads to loose bounds due to the loss of dependency information caused by Sort operation. In contrast, the proposed VR-Ind method preserves input-dependent relationships and yields significantly tighter bounds.

B Proofs

Proposition 1. Any continuous function is linearly over-approximable.

Proof. We present a simple proof for scalar f and this can be extended to higher dimensions. We fix x 0 ∈ R and a finite slope k ∈ R . Since f ( x ) -k ( x -x 0 ) is continuous, for any ε > 0 , there must exist an open neighborhood U ( x 0 ) around x 0 , such that ∀ x ∈ R ,

Equivalently,

We construct candidate linear upper and lower bounds over U as:

We can check that these linear functions l U ( x ) and u U ( x ) are valid lower and upper bounds for f ( x ) over U , because ∀ x ∈ U ( x 0 ) :

Additionally, difference between l U ( x ) and u U ( x ) is tightly bounded, as ∀ x ∈ U ( x 0 ) :

Now, for any compact set B , the collection U = { U ( x ) | x ∈ B } forms an open cover of B . By the definition of compactness, there exists a finite subcover of B , denoted by { U i = U ( x i ) | i = 1 , . . . , s } . We can now define piecewise linear functions on B as follows:

where I is the index set of U i that cover x . Consequently, we have

and

Thus, f is linearly over-approximable.

Lemma 1. Given any non-singular matrix X , the output of Algorithm MATRIXINV ⟨ lXinv , uXinv ⟩ satisfies that:

Proof. For any non-singular n × n matrices X and X ref , denote their difference as ∆X = -( X -X ref ) . The k th order Taylor Polynomial Xp and remainder X R of matrix inverse of X , estimated at X ref , can be written as follows:

Assuming that ∥ ∆X · X -1 ref ∥ < 1 , the remainder X R can be bounded by:

Using the bound in inequality 11, we obtain bounds on X -1 :

Given the definition of lXinv = Xp -X R and uXinv = Xp + X R , we obtain the lower and upper bound for X -1 as:

Lemma 2. For any given inputs - effective opacities a , colors c and depth d , VR-IND outputs the same value as computation from Line 13 to Line 18 in GAUSSIANSPLAT .

Proof. We first derive the math expression of pixel color pc from GAUSSIANSPLAT. By denoting arg sort( d ) as a mapping σ : I → I , where I refers to the index set of Gaussians, as at Line 13, cs at Line 14 and oc at Line 16 can be written as follows.

According to the definition of Ind operation, we have:

Then by replacing i , j with σ ( i ) , σ ( j ) , and multiplying a σ ( i ) , (17) becomes:

For any fixed index i , by multiplying all the case in the second branch of (18), it follows:

By applying Equation 16, Equation 18 and Equation 19 into Line 16, oc i can be written as follows:

Then by applying Equation 20 into Line 18, the pixel color pc computed via Algorithm BLENDSORT can be expressed as:

Since the summation and product operations are invariant to the reordering of input elements, and both indices i and j iterate over the entire index set I , the reordering mapping σ in Equation 21 can be eliminated, resulting in:

The last equation comes from 17.

Next, we derive the math expression for pc in Algorithm VR-IND. oc i at Line 2 can be written as:

Then, by applying Equation 23 into Line 4 of Algorithm VR-IND, the expression for pc becomes:

Note that the expressions for pc are identical in both Equation 22 and Equation 24. This demonstrates that Algorithms BLENDSORT and VR-IND yield the same input-output relationship while applying different operations, thus completing this proof.

C Supplementary Experiment Results

C.1 Detailed Scenario Information

Table 5 summarizes scene configurations and rendering quality metrics for the scenarios discussed in Section 5. Figure 6 depicts representative GAUSSIANSPLAT-rendered images for each scene.

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C.2 Supplementary Results for Certified Classification

Figure 7 compares certified classification regions with those verified by sampling.

C.3 Supplementary Results for Certified Pose Estimation

Figure 8 compares certified estimated pose regions with those verified by sampling.

C.4 Supplementary Results for Certified Object Detection

Figure 9 compares certified detection with those verified by sampling.

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C.5 ABSTRACTRENDER Results for Scene Variations

In addition to handling camera pose variations as discussed in Section 5, ABSTRACTRENDER can also accommodate scene variations. For GAUSSIANSPLAT scenes, we consider variations in meaningful sets of 3D Gaussian parameters-such as color, mean position, and opacity of objects like trees in a street scene. For NERF scenes, we consider variations in hue or saturation across all 3D point colors in the scene. Table 6 presents ABSTRACTRENDER results under these scene variations for both GAUSSIANSPLAT and NERF representations. Figure 10 visualizes the abstract image of the PineTree (a GAUSSIANSPLAT scene) scene under uncertainty in color, mean position, and opacity of two tree objects. Figure 11 shows the abstract image of the Lego scene (a NERF scene) under uncertainty in hue and saturation across all 3D points.

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposed Abstract Rendering, which is a framework that abstracts NeRF or 3DGS as a collection of operations, where most operations are linearly over-approximable and can be provably bounded. The paper also designs techniques to handle nonlinear operations (Matrix inverse, sort, cumulative product) by designing equivalent boundable algorithms. The paper verifies the soundness of the framework by comparing the results with empirical sampling, and further conducts experiments to demonstrate various applications of this certification framework.\n\nSTRENGTHS AND WEAKNESSES:\n# Strengths\nThe proposed method is generally novel. It is the first work that interprets the NeRF and 3DGS algorithms using basic operations, provides provable bounds for them, and correctly handles nonlinear operations by further designing tailored techniques to bound them. The soundness of the proposed method is well-validated by experiments, and application-side experiments are conducted to demonstrate the potential of the proposed method.\n\n# Weaknesses\n1. Dependent on existing frameworks: After decomposing the rendering algorithm into operations, it requires an existing framework (CROWN) to provide the certification, which is not the paper's contribution.\n2. Possible refinements in the examples in Sec. 4: In Example 1 in Sec. 4.1, the original matrix $X$ is not given; is this a mistake? In Example 2 and Fig. 2, it is unclear how the mid-left image's (\"via GaussianSplat\") upper bound is generated, since the \"Sort\" operation in GaussianSplat is not linearly over-approximable.\n3. Missing comparisons: The experimental results look good, but there is no comparison or verification for these results. A possible line of work that is worth comparing to is the Bayesian uncertainty quantification in NeRFs (e.g., [1, 2, 3]). How does the uncertainty obtained by the certified bounds (via CROWN) compare to the Bayesian uncertainty? Are they similar or different in purpose or applications?\n\n```\n[1] Goli, Lily, et al. \"Bayes' rays: Uncertainty quantification for neural radiance fields.\" Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. 2024.\n[2] Jiang, Wen, Boshu Lei, and Kostas Daniilidis. \"Fisherrf: Active view selection and mapping with radiance fields using fisher information.\" European Conference on Computer Vision. Cham: Springer Nature Switzerland, 2024.\n[3] Lyu, Linjie, et al. \"Manifold Sampling for Differentiable Uncertainty in Radiance Fields.\" SIGGRAPH Asia 2024 Conference Papers. 2024.\n```\n\nQUESTIONS:\n1. Can you explain whether the framework is dependent on a certain certification framework (CROWN) or can be generalized?\n2. Can you refine your examples in Sec. 4 by adding more explanations?\n3. Can you compare the uncertainty obtained by the certified bounds with Bayesian uncertainty from a probabilistic perspective, and, if possible, provide experimental comparisons?\n4. In Fig. 4, it is surprising that the results of the car and the truck are asymmetric. Intuitively, the result should be symmetric about the axis of $\\frac{3\\pi}{2}$.\n5. In Fig. 5, I can't infer anything from simply looking at the red-green axis. Can you provide some explanations about why this result makes sense?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-11"}

comments: SUMMARY:
This paper proposes a verification framework that computes provable bounds on all images rendered by neural rendering methods such as NeRF and 3DGS under perturbed camera poses. This is achieved through three approximations of non-linear operations, such as matrix inversion and sorting.

STRENGTHS AND WEAKNESSES:
### Strength
- The application of formal verification to neural rendering is novel.
- The proposed approximations are methodologically and mathematically sound, and clearly written.

### Weakness
- It is unclear whether the verification, which is essential for assessing the robustness of general neural networks, is also meaningful for 3D rendering models. In rendering, pixel-level differences caused by camera perturbation are expected and do not indicate a failure or lack of robustness. Moreover, the practical use of rendered images from an already trained model for downstream tasks remains unclear.
- Despite using small, object-centric scenes, the verification runtime is reported to take hours for downstream evaluations, raising concerns about scalability.
- The evaluation setup appears inconsistent. For example, comparisons across methods (NeRF vs. 3DGS) are not controlled under the same configuration.
- Adding the perS to the main paper, rather than only the appendix, would help clarify the extent of over-approximation introduced by the method.

QUESTIONS:
- Could the authors clarify in what practical scenarios that the verification of 3D rendering would be effective?
- Can the authors provide more detailed results for real-world scenes?
- Can it be possible to provide several results where all settings are kept the same, except for the rendering method (NeRF vs. 3DGS)?

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 3
IMPACT: 2
ORIGINALITY: 3
DATE: 2025-06-27

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper first proposes AbstractRendering, an approximate way to estimate the lower and upper bound of the rendered images from a 3D scene given a linear set of camera poses.\nThe paper studies the math operations used in the 3D-GS and NeRFs, especially the (1) matrix inversion (2) sorting-based summation (3) and cumulative-product-based summation, and proposes better estimation for these operators.\nExperiments are conducted to examine the effectiveness of the proposed method.\n\nSTRENGTHS AND WEAKNESSES:\nStrength:\n- The idea of the rendering bound under the noisy camera pose is novel and insightful.\n- The paper is well-organized and easy to follow.\n- The experiments support the claims well.\n\nWeakness:\nI am not an expert in approximation and have not kept up with recent progress in that field. The motivation, theories, and results look reasonable to me. I have only some questions that are not necessarily the weakness of the paper:\n- The 3D-GS scene model described in Section 3.2 L.137, directly uses the RGB vector $\\mathbf{c}$ to model the color instead of the wide-used SH coefficient. Is there any specific reason that the SH coefficient is not used? To my understanding, similar approximation can be made to estimate the procedure of calculating the color from the SH coefficients.\n- In L.159, the linear set of cameras is easy to understand by flatting the camera parameters into one single vector and applying the definition of linear set in L.116 (or by the spatial distribution of the noisy cameras). However, as both Gaussian and NeRF scenes have lots of parameters, how to properly define a linear set of the Scene $\\mathbf{Sc}$?\n- I am a little confused about Table 1. The paper mentioned in L.282 that *the NeRF can leverage the linear bound relaxation for cumulative product summation* so it shows smaller MPG/XPG than the 3D-GS. Is it the only reason to cause such a difference? Will the intrinsic robustness (to viewpoint changes) of 3D representations affect the MPG/XPG metric? For example, if one representation reconstructs the scene better (e.g. with more dense source views, or with the modeling of viewpoint effect such as the ray direction encoding in NeRF or SH in 3D-GS), will it have a smaller MPG/XGP metric for the same scene, compared to the other 3D representation of the same kind whose reconstruction quality is worse? Furthermore, is there some quantitative comparison between the proposed method and the sampling-based method (as Fig. 3)?\n\nQUESTIONS:\n- Why there are N/A values for PSNR/SSIM in Table 5 of the updated appendix?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes an “abstract rendering” framework for scenes represented by neural radiance fields, aiming to compute conservative bounds that encompass the set of all possible rendered images produced over a continuous range of camera poses. The authors design algorithms for propagating input uncertainty through the rendering pipeline, constructing conservative bounds on the output images for verification.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n1. The paper extends abstract image computation techniques from mesh-based rendering to Gaussian splatting and NeRF, which are emerging and popular scene representations.\n2. Adapting abstract image computation to the rendering process for Gaussian splats and NeRF is mathematically nontrivial.\n3. Empirical results show that the proposed method can approximate the pixel-color bounds on the rendered images, which are affected by uncertainty in camera poses.\n\n\n**Weaknesses**\n1. The practical value of the proposed verification is questionable, as the method operates on fixed scene representations obtained via inverse rendering pipelines that already assume known camera parameters. In practice, 3D Gaussian splats or neural radiance fields are optimized to be **consistent** with training camera poses; if the camera poses used during scene capture were uncertain, that uncertainty would already affect the scene representation, not just the rendered images. Thus, verifying possible rendered images from a fixed scene under varying cameras does not reflect real-world verification needs.\n2. Questionable Need Over Empirical Approaches: Robustness to camera pose or scene perturbations could be empirically evaluated by sampling (as Table 1 illustrated), given that 3DGS and NeRF rendering are already **real-time**. The paper does not convincingly justify the necessity of complex and approximate abstraction over simple sampling for practical applications.\n3. The formal bounds produced are over low-level pixel values, whereas safety properties are typically defined at the level of object detection, semantics, or control decisions. The paper does not address end-to-end safety or correctness of downstream tasks.\n\nQUESTIONS:\n1. Please see the weakness section.\n2. What is the practical benefit of the proposed method, compared to dense empirical sampling of the rendering pipeline, especially given the real-time performance of modern 3DGS/NeRF?\n3. Colors in Algorithm 1 should be view-dependent, i.e. perturbing the camera may yield different colors for the same Gaussian, which is not captured in the proposed algorithm.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-03"}