Cascaded Non-Line-of-Sight Imaging

Diego Royo; María Peña; Forrest B. Peterson; Andreas Velten; Julio Marco; Diego Gutierrez

cs.GRSep 3, 2026

Cascaded Non-Line-of-Sight Imaging

Authors: Diego Royo, María Peña, Forrest B. Peterson, Andreas Velten, Julio Marco, Diego Gutierrez

Organizations: Universidad de Zaragoza–I3A, Spain · University of Wisconsin–Madison, United States of America

Abstract

Time-of-flight non-line-of-sight (NLOS) imaging recovers information from hidden objects by analyzing the time of flight of indirect photons scattered on a visible (relay) wall. Most methods make the simplifying assumption that photons travel exclusively three-bounce paths, thus ignoring other useful information encoded in higher-order photons (with, e.g., four- or five-bounce paths). We present a novel cascaded NLOS imaging approach that leverages higher-order information and allows imaging a broader range of single- and multi-corner scenarios. We combine ultra-fast laser scanning with recent time-gated 2D sensor arrays to capture the scene's impulse response on a visible relay wall. From the captured impulse response, our method computes an analogous virtual impulse response at any other hidden wall. This effectively allows us to concatenate a second, virtual NLOS imaging system that leverages higher-order illumination. We validate our cascaded imaging method both in simulation and with a real prototype, demonstrating NLOS imaging with fourth- and fifth-bounce illumination of objects in challenging orientations and hidden around two corners. We also analyze how wave-based NLOS imaging interacts with rough hidden walls, which explains and helps overcome existing visibility limitations. We further illustrate how to image hidden objects from different perspectives, thus observing previously unseen features, by relying on multiple hidden walls.

Figures & tables

Panel of Figure 1 — Figure 1. (a) Classic time-of-flight NLOS imaging has shown how to image objects hidden around one corner, using time-resolved measurements of third-bounce indirect light (purple) from an ultra-fast laser (red) and sensor (blue). (b) Our cascaded NLOS imaging method allows imaging objects hidden around two corners by concatenating a second, virtual NLOS imaging system and without the need to capture any additional data, effectively using higher-order illumination (e.g., fifth-bounce light in yellow). Our work enables NLOS imaging over a wider, more general set of configurations, helping overcome visibility issues suffered by previous methods.

Panel of Figure 2 — Figure 2. (a) A laser device emits delta light pulses $\delta(\mathbf{l}_{\scriptstyle 1},t)$ , illuminating each point $\mathbf{l}_{\scriptstyle 1}\in{\mathcal{L}_{1}}$ on the relay wall $\mathfrak{R}_{1}$ . (b) An ultra-fast sensing device captures the time-resolved impulse response $H(\mathbf{l}_{\scriptstyle 1},\mathbf{s}_{\scriptstyle 1},t)$ on multiple visible points $\mathbf{s}_{\scriptstyle 1}\in{\mathcal{S}_{1}}$ of the imaging aperture ${\mathcal{S}_{1}}$ in $\mathfrak{R}_{1}$ . (c) Imaging operators such as the confocal camera $\Phi_{\text{cc}}(\mathbf{v};H)$ transform $H(\mathbf{l}_{\scriptstyle 1},\mathbf{s}_{\scriptstyle 1},t)$ to obtain an image $f(\mathbf{v})$ of the hidden bunny at points $\mathbf{v}$ on a bounding volume $\mathcal{V}$ .

Symbol	Description

$\mathfrak{R}_{n},\ n=1$	The visible relay wall
$\mathfrak{R}_{n},\ n>1$	A hidden, secondary relay wall
$\mathbf{l}_{\scriptstyle a},\mathbf{s}_{\scriptstyle b}$	Illuminated and measured points on $\mathfrak{R}_{a}$ and $\mathfrak{R}_{b}$
$\mathcal{L}_{a},\mathcal{S}_{b}$	Illumination and camera apertures on $\mathfrak{R}_{a}$ and $\mathfrak{R}_{b}$
$H(\mathbf{l}_{\scriptstyle 1},\mathbf{s}_{\scriptstyle 1},t)$	Captured impulse response

Table 1. Symbols used throughout this paper. A hat (

\hat{\cdot}

) denotes the temporal Fourier transform, with the time

t

replaced by the frequency

\Omega

Figure 3. (a) The confocal camera operator

\Phi_{\text{cc}}

computationally focuses light at the illumination aperture

{\mathcal{L}_{1}}

and at the camera aperture

{\mathcal{S}_{1}}

\mathfrak{R}_{1}

on the same point

\mathbf{v}

. We separate this procedure into (b) the transient camera

\Phi_{\text{tc}}

and (c) our transient projector

\Phi_{\text{tp}}

operators. Each focuses the light at

{\mathcal{S}_{1}}

{\mathcal{L}_{1}}

to individual points

\mathbf{s}_{\scriptstyle b}

and

\mathbf{l}_{\scriptstyle a}

respectively.

Panel of Figure 4 — Figure 4. Cascading NLOS imaging systems. (a) A laser emits pulses to each point $\mathbf{l}_{\scriptstyle 1}\in{\mathcal{L}_{1}}$ on $\mathfrak{R}_{1}$ and an ultra-fast camera captures $H(\mathbf{l}_{\scriptstyle 1},\mathbf{s}_{\scriptstyle 1},t)$ at points $\mathbf{s}_{\scriptstyle 1}\in{\mathcal{S}_{1}}$ also on $\mathfrak{R}_{1}$ . Here $H(\mathbf{l}_{\scriptstyle 1},\mathbf{s}_{\scriptstyle 1},t)$ contains third- and fifth-bounce illumination from $\mathfrak{R}_{2}$ and T, respectively. (b) We compute our virtual impulse response $H^{\prime}_{a,b}(\mathbf{l}_{\scriptstyle a},\mathbf{s}_{\scriptstyle b},t)$ on $\mathfrak{R}_{2}$ from $H$ , i.e., $H^{\prime}_{2,2}(\mathbf{l}_{2},\mathbf{s}_{2},t)$ , using the transient projector $\Phi_{\text{tp}}$ and camera $\Phi_{\text{tc}}$ operators to focus light at ${\mathcal{L}_{1}}$ or ${\mathcal{S}_{1}}$ to each point $\mathbf{l}_{2}\in{\mathcal{L}_{2}}$ or $\mathbf{s}_{2}\in{\mathcal{S}_{2}}$ in $\mathfrak{R}_{2}$ respectively. Here $H^{\prime}_{2,2}$ shows light reflected by the T as third-bounce illumination. (c) Finally we concatenate a confocal camera operator $\Phi_{\text{cc}}$ with $H^{\prime}_{2,2}$ , resulting in an image $f_{2,2}(\mathbf{v})$ for $\mathbf{v}\in\mathcal{V}$ of the T effectively computed using fifth-bounce illumination.

Panel of Figure 5 — Figure 5. Cascaded NLOS imaging methods. (a) The simulated impulse response $H(\mathbf{l}_{\scriptstyle 1},\mathbf{s}_{\scriptstyle 1},t)$ at $\mathfrak{R}_{1}$ contains light that has scattered on the relay wall $\mathfrak{R}_{1}$ , the three letters A, B, and C (oriented differently), and on the left $\mathfrak{R}_{2}$ and right $\mathfrak{R}_{3}$ walls. Hence, $H$ contains third- (purple), fourth- (cyan) and fifth-bounce (yellow) illumination. (b) Third-bounce imaging methods use apertures ${\mathcal{L}_{1}}$ and ${\mathcal{S}_{1}}$ and thus can only use part of $H$ . (c) In this example, our fourth-bounce imaging method concatenates our transient projector operator $\Phi_{\text{tp}}$ focused at $\mathbf{l}_{2}\in\mathcal{L}_{2}$ on the left wall $\mathfrak{R}_{2}$ along with the confocal camera operator $\Phi_{\text{cc}}$ . (d) Our fifth-bounce method concatenates our transient projector $\Phi_{\text{tp}}$ and camera $\Phi_{\text{tc}}$ operators focused at $\mathbf{l}_{3}\in\mathcal{L}_{3}$ and $\mathbf{s}_{3}\in\mathcal{S}_{3}$ respectively on the right wall $\mathfrak{R}_{3}$ along with the confocal camera operator $\Phi_{\text{cc}}$ . (e) Combined result.

Figure 6. Virtual surface reflectance. Top: planar hidden wall. Bottom: rough hidden wall with $3\text{\times}3\text{\,}\mathrm{c}\mathrm{m}$ facets. (a) Scene setup: we illuminate one point $\mathbf{l}_{\scriptstyle 1}$ , and capture the impulse response in ${\mathcal{S}_{1}}$ . We mark the imaged region as $\mathcal{V}$ . (b) Simulated propagation of a phasor wave of wavelength $\lambda=$3\text{\,}\mathrm{c}\mathrm{m}$$ emitted from $\mathbf{l}_{\scriptstyle 1}$ and reflected by the hidden wall. The planar wall reflects light specularly (cyan lines); only the area $A_{p}$ returns light to the aperture ${\mathcal{S}_{1}}$ . In contrast, the rough wall reflects diffusely, returning light to the aperture from a larger area $A_{r}$ . (c) Resulting image $f_{1,1}(\mathbf{v})=\Phi_{\text{cc}}(\mathbf{v};H)$ over $\mathcal{V}$ . The orange dotted line marks the expected wall location. — Figure 6. Virtual surface reflectance. Top: planar hidden wall. Bottom: rough hidden wall with $3\text{\times}3\text{\,}\mathrm{c}\mathrm{m}$ facets. (a) Scene setup: we illuminate one point $\mathbf{l}_{\scriptstyle 1}$ , and capture the impulse response in ${\mathcal{S}_{1}}$ . We mark the imaged region as $\mathcal{V}$ . (b) Simulated propagation of a phasor wave of wavelength $\lambda=$ 3\text{,}\mathrm{c}\mathrm{m}$$ emitted from $\mathbf{l}_{\scriptstyle 1}$ and reflected by the hidden wall. The planar wall reflects light specularly (cyan lines); only the area $A_{p}$ returns light to the aperture ${\mathcal{S}_{1}}$ . In contrast, the rough wall reflects diffusely, returning light to the aperture from a larger area $A_{r}$ . (c) Resulting image $f_{1,1}(\mathbf{v})=\Phi_{\text{cc}}(\mathbf{v};H)$ over $\mathcal{V}$ . The orange dotted line marks the expected wall location.

Figure 7. Effect of the filter parameters $\lambda_{c}$ and $\sigma$ of $K(t;\lambda_{c},\sigma)$ (Equation 3 ) for the setups in Figure 6 . (a) The hidden wall is planar with respect to every $\lambda_{c}$ and is thus always virtually specular; its edges have limited visibility. (b) The rough wall is virtually diffuse for $\lambda_{c}$ around $3\text{\,}\mathrm{c}\mathrm{m}$ (the size of its facets), and its full surface is imaged. For larger $\lambda_{c}$ , it becomes planar with respect to the wavelength and converges to the same virtually specular behavior. Larger $\sigma$ , corresponding to a narrower bandwidth of $K$ , makes the NLOS imaging result more uniform, suggesting increased diffuseness.

Panel of Figure 8 — Figure 8. Enhanced multi-view imaging of complex geometry. (a) The simulated capture $H(\mathbf{l}_{\scriptstyle 1},\mathbf{s}_{\scriptstyle 1},t)$ contains third-bounce illumination from the target object ( Bunny ) and also higher-order interreflections that arise from interactions with the top $\mathfrak{R}_{2}$ and right $\mathfrak{R}_{3}$ walls. (b) We show that fourth-bounce illumination extends the coverage of conventional NLOS imaging methods. Here we focus the camera aperture ${\mathcal{S}_{1}}$ of the first relay wall $\mathfrak{R}_{1}$ on the top $\mathfrak{R}_{2}$ and right $\mathfrak{R}_{3}$ walls (creating virtual imaging devices with apertures ${\mathcal{S}_{2}}$ and $\mathcal{S}_{3}$ , respectively). Each yields a different image $f_{1,1}(\mathbf{v}),f_{1,2}(\mathbf{v}),f_{1,3}(\mathbf{v})$ of the target object, and (c) we combine all of them in $f_{\text{all}}(\mathbf{v})$ to show the enhanced visibility of our cascaded NLOS imaging paradigm.

Panel of Figure 9 — Figure 9. We simulate tracking an object in a region $\mathcal{V}_{\text{target}}$ hidden around two corners, not visible from the relay wall $\mathfrak{R}_{1}$ . (a) Light from the target object is reflected on the rough hidden wall $\mathfrak{R}_{2}$ before reaching $\mathfrak{R}_{1}$ , resulting in fifth-bounce illumination (yellow arrow). (b) First, we image the secondary wall $\mathfrak{R}_{2}$ from $\mathfrak{R}_{1}$ using third-bounce illumination (purple arrow). (c) The virtual mirrors (VM) approach ( Royo et al., 2023 ) fails due to the roughness of $\mathfrak{R}_{2}$ . Our cascaded NLOS imaging yields a good approximation of the trajectory of the hidden object by implementing a secondary imaging system at $\mathfrak{R}_{2}$ .

Panel of Figure 10 — Figure 10. Our hardware prototype. We use a $16\times 16$ SPAD array, focused on the relay wall using a $6\times$ Edmund Optics lens, and a two-mirror galvanometer (Thorlabs GVS012) to guide an ultra-fast laser emitter towards a $190\times 190$ grid of points on the relay wall.

Panel of Figure 11 — Figure 11. Real-world demonstration of NLOS imaging with fourth-bounce light. (a) We use a setup similar to Figure 5 . (b) While the 2 can be imaged using third-bounce methods ( $f_{1,1}(\mathbf{v})$ ), the orientation of the A requires fourth-bounce illumination ( $f_{1,2}(\mathbf{v})$ ), enabled by our cascaded approach (analogous to our simulations in Figure 5 b and Figure 5 c). (c) Combined image $f_{\text{all}}(\mathbf{v})=f_{1,1}(\mathbf{v})+f_{1,2}(\mathbf{v})$ .

Panel of Figure 12 — Figure 12. We repeat in simulation the same imaging procedure from Figure 5 e, changing the size and resolution of the illumination aperture ${\mathcal{L}_{1}}$ to match the SPAD array from our prototype. (a) Original result. (b) Reducing both the resolution to $16\times 16$ points and covered area to $50\text{\times}32.5\text{\,}\mathrm{c}\mathrm{m}$ . The fourth-bounce imaging results of the A letter match the resolution of our real capture in Figure 11 . Moreover, as explained in Section 8.2 , the fifth-bounce results of the C letter also match our real-world, fifth-bounce experiments.

Panel of Figure 13 — Figure 13. Overcoming two-corner visibility limits using our hardware prototype. (a) The left $\mathfrak{R}_{2}$ and right $\mathfrak{R}_{3}$ walls reflect higher-order illumination to and from the target object. (b) Due to the walls’ position and orientation, neither third-bounce methods nor the virtual mirrors approach ( Royo et al., 2023 ) can image the target object (marked by $\mathcal{V}$ and $\mathcal{V}^{m}$ ). (c) Our cascaded NLOS fifth-bounce method images the object at $\mathcal{V}$ using $\mathcal{L}_{2}$ and $\mathcal{S}_{3}$ .

Figure 14. We compare our real-world experiment (top) from Figure 13 with its simulated counterpart (bottom), mimicking our $16\times 16$ SPAD. Both the captured $H$ and virtual $H^{\prime}_{2,3}$ impulse responses (a), as well as the resulting image $f_{2,3}(\mathbf{v})$ (b) closely match, with small differences due to geometric discrepancies between the real and the simulated scenes.

Panel of Figure 15 — Figure 15. Imaging around two corners using our hardware prototype. (a) The rough hidden wall $\mathfrak{R}_{2}$ reflects light to and from the target object. (b) Both conventional methods and virtual mirrors ( Royo et al., 2023 ) fail to image the target object (marked by $\mathcal{V}$ and $\mathcal{V}^{m}$ , respectively). (c) In contrast, our method identifies the target at the correct location $\mathcal{V}$ by creating a secondary virtual imaging system at $\mathfrak{R}_{2}$ .

Panel of Figure 16 — Figure 16. (a) We repeat the setup from Figure 13 using an L-shaped target. (b) Front view of our imaging result using our real prototype ( $16\times 16$ SPAD covering a $50\;\text{cm}\times 32.5\;\text{cm}$ aperture). (c) Top view of the same result.

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 18 — Figure S.1 . Comparison of our reconstructed virtual impulse response $H^{\prime}_{2,2}$ at the wall $\mathfrak{R}_{2}$ with the equivalent captured impulse response $H_{2,2}$ for the setup in Figure 4 of the main paper. (a) Imaging procedures to obtain each impulse response. The top row virtually illuminates and captures points $\mathbf{l}_{2}$ and $\mathbf{s}_{2}$ of $\mathfrak{R}_{2}$ using our transient projector $\Phi_{\text{tp}}$ and camera $\Phi_{\text{tc}}$ operators, respectively. The virtual impulse response uses fifth-bounce illumination from the T letter, highlighted in yellow. In the bottom row, we simulate an equivalent scene where we move the laser and SPAD to point directly towards $\mathfrak{R}_{2}$ . This results in third-bounce illumination from the T letter, highlighted in purple. (b) Resulting impulse responses for each case.

Figure S.2 . (a) We replicate the scene from Figure 15 in the main paper. (b) We compare our estimation of the secondary relay wall $\mathfrak{R}_{2}$ using a third-bounce method against a known layout obtained via 3D scans. (c) We analyze the impact of estimation error in the final cascaded imaging results $f_{2,2}(\mathbf{v})$ , and observe both methods show very similar results.

Figure S.3 . We extend the missing cone analysis from Liu et al. (2019a) to our scene in Figure 13 of the main paper, where we rotate the target object. The virtual mirrors method from Royo et al. (2023) and our method use different apertures

\mathcal{L}

and

\mathcal{S}

, and as a result each samples different Fourier cones of the NLOS measurement function. (a) The virtual mirrors method is fixed to

{\mathcal{L}_{1}}

and

{\mathcal{S}_{1}}

. In this case, the sampled Fourier cone only works for setups with specific orientations, and thus fails on the bottom row. (b) Our method can create apertures at different parts of the hidden scene (

\mathcal{L}_{2}

\mathcal{L}_{3}

and

\mathcal{S}_{3}

), which in turn sample different parts of the NLOS measurement function. Note how our method can adapt to each configuration, lifting previous restrictions about specific orientations.

Figure S.4 . (a) We compare the resolution of our fifth-bounce imaging using $\mathfrak{R}_{2}$ and (b) the corresponding third-bounce imaging where the object at $\mathcal{V}$ is at a similar distance from the relay wall $\mathfrak{R}_{1}$ . We simulate both captures with and without noise. The imaging resolution loss from using multiple apertures in $\mathfrak{R}_{1}$ and $\mathfrak{R}_{2}$ in the two-corner scene (compared to the single-corner scene) is much more notable than the effect of noise.

Panel of Figure 22 — Figure S.5 . 3D scans and photographs for the setups using our real prototype in Section 8. Our hardware prototype, visible in (a), is pointed towards the relay wall $\mathfrak{R}_{1}$ . In the 3D scans, we highlight the hidden target objects in orange, and the areas where the SPAD array (red, ${\mathcal{L}_{1}}$ ) and laser (blue, ${\mathcal{S}_{1}}$ ) are focused (b, c).

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Time-of-Flight non-line-of-sight (ToF NLOS) imaging techniques provide state-of-the-art reconstructions of scenes hidden around corners by inverting the optical path of indirect photons scattered by visible surfaces and measured by picosecond resolution sensors. The emergence of a wide range of ToF NLOS imaging methods with heterogeneous formulae and hardware implementations obscures the assessment of both their theoretical and experimental aspects. We present a comprehensive study of a representative set of ToF NLOS imaging methods by discussing their similarities and differences under common formulation and hardware. We first outline the problem statement under a common general forward model for ToF NLOS measurements, and the typical assumptions that yield tractable inverse models. We discuss the relationship of the resulting simplified forward and inverse models to a family of Radon transforms, and how migrating these to the frequency domain relates to recent phasor-based virtual line-of-sight imaging models for NLOS imaging that obey the constraints of conventional lens-based imaging systems. We then evaluate performance of the selected methods on hidden scenes captured under the same hardware setup and similar photon counts. Our experiments show that existing methods share similar limitations on spatial resolution, visibility, and sensitivity to noise when operating under equal hardware constraints, with particular differences that stem from method-specific parameters. We expect our methodology to become a reference in future research on ToF NLOS imaging to obtain objective comparisons of existing and new methods.

Abstract

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