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Checkerboard rendering

Checkerboard rendering is a technique designed to approximate high-resolution output, such as , by rendering approximately half the pixels in an alternating checkerboard pattern across the frame and reconstructing the missing pixels through spatial and methods. This approach leverages data from previous frames, including object ID buffers that track geometry boundaries, to propagate colors and details, enabling efficient upscaling without the full computational load of native rendering. Originally developed to optimize performance on hardware-limited systems, it produces images with crisp edges, enhanced texture detail, and reduced artifacts that closely resemble full-resolution renders. The technique gained prominence with the launch of the PlayStation 4 Pro in 2016, where integrated hardware support for checkerboard rendering to deliver enhanced visuals on displays without requiring developers to fully render at 2160p resolution. According to 's lead system architect , the method uses a 1920x2160 render buffer—effectively half of —combined with custom hardware that writes an ID buffer alongside the Z-buffer at the same resolution, facilitating accurate reconstruction even during camera or object motion. Early adopters included titles like and , which employed checkerboarding to achieve dynamic resolutions up to near- while maintaining stable frame rates. similarly supported the approach on the Xbox One X, extending its use across console gaming ecosystems. In practice, checkerboard rendering requires deep integration into a game's rendering pipeline, particularly with post-processing effects like and dynamic resolution scaling, as demonstrated in EA's engine for games such as Battlefield 1 and Mass Effect: Andromeda. Key features include object ID tracking for disocclusion handling, alpha unrolling for transparency, and gradient adjustments to minimize visual artifacts, though it doubles the pixel shader workload compared to standard rendering. More recent implementations, such as Intel's Checkerboard Rendering (CBR) for integrated graphics from 2018, emphasize its adaptability to real-time upsampling on lower-end hardware using quarter-resolution rendering with temporal reconstruction, motion vectors, and . Despite its efficiency, checkerboard rendering can introduce minor temporal instabilities, such as shimmering in fine details during motion, but these are often mitigated through advanced reconstruction algorithms that outperform traditional upscaling methods like . It remains a foundational upscaling in modern game development as of 2025, with continued use on platforms like the (e.g., in titles such as the 2024 beta of Wilds) and influencing subsequent technologies like NVIDIA's DLSS, AMD's , and Sony's PlayStation Spectral Super Resolution (PSSR).

Introduction

Definition

Checkerboard rendering is a technique that renders in a non-overlapping checkerboard pattern, covering approximately 50% of the target 's pixel centers, while the remaining are generated through to achieve an effective higher output. This method allows graphics processing units to compute full- and depth information for all but shade only the selected subset, reducing computational load without significantly compromising visual fidelity. As a form of selective , checkerboard rendering samples geometry and shading at full target resolution but applies shading computations solely to alternate arranged in a grid-like , such as every other along diagonals. This approach leverages spatial and temporal reconstruction to fill in unshaded , effectively simulating benefits at a fraction of the cost of traditional full-resolution rendering. The technique is particularly suited for applications where hardware constraints limit native high-resolution performance. A representative example involves rendering a scene at an internal resolution of 3200×1800 (often denoted as 1800p) in a pattern to produce a (3840×2160) output, where only half the pixels are fully shaded per frame, and the rest are interpolated using neighboring data and depth buffers. This pattern ensures even coverage across the image, minimizing artifacts like while approximating the detail of native rendering.

Purpose and Benefits Overview

Checkerboard rendering primarily serves to deliver high-resolution output, such as , on graphics hardware that lacks the power for native full-resolution rendering, thereby alleviating computational demands while preserving acceptable image quality. This technique emerged as a solution for consoles like the PlayStation 4 Pro, where it allows demanding games to target ultra-high-definition displays without overwhelming the GPU. By rendering only a subset of pixels in a staggered pattern, it effectively halves the shading workload compared to native rendering, enabling smoother performance in complex scenes. Among its key benefits, checkerboard rendering strikes a balance between performance gains and visual fidelity by optimizing GPU utilization for partial frame generation, which supports stable frame rates even in resource-intensive environments. It provides efficiency on integrated or mid-range graphics hardware, offering speed-ups of approximately 1.14x to 1.4x in shading-heavy workloads, while integrating seamlessly with existing pipelines to minimize artifacts. This approach has been adopted in production engines like for titles such as , demonstrating its scalability across various game scenarios without requiring extensive hardware upgrades. Conceptually, the technique achieves supersampling-like effects—enhancing edge smoothness and detail—without the full overhead of rendering every pixel, as the pixel pattern facilitates efficient reconstruction to a complete high-resolution frame. This trade-off prioritizes broader accessibility to high-fidelity graphics, particularly on consumer-grade systems, by leveraging temporal data and post-processing to fill gaps with minimal quality degradation.

History

Early Concepts

The origins of checkerboard rendering concepts trace back to advancements in and sampling theory during the and , where researchers applied the Nyquist-Shannon sampling theorem to for efficient image reconstruction on non-uniform sampling grids. The theorem posits that a continuous signal can be perfectly reconstructed from its samples if the sampling rate exceeds twice the highest frequency component, preventing artifacts; in graphics, this principle was adapted to handle discrete pixel grids by exploring non-uniform distributions that avoid the need for dense, uniform while minimizing visible distortions. Early work emphasized how non-uniform grids could redistribute high-frequency energy into less perceptible noise, laying the groundwork for sparse sampling strategies that prioritize computational efficiency over exhaustive coverage. Academic discussions in the late introduced practical nonuniform sampling techniques for , focusing on generating patterns at low densities to produce antialiased images without excessive computational cost. For instance, point-diffusion algorithms were proposed to create Poisson-disk distributions, which satisfy blue-noise properties by concentrating power in mid-to-high frequencies and reducing low-frequency clumps, outperforming jittered uniform grids in perceptual quality. These methods extended analysis to image , demonstrating that strategic sampling patterns could emulate higher-resolution outputs by scattering into high-frequency noise that filters could suppress effectively. Such approaches predated applications but highlighted the potential for grid-based variations to optimize sampling for complex scenes. By the early 1990s, these ideas evolved to incorporate multidimensional parameters in ray tracing, further refining sparse sampling for anti-aliasing in distribution effects like motion blur and depth of field. Researchers extended two-dimensional nonuniform sampling to higher dimensions, using spectrally optimal patterns to ensure aliasing manifests as random, high-frequency noise rather than structured artifacts, aligning with Nyquist principles by avoiding bandlimiting challenges in uniform setups. Concepts of rotated or stratified grid sampling emerged in these contexts as ways to enhance frequency coverage without proportional increases in sample count, enabling efficient representation of high-frequency details in pre-rendered imagery. This theoretical foundation influenced later adaptations for dynamic rendering, though initial focus remained on offline graphics research.

Commercial Adoption

Checkerboard rendering gained significant commercial traction with the launch of the PlayStation 4 Pro in November 2016, where promoted the technique as a hardware-accelerated method to achieve enhanced visuals in games without full native rendering. This was particularly evident in titles like , developed using the Decima engine by , which utilized 2160p checkerboard rendering to deliver high-fidelity open-world visuals on displays. Microsoft followed suit with the X release in November 2017, incorporating support for checkerboard rendering to enable output in a range of enhanced titles. Games such as employed the technique, rendering at native 1800p and upscaling via checkerboard to for improved performance and image quality. Other examples included , demonstrating the method's integration across third-party ports. Subsequent developments expanded checkerboard rendering beyond consoles. In August 2018, published a whitepaper detailing its implementation for real-time upscaling on integrated graphics, highlighting the technique's potential for lower-power devices following its console success. On the PC side, adoption remained limited, with engines like 4 enabling experimental support through plugins and custom implementations, though native higher resolutions were preferred due to variable hardware capabilities. A key milestone came at 2017, where presented details on Decima's checkerboard approach, influencing broader interest. By 2020, with the and Xbox Series X/S launches, checkerboard rendering saw growing use for dynamic resolution scaling in performance modes, allowing developers to balance output with stable frame rates in demanding titles like . This evolution built on earlier theoretical concepts from academic work, adapting them for next-generation hardware efficiency. Following the ninth-generation console launches, checkerboard rendering continued to be employed in updates and new titles. In 2021, received a patch for enabling checkerboard rendering at 60 in high-resolution mode. As of 2024, it remained in use on Pro for select games, including the series at /60 via checkerboard, and in performance mode at approximately 1800p checkerboard, often alongside emerging upscaling methods like PlayStation Spectral Super Resolution (PSSR).

Technical Fundamentals

Rendering Pattern

Checkerboard rendering employs a spatial arrangement where pixels are rendered in a diagonal grid across the target resolution, such that rendered pixels—analogous to the black squares on a —alternate with non-rendered pixels that will later be interpolated, collectively covering exactly 50% of the grid for the initial pass. This pattern ensures that no two horizontally or vertically adjacent pixels are both rendered in the same frame, promoting a balanced distribution of computational load while maintaining high-fidelity sampling. In terms of sampling, full , including depth and primitive coverage, is performed at the target resolution (e.g., or 3840×2160), but , lighting, and texturing computations are restricted to the rendered s only, halving the shading workload compared to full-resolution rendering. The pattern's design facilitates an even distribution of high-frequency samples throughout the image, avoiding clustered that could degrade detail in specific regions. For instance, in a target, samples are selected at positions where the row index i and column index j satisfy (i + j) \mod 2 = 0, creating the alternating offsets that define the grid. This mathematical basis, rooted in , guarantees a quasi-random yet deterministic placement that captures fine details effectively across the frame. The choice of pattern—standard diagonal checkerboard versus rotated variants—significantly influences artifact susceptibility, particularly around edges. The standard checkerboard, which aligns with axis-oriented offsets, preserves horizontal and vertical edge details at near-native resolution by ensuring samples straddle these boundaries, outperforming uniform subsampling that might blur or alias such features due to consistent offsets. In contrast, rotated patterns (e.g., at 45 degrees) can better handle diagonal edges but may introduce more aliasing on axis-aligned ones if not carefully tuned; however, the standard approach is preferred in many implementations for its compatibility with hardware-accelerated interpolation and reduced edge crawling artifacts under motion. Subsequent interpolation fills the gaps to reconstruct the full image.

Interpolation Process

In checkerboard rendering, the interpolation process reconstructs unrendered pixels by estimating their values from the sparse set of rendered samples, typically using spatial or temporal methods to fill in the gaps while preserving image quality. The core approach relies on , where each missing pixel is computed as a weighted average of its four nearest rendered neighbors, which are positioned diagonally due to the alternating sampling pattern. This method is efficient and widely adopted in applications, as it leverages hardware-accelerated texture sampling to blend the samples based on sub-pixel offsets. The formula for a target at position (x, y) with fractional offsets a = x - \lfloor x \rfloor and b = y - \lfloor y \rfloor is given by: P(x,y) = (1-a)(1-b) \cdot P_{00} + a(1-b) \cdot P_{10} + (1-a)b \cdot P_{01} + ab \cdot P_{11} where P_{00}, P_{10}, P_{01}, and P_{11} represent the diagonally adjacent rendered pixels. To mitigate blurring and enhance edge preservation, this spatial interpolation is often combined with edge-aware filtering techniques, such as FXAA applied prior to resolution, which detects and sharpens high-contrast boundaries while smoothing low-contrast areas. serves as an advanced spatial variant, extending bilinear by incorporating additional neighboring samples for smoother gradients and reduced , though it increases computational cost. Temporal addresses limitations of purely spatial methods by incorporating data from previous , using motion vectors to reproject and blend historical samples into the current unrendered positions, thereby reducing flickering and improving over time. This involves comparing depth values from prior and current —e.g., rejecting history if |\text{prev_depth} - \text{current_depth_avg}| \geq \text{tolerance}—and blending accepted samples (often at a 50/50 ) to resolve up to four samples per after two . For higher-quality spatial reconstruction, Lanczos filtering can be employed as an alternative to bilinear, using a sinc-based with a support radius (typically 2–3) to better preserve high-frequency details, though it is less common in scenarios due to its higher . These interpolation techniques excel at handling low-frequency content, such as broad gradients and uniform areas, by effectively averaging samples without introducing noticeable artifacts, but they can cause blurring on fine details like thin lines or textures unless paired with methods like temporal . In practice, the combination of spatial with temporal reprojection achieves near-native resolution quality, with minor softening as a for .

Implementation

Hardware Considerations

Checkerboard rendering imposes specific demands on graphics processing units (GPUs), primarily requiring hardware support for dynamic resolution rendering and efficient pixel discard mechanisms, such as viewport scaling or scissor tests to render only a subset of pixels in a patterned fashion. Mid-range GPUs, exemplified by the AMD Polaris-based architecture in the PlayStation 4 Pro with its 36 compute units and 4.2 teraflops of compute performance, can achieve approximately 50% reduction in pixel shading load when targeting resolutions like 1440p checkerboard upscaled to 4K, making it feasible without excessive developer overhead. This approach leverages fixed-function hardware features, including dedicated ID buffers for pattern reconstruction, to minimize computational cost. In terms of memory and bandwidth, checkerboard rendering reduces video RAM (VRAM) usage by shading only half the pixels of the target resolution, thereby lowering and storage needs compared to native rendering. This is particularly beneficial in bandwidth-constrained environments like consoles, where the PlayStation 4 Pro's 8 GDDR5 memory operates at 218 /s— a 24% increase over the base model—allowing more efficient data flow for high-resolution outputs without proportional increases in memory pressure. Platform-specific optimizations highlight its suitability for fixed hardware in consoles, such as the Pro's custom GPU enhancements that enable seamless checkerboard targeting for near-4K visuals at stable frame rates. On PC platforms, however, implementation faces challenges due to hardware variability and inconsistent driver support across vendors, requiring custom solutions that may not scale uniformly. For , the technique supports adjustable pattern densities, such as rendering at 25% of full for even greater load reduction, which proves effective on weaker integrated graphics processors like Intel's, delivering speed-ups of 1.14x to 1.4x in pixel-bound scenarios through quarter-resolution render targets. This flexibility allows adaptation to diverse hardware without fundamental redesigns.

Software Techniques

Developers integrate checkerboard rendering into game engines by configuring render targets at half or quarter resolution and applying alternating offsets to create the sparse pixel pattern across frames. In 12, this involves jittering the by one full-resolution between frames to shift the rendered areas, ensuring efficient use of the . Custom viewports and scissor tests in APIs like 11/12 or further enable masking of non-rendered regions, allowing precise control over the checkerboard layout without additional hardware dependencies. The rendering pipeline proceeds in multiple passes: the first renders the scene geometry and to the pixels in a low-resolution buffer, often with 2x MSAA for edge quality. A subsequent pass employs a dedicated , typically in HLSL, to fill missing pixels by blending data from the current frame and the previous frame's buffer, guided by motion vectors and depth comparisons to resolve occlusions. Post-processing effects, such as bloom, are applied exclusively to the reconstructed full-resolution image to avoid inconsistencies; in the engine, this necessitates adjustments to the post-processing chain, including efficient depth resolves and integration with . Game engines offer specific tools and libraries to streamline implementation. supports dynamic resolution scaling through console variables like r.ScreenPercentage, which developers adapt for checkerboard patterns on supported platforms. In , custom shaders handle pattern generation and texture merging, using render textures at half height to alternate between even and odd frames before . Debugging these setups involves visualization tools that overlay rendered versus interpolated pixels, helping identify artifacts like ghosting; the DynamicCheckerboardRendering sample provides in-engine toggles to display occluded or missing pixels for targeted fixes.

Applications

In Console Gaming

Checkerboard rendering emerged as a cornerstone technique in console gaming with the launch of the PlayStation 4 Pro in November 2016, which introduced hardware-accelerated support for the method to upscale lower-resolution renders to output, enabling compatibility with emerging televisions without requiring full native rendering. The Xbox One X followed in November 2017, adopting checkerboard rendering as one of several upscaling options, though it was used less frequently than on the PS4 Pro due to the console's greater computational power for native in select titles. By 2018, over 50 PS4 Pro-enhanced titles had implemented the technique, including high-profile releases such as and Marvel's Spider-Man, which leveraged it to deliver sharp visuals on displays while maintaining performance targets. A notable example is , released in 2017, which utilized dynamic checkerboard rendering to ensure open-world stability on the PS4 Pro, dynamically adjusting the render resolution to preserve a consistent 30 frames per second at an effective output. This implementation was integrated into ' Decima engine, allowing for custom per-pixel data processing that minimized artifacts and supported complex environmental details without drops. The technique's use has evolved on next-generation consoles, with the and Xbox Series X/S supporting checkerboard rendering to enable hybrid resolution approaches in select titles, balancing visual fidelity and efficiency. The PlayStation 5 Pro, launched in November 2024, introduces PlayStation Spectral Super Resolution (PSSR) for advanced upscaling, though checkerboard rendering persists in select base PS5 modes and titles as of 2025. These systems also enhance for PS4 Pro and Xbox One X games, applying automatic boosts to stabilize dynamic resolutions and improve interpolation for smoother playback. By facilitating effective 4K visuals at reduced computational cost, checkerboard rendering empowers console developers to prioritize high-resolution assets and effects in action-intensive scenes, avoiding the frame rate penalties associated with native 4K rendering on hardware with fixed resources.

In PC and Other Platforms

Checkerboard rendering on personal computers is typically implemented as an optional feature in select games, often through in-game settings or engine-specific options, allowing players to balance performance and resolution on varied hardware. For instance, in F1 2019, the game includes a TAA Checkerboard mode that renders approximately half the pixels in a pattern each , reconstructing the full to improve s while approximating higher resolutions. This approach contrasts with console implementations by relying on developer integration rather than standardized support. Intel pioneered a dedicated checkerboard rendering (CBR) technique in 2018 specifically for low-end integrated graphics processing units, such as those in UHD Graphics, to enable real-time upscaling on budget systems. The method generates full-resolution output with reduced shading computations, making it suitable for entry-level PCs where native high-resolution rendering is infeasible. Similarly, NVIDIA introduced experimental checkerboard support in drivers starting in late 2019, primarily for multi-GPU (SLI) configurations, to enhance performance in compatible titles without requiring extensive game modifications. Adoption on PC remains limited due to inconsistencies in graphics driver support across vendors like and , which often require custom extensions or profiles for activation. High-end PCs favor scalable rendering, reducing the need for such techniques amid alternatives like DLSS or .

Advantages and Limitations

Performance Gains

Checkerboard rendering achieves significant compute reductions by approximately half the pixels per frame through an alternating pattern, effectively halving the workload compared to rendering. This leads to 30-50% savings in GPU utilization, depending on the rendering and scene complexity. For instance, on the PlayStation 4 Pro, games like render at an effective 1800p checkerboard resolution (yielding output) with a total frame time of 15.99 ms, compared to 21.07 ms for native 1800p, representing a 24% reduction primarily from savings in G-buffer generation (1.14 ms saved) and (1.50 ms saved). Similarly, Mass Effect: Andromeda sees a 36% frame time reduction to 23.42 ms from 36.82 ms at native 1800p, with major gains in lighting computations (6.18 ms saved). These efficiencies translate to substantial frame rate benefits, allowing games to maintain higher target frame rates at elevated resolutions. In pixel shader-bound workloads, checkerboard rendering can reduce frame times by up to 30%, enabling 60 performance in titles that would otherwise be limited to 30 at native resolutions. Benchmarks demonstrate this, such as 1440p checkerboard rendering to yielding approximately 1.8x speedup over native due to the halved pixel shading. On the PS4 Pro, this approach supports -like output at 2.88 million effective pixels (half of 1800p's approximately 5.76 million), facilitating stable 60 in performance-oriented modes for games like those using the engine. Resource efficiency is another key advantage, with reduced computational demands on consoles. This scalability benefits battery-powered devices, such as laptops with integrated graphics, where average speedups of 1.14x to 1.2x (up to 1.4x in shading-dominant scenes) minimize energy use while supporting higher resolutions. In deferred rendering pipelines, checkerboard rendering improves fill rate efficiency by minimizing overdraw, as fewer pixels are processed in geometry and lighting passes. For example, in Frostbite-based titles, this results in targeted savings like 15% in ambient shadowed and 12% in clustered over full-resolution equivalents, enhancing overall GPU throughput without proportional increases in demands. The interpolation step contributes minimally to overhead, typically under 2 ms, preserving the net efficiency gains.

Visual Artifacts and Drawbacks

Checkerboard rendering, while enabling higher apparent resolutions, introduces several visual artifacts primarily stemming from its alternating sampling and temporal process. Temporal manifests as shimmering on moving edges, where mismatches between frames cause flickering or crawling artifacts, particularly noticeable in dynamic scenes with . Moiré patterns can also emerge in fine textures when the checkerboard aligns poorly with repetitive details, exacerbating in fast-paced environments like . The technique does not achieve true , with a loss of high-frequency details in interpolated areas, reducing overall and , especially in static or low-motion shots where the compromises become more apparent, as seen in implementations rendering at intermediate resolutions like 3360x1890 before upscaling. Additional drawbacks include increased on diagonal edges without supplementary , leading to fuzzy fringing on elements like hair in Uncharted 4 or alpha effects in . These issues are highly dependent on scene complexity, with artifacts worsening in high-motion scenarios due to occlusion-related ghosting, where reprojected pixels from prior frames introduce erroneous shading. To mitigate these artifacts, checkerboard rendering requires careful tuning, often combined with (TAA) to stabilize output and reduce instability from subtle camera movements or edge crawling on primitives. Such integration helps preserve detail but demands deep post-processing adjustments to avoid amplifying problems like severe sharpening filters in motion-heavy titles such as .

Comparisons to Alternatives

Versus Native Resolution Rendering

Checkerboard rendering significantly reduces computational demands compared to native high-resolution rendering. Rendering at native requires processing four times the pixels of , whereas checkerboard rendering typically shades approximately half the pixels of native —equivalent to about twice the workload—by alternating pixel samples across frames. This makes checkerboard viable on mid-tier like the PS4 Pro, where native 4K would exceed performance budgets, enabling higher frame rates or additional graphical effects. In terms of image quality, provides perfect per-pixel sampling without approximation, but at a 2-4x performance penalty relative to baselines. approximates the full through temporal , resulting in minor detail loss such as subtle artifacts in fine textures or motion, though it inherently offers superior due to its staggered sampling pattern. Analyses show images are sharper than simple upscaling from equivalent lower resolutions but fall short of native fidelity, with occasional flickering or in dynamic scenes. Native rendering suits high-end systems, such as the PS5 achieving at 60 fps in titles like , where ample GPU power supports full pixel computation without compromises. In contrast, checkerboard rendering was employed on the PS4 Pro for resolution boosts in base modes of games like and Mass Effect: Andromeda, targeting output on hardware incapable of native equivalents. Overall, checkerboard rendering trades a modest reduction in visual fidelity—often manifesting as slightly softened details—for roughly 50% computational efficiency over native , as evidenced by halved shading costs and performance gains of 24-36% in tested titles. This balance allowed broader access to high-resolution visuals on limited during the PS4 Pro era.

Versus AI-Based Upscaling

Checkerboard rendering employs a fixed geometric , where the scene is rendered at approximately half the target in a staggered, alternating , followed by deterministic to reconstruct the full image. This approach relies on spatial and temporal , such as expanding 2x2 blocks into 4x4 structures using edge-directed filters and leveraging data from previous frames, including motion vectors, for and disocclusion handling. In contrast, AI-based upscaling techniques like NVIDIA's DLSS 2.0 utilize convolutional neural networks trained on high-resolution reference images to predict and generate missing details from lower-resolution inputs, incorporating temporal information from multiple frames for enhanced . In terms of quality and performance, AI methods generally outperform checkerboard rendering in detail preservation and artifact reduction; for instance, can upscale from to with sharper textures and fewer shimmering artifacts in dynamic scenes, achieving visual fidelity closer to while delivering 1.5-2x frame rate gains on compatible hardware. However, DLSS requires extensive training datasets of game-specific or generic high-quality frames and dedicated tensor cores for inference, introducing computational overhead and potential variability across games. Checkerboard rendering, being fully deterministic and free of dependencies, avoids such requirements and runs efficiently on standard GPU shaders, though it may exhibit more noticeable or grid-like artifacts in high-contrast areas. Checkerboard rendering first appeared in 2016 with the PlayStation 4 Pro, enabling near-4K output on mid-range hardware without specialized accelerators. DLSS debuted in 2018 alongside NVIDIA's RTX 20-series GPUs, restricting its use to hardware with tensor cores, while equivalents like AMD's offer broader compatibility but lack DLSS's AI-driven precision in early versions. This hardware-agnostic nature makes checkerboard suitable for legacy systems or integrated graphics, whereas AI upscaling demands modern, vendor-specific GPUs. Checkerboard rendering fits scenarios requiring straightforward resolution boosts on resource-constrained consoles or older PCs, providing reliable performance without per-title optimization. AI-based methods like DLSS excel in demanding applications, such as ray-traced games, where their temporal accumulation yields superior stability and detail recovery, often at higher effective resolutions than checkerboard equivalents.

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