Shape from Shading: A Survey

Ruo ZhangPing-Sing TsaiJ. CryerM. Shah

article1999TPAMI1,833 citations

Presents a comprehensive empirical benchmark of six leading shape-from-shading algorithms, providing standardized implementations and quantitative performance comparisons across depth accuracy, gradient error, and computational efficiency on synthetic and real imagery.

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Recovering three-dimensional surface shape from variations in image shading is a foundational challenge in computer vision with significant potential across industrial inspection, robotics, and digital modeling. While dozens of algorithms have emerged since the field began, real-world deployment remains constrained because existing techniques frequently assume idealized illumination and reflection properties that rarely occur in natural environments.

The article systematically compares and evaluates six representative shape-from-shading methods to assess their computational speed and geometric accuracy across diverse synthetic and real-world test images.

To perform this assessment, the authors implemented six established algorithms spanning four broad methodology categories: minimization, propagation, local, and linear approaches. The experimental benchmark evaluated these implementations on synthetic test images generated under controlled lighting conditions with known ground-truth depth maps, as well as on real-world photographs. Performance was measured using numerical depth and surface gradient errors alongside standardized CPU processing times.

The primary finding is that no single algorithm delivers consistently reliable performance across different image types. Overall, all tested methods produced generally poor surface reconstructions on synthetic benchmarks, and performance degraded even further when applied to real-world images. Among the evaluated categories, minimization approaches achieved the highest overall accuracy and robustness, whereas linear and local methods executed significantly faster at the expense of fidelity and stability. In terms of overall error ranking, optimization-based techniques by Lee and Kuo, followed by Zheng and Chellappa, showed superior accuracy, but their computational demands were several orders of magnitude higher than fast alternatives like Tsai and Shah or Lee and Rosenfeld.

These outcomes demonstrate that standalone shape-from-shading algorithms relying on traditional assumptions are insufficient for mission-critical or precision-driven computer vision tasks. The substantial computational cost of high-accuracy algorithms poses latency risks for real-time applications, while fast approximations introduce unacceptable geometric inaccuracies and noise vulnerability.

Organizations developing practical vision systems should avoid deploying isolated shape-from-shading techniques based on simplistic reflectance assumptions. Instead, future implementations should integrate shading cues with complementary data sources, such as stereo vision, structured range data, or shadow analysis. Where feasible, engineering pipelines should incorporate multi-image approaches that vary illumination or camera viewpoints to successively refine surface depth estimates.

The findings are bounded by the specific single-point light source configurations, smooth surface geometries, and idealized reflection models evaluated. Decision-makers should exercise caution when extrapolating synthetic performance metrics to complex operational environments where interreflections, shadows, and non-uniform surface textures dominate.

Carnegie Mellon University ETH Zurich University of Central Florida cpu high resolution image adaptive algorithms shape from shading photometric stereo depth maps

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Abstract

Since the first shape-from-shading (SFS) technique was developed by Horn in the early 1970s, many different approaches have emerged. In this paper, six well-known SFS algorithms are implemented and compared. The performance of the algorithms was analyzed on synthetic images using mean and standard deviation of depth (Z) error, mean of surface gradient (p, q) error and CPU timing. Each algorithm works well for certain images, but performs poorly for others. In general, minimization approaches are more robust, while the other approaches are faster. The implementation of these algorithms in C, and images used in this paper, are available by anonymous ftp under the pub/tech-paper/survey directory at eustis.cs.ucf.edu (132.170.108.42). These are also part of the electronic version of paper.

Topics

cpu high resolution image adaptive algorithms optimization problems anonymous communication central processing unit

Keywords

shape from shading photometric stereo depth maps rough surfaces surface depth spherical surfaces

1 Introduction
2 Literature Review
2.1 Minimization Approaches
2.2 Propagation Approaches
2.3 Local Approaches
2.4 Linear Approaches
Interreflections
Convergence, Uniqueness and Existence
The Organization of the Paper
3 Reflectance Models
3.1 Lambertian and Specular Reflectance Models
4 Selected Shape from Shading Algorithms
4.1 Minimization Approaches
4.1.1 Zheng and Chellappa (91)
4.1.2 Lee and Kuo (91)
4.2 Propagation Approaches
4.2.1 Bichsel and Pentland (92)
4.3 Local Approaches
4.3.1 Lee and Rosenfeld (85)
4.4 Linear Approaches
4.4.1 Pentland (88)
4.4.2 Tsai and Shah (92)
5 Experimental Images
5.1 Synthetic Images
5.2 Real Images
6 Experimental Results
6.1 Zheng and Chellappa
6.2 Lee and Kuo
6.3 Bichsel and Pentland
6.4 Lee and Rosenfeld
6.5 Pentland
6.6 Tsai and Shah
7 Error Analysis
8 Timing
9 Conclusions and Future Research
Acknowledgments
References

Knowls

Knowl 1 — Taxonomy of Shape from Shading Approaches
model/method
Shape from Shading (SFS) methods reconstruct a 3-D surface shape from the intensity variations of a single 2-D image. Based on their algorithmic formulation, SFS algorithms are categorized into four principal groups:
1. Minimization Approaches: Formulate surface recovery as a global optimization problem by minimizing an energy functional that combines a brightness constraint with regularization terms (such as surface smoothness, integrability, or intensity gradient constraints).
2. Propagation Approaches: Propagate surface depth or orientation outward across the image starting from singular points (points of local maximum brightness where surface orientation is uniquely determined) or initial boundary curves.
3. Local Approaches: Calculate surface slant and tilt locally at each pixel using image intensity and its spatial derivatives under simplifying local shape assumptions, such as spherical patches.
4. Linear Approaches: Linearize the nonlinear reflectance map with respect to surface gradients or depth directly, yielding closed-form frequency-domain solutions or simple linear relaxation schemes that avoid matrix inversions.
Knowl 2 — Variational Energy Constraints for Shape from Shading
equation
Variational minimization approaches in Shape from Shading determine surface geometry by minimizing an energy functional combining data fidelity and regularizing constraints over the image domain $\Omega$ :
1. Brightness Constraint: Penalizes irradiance error between measured intensity $I(x, y)$ and estimated reflectance $R(p, q)$ : $E_{\text{brightness}} = \iint_\Omega (I(x, y) - R(p, q))^2 \, dx \, dy$ where $(p, q) = \left(\frac{\partial Z}{\partial x}, \frac{\partial Z}{\partial y}\right)$ represents the surface gradient of depth $Z(x, y)$ .
2. Smoothness Constraint: Enforces gradual variation in surface gradients or surface normals $\vec{N}$ to regularize ill-posedness: $E_{\text{smooth}} = \iint_\Omega (p_x^2 + p_y^2 + q_x^2 + q_y^2) \, dx \, dy \quad \text{or} \quad \iint_\Omega (\|\vec{N}_x\|^2 + \|\vec{N}_y\|^2) \, dx \, dy$ where subscripts denote partial differentiation.
3. Integrability Constraint: Enforces that the recovered gradient field corresponds to a continuous, integrable surface ( $Z_{xy} = Z_{yx}$ ): $E_{\text{integ}} = \iint_\Omega (p_y - q_x)^2 \, dx \, dy \quad \text{or} \quad \iint_\Omega \left((Z_x - p)^2 + (Z_y - q)^2\right) \, dx \, dy$
4. Intensity Gradient Constraint: Aligns the spatial derivatives of the simulated reflectance $(R_x, R_y)$ with the measured image derivatives $(I_x, I_y)$ : $E_{\text{grad}} = \iint_\Omega \left((R_x - I_x)^2 + (R_y - I_y)^2\right) \, dx \, dy$
5. Unit Normal Constraint: Constrains the recovered normal vectors $\vec{N}$ to unit length: $E_{\text{unit}} = \iint_\Omega (\|\vec{N}\|^2 - 1)^2 \, dx \, dy$
Knowl 3 — Zheng and Chellappa Variational SFS Method
model/method
The Zheng and Chellappa algorithm reconstructs surface shape by minimizing an energy functional that replaces the conventional smoothness constraint with an intensity gradient constraint alongside brightness and integrability terms: $E = \iint_\Omega \left( (I - R)^2 + (R_x - I_x)^2 + (R_y - I_y)^2 + \mu \left( (Z_x - p)^2 + (Z_y - q)^2 \right) \right) dx \, dy$ where $I(x, y)$ is image brightness, $R(p, q)$ is the reflectance map, $I_x, I_y, R_x, R_y$ are first-order spatial derivatives, $Z(x, y)$ is surface depth, $(p, q)$ is the surface gradient, and $\mu$ is a weighting constant set to $\mu = 1$ .

The resulting Euler-Lagrange differential equations are discretized and simplified using a first-order Taylor series expansion of the reflectance map $R$ , leading to an iterative update scheme that optimizes depth $Z$ and gradients $(p, q)$ simultaneously. The computation is executed over a multi-resolution image pyramid to improve convergence speed and prevent local minima traps. Depth and gradient fields are initialized to zero everywhere without requiring boundary condition initialization. Finite differences are used for derivatives (forward differences in the interior and backward differences at image boundaries). When the illuminant aligns with the viewing direction $(0, 0, 1)$ , terms in the iterative update vanish, requiring a small perturbation such as $(0.01, 0.01, 1)$ .
Knowl 4 — Lee and Kuo Triangular Element Multigrid SFS Method
model/method
The Lee and Kuo algorithm approximates a 3-D surface as a union of planar triangular surface patches whose vertices form nodal points. The method solves directly for depths at the nodal points, obtaining intermediate pixel depths via linear interpolation.

For each triangular patch, the average intensity across pixels in the triangle is equated to the reflectance function, with the surface gradient $(p, q)$ computed from the cross product of two adjacent patch edges. Linearizing the reflectance map $R(p, q)$ with respect to $(p, q)$ via a first-order Taylor expansion yields a linear relation between triangular patch intensity and the depths at its three nodal vertices. Combining this linearized irradiance constraint with a quadratic smoothness regularizer reduces shape recovery to a large-scale sparse linear system.

The linear system is solved using a V-cycle multigrid numerical scheme:
- A grid hierarchy of $L = \log_2(M) - 1$ levels is used for an $M \times M$ image.
- Gauss-Seidel relaxation serves as the smoothing operator on coarse grids and exact solver on the finest grid.
- Full-weighting restriction transfers residuals to coarser grids, and bilinear interpolation performs prolongation.
- The algorithm executes up to 10 successive outer linearizations, using 10 V-cycles in the first iteration, 2 in the second, and 1 in subsequent iterations, initialized with zero depth everywhere.
Knowl 5 — Bichsel and Pentland Minimum Downhill Propagation SFS Algorithm
algorithm
The Bichsel and Pentland propagation algorithm reconstructs depth directly from singular points without global variational optimization, enforcing surface continuity by propagating downhill along illumination trajectories:
Input: Image $I(x, y)$ , light source direction vector $S = (s_x, s_y, s_z)$
Output: Recovered depth map $Z(x, y)$
Rotate image coordinates so that light source direction $S$ aligns with one of eight discrete directions.
Identify singular points (pixels with maximum gray level).
Initialize depth at singular points to a fixed positive height (e.g., 55).
Initialize depth at all non-singular pixels to a large negative number (e.g., $-1.0 \times 10^{10}$ ).
Precompute surface gradients $(p, q)$ under the parallel slope constraint:
   $p = \frac{-s_x s_z \pm \sqrt{(1 - R^2)(R^2 - s_y^2)}}{R^2 - s_x^2 - s_y^2}$
   $q = \frac{p s_y s_x - s_y s_z}{R^2 - s_y^2}$
for iteration = 1 to max_iterations (typically 5 to 8) do
  Alternate the traversal direction across the 8 discrete directions.
  for each non-singular pixel $(i, j)$ do
    Compute local candidate surface heights from neighboring pixels along downhill directions.
    Update $Z(i, j)$ by maximizing local height via Gauss-Seidel relaxation.
  end for
end for
Apply inverse rotation to restore depth map $Z(x, y)$ to viewer coordinates.
return $Z(x, y)$
The method terminates rapidly (typically in 5 to 8 iterations) because distance to the light source is a monotonically increasing function of local height when the angle between illumination and the optical axis is under $90^\circ$ .
Knowl 6 — Lee and Rosenfeld Local Spherical Shape from Shading Method
model/method
The Lee and Rosenfeld local method estimates surface orientation at each pixel under the assumption that local surface patches are spherical. Under Lambertian reflectance ( $I = \rho \vec{N} \cdot \vec{S}$ , with surface albedo $\rho$ , unit surface normal $\vec{N}$ , and unit light source vector $\vec{S}$ ), surface slant and tilt are evaluated in light-source coordinates and mapped to viewer coordinates.

The tilt angle $\tau$ of the surface normal in viewer coordinates is computed analytically from first-order spatial intensity derivatives: $\tau = \arctan\left( \frac{I_y \cos \tau_S - I_x \sin \tau_S}{I_x \cos \tau_S \cos \sigma_S + I_y \cos \sigma_S \sin \tau_S} \right)$ where $I_x = \frac{\partial I}{\partial x}$ and $I_y = \frac{\partial I}{\partial y}$ are image intensity derivatives, and $\sigma_S, \tau_S$ are the slant and tilt angles of the light source vector $\vec{S} = (\sin \sigma_S \cos \tau_S, \sin \sigma_S \sin \tau_S, \cos \sigma_S)$ .

The cosine of surface slant is determined from the ratio of local image intensity to the maximum intensity $\rho$ (obtained at singular points where $\vec{N} = \vec{S}$ ). Because the method is non-iterative and uses only first derivatives of intensity, it is computationally fast but sensitive to image noise and geometric departures from local sphericity.
Knowl 7 — Pentland Linear Closed-Form Fourier Shape from Shading Method
model/method
Pentland's linear approach computes a non-iterative, closed-form depth solution by expanding the Lambertian reflectance map $R(p, q) = \frac{\cos \sigma_S + p \cos \tau_S \sin \sigma_S + q \sin \tau_S \sin \sigma_S}{\sqrt{1 + p^2 + q^2}}$ into a first-order Taylor series around the planar state $(p_0, q_0) = (0, 0)$ : $I(x, y) \approx \cos \sigma_S + p \cos \tau_S \sin \sigma_S + q \sin \tau_S \sin \sigma_S$ where $\sigma_S$ and $\tau_S$ are light source slant and tilt angles, and $(p, q) = \left(\frac{\partial Z}{\partial x}, \frac{\partial Z}{\partial y}\right)$ .

Dropping the constant DC component $\cos \sigma_S$ and taking the 2-D Fourier transform of both sides using derivative Fourier identities: $\frac{\partial Z(x, y)}{\partial x} \longleftrightarrow F_Z(\omega_1, \omega_2)(i \omega_1), \qquad \frac{\partial Z(x, y)}{\partial y} \longleftrightarrow F_Z(\omega_1, \omega_2)(i \omega_2)$ yields the frequency-domain relationship: $F_I(\omega_1, \omega_2) = F_Z(\omega_1, \omega_2) \left( i \omega_1 \cos \tau_S \sin \sigma_S + i \omega_2 \sin \tau_S \sin \sigma_S \right)$ where $F_I$ and $F_Z$ are the Fourier transforms of image intensity $I(x, y)$ and surface depth $Z(x, y)$ , respectively. Surface depth $Z(x, y)$ is recovered directly by algebraic division in the frequency domain followed by the 2-D inverse Fast Fourier Transform.
Knowl 8 — Tsai and Shah Discrete Linear Shape from Shading Algorithm
algorithm
Tsai and Shah developed an iterative SFS algorithm that linearizes the discrete Lambertian reflectance map directly in terms of depth $Z(x, y)$ using finite differences:
Input: Image $I$ of size $M \times M$ , light source vector $S = (s_x, s_y, s_z)$
Output: Depth map $Z$
Initialize $Z_{i,j}^0 = 0$ for all $1 \le i, j \le M$ .
for iteration $n = 1, 2, \dots$ until convergence do
  for each pixel $(i, j)$ do
    Compute discrete gradient approximations:
       $p_{i,j} = Z_{i,j}^{n-1} - Z_{i-1,j}^{n-1}$
       $q_{i,j} = Z_{i,j}^{n-1} - Z_{i,j-1}^{n-1}$

    Compute Lambertian reflectance and intensity residual:
       $R(p_{i,j}, q_{i,j}) = \frac{-s_x p_{i,j} - s_y q_{i,j} + s_z}{\sqrt{1 + p_{i,j}^2 + q_{i,j}^2}}$
       $f(Z_{i,j}^{n-1}) = I_{i,j} - R(p_{i,j}, q_{i,j})$

    Evaluate derivative with respect to $Z_{i,j}$ :
       $\frac{d}{d Z_{i,j}} f(Z_{i,j}^{n-1}) = -\frac{(-s_x - s_y)\sqrt{1 + p^2 + q^2} - (-s_x p - s_y q + s_z)\frac{p + q}{\sqrt{1 + p^2 + q^2}}}{1 + p^2 + q^2}$

    Update depth via Jacobi relaxation step:
       $Z_{i,j}^n = Z_{i,j}^{n-1} + \frac{-f(Z_{i,j}^{n-1})}{\frac{d}{d Z_{i,j}} f(Z_{i,j}^{n-1})}$
  end for
end for
Apply Gaussian smoothing filter to depth map $Z$ .
return $Z$
The algorithm recovers depth through pointwise scalar division without matrix inversions, but requires safeguards against division by zero and degrades in regions with self-shadows.

Knowl 9 — Quantitative Reconstruction Accuracy of Six SFS Algorithms

data/table

Six representative SFS algorithms were evaluated quantitatively on synthetic images ( $128 \times 128$ ) generated from two known geometric models (Synthetic Vase and Mozart) illuminated from two directions: $S_1 = (0, 0, 1)$ and $S_2 = (1, 0, 1)$ . Output depths were normalized against true range data before computing mean depth error ( $\bar{e}_Z$ ), depth standard deviation ( $\sigma_Z$ ), and mean surface gradient error ( $\bar{e}_{pq}$ ).

Algorithm	Mean $Z$ Error ( $\bar{e}_Z$ )				Std Dev $Z$ Error ( $\sigma_Z$ )				Mean $(p, q)$ Error ( $\bar{e}_{pq}$ )				Sum $\bar{e}_Z$	Rank
	Vase		Mozart		Vase		Mozart		Vase		Mozart		Sum $\bar{e}_Z$
	$S_1$	$S_2$	$S_1$	$S_2$	$S_1$	$S_2$	$S_1$	$S_2$	$S_1$	$S_2$	$S_1$	$S_2$
Lee Kuo (1993)	10.0	7.9	13.7	9.77	13.2	15.39	19.2	22.1	1.6	0.9	1.7	0.6	41.37	1
Zheng Chellappa (1991)	8.5	8.5	15.1	10.56	11.1	13.9	18.4	15.9	2.2	1.3	2.3	1.1	42.66	2
Bichsel Pentland (1992)	10.08	7.9	20.5	7.7	13.8	16.9	37.4	14.6	2.7	1.9	3.1	1.9	46.18	3
Pentland (1988)	11.2	9.0	15.7	19.7	12.6	11.1	18.2	20.56	1.8	1.2	1.3	1.3	55.60	4
Lee Rosenfeld (1985)	8.4	18.3	17.8	11.3	14.6	22.3	33.0	30.3	3.3	6.8	13.7	4.3	55.80	5
Tsai Shah (1994)	8.3	12.7	18.5	20.0	15.0	19.7	33.3	30.5	1.4	6.7	5.5	4.2	59.50	6

The data shows that variational minimization approaches (Lee & Kuo, Zheng & Chellappa) achieve the highest overall reconstruction accuracy, followed by downhill propagation (Bichsel & Pentland). Linear and local methods exhibit higher overall error, especially when oblique illumination ( $S_2$ ) introduces stronger nonlinear intensity variations.

Knowl 10 — Computational Runtime and Efficiency Across SFS Algorithms

data/table

Computational execution times (excluding disk I/O) were benchmarked on a SUN SPARC 4 workstation for $128 \times 128$ synthetic images under light source vectors $S_1 = (0, 0, 1)$ and $S_2 = (1, 0, 1)$ :

Algorithm	Vase ( $128 \times 128$ )		Mozart ( $128 \times 128$ )
Algorithm	$S_1$ (seconds)	$S_2$ (seconds)	$S_1$ (seconds)	$S_2$ (seconds)
Tsai Shah (1994)	0.9	1.2	3.8	4.8
Lee Rosenfeld (1985)	1.5	1.7	6.2	6.9
Pentland (1988)	4.2	4.2	17.7	18.4
Bichsel Pentland (1992)	7.3	6.1	27.6	26.9
Zheng Chellappa (1991)	18.1	70.8	49.6	409.5
Lee Kuo (1993)	188.4	2584.9	771.5	20249.5

The timing data demonstrates an inverse trade-off between computational cost and reconstruction accuracy:

Variational minimization techniques (Lee & Kuo, Zheng & Chellappa) are computationally intensive, taking up to thousands of seconds because runtime varies with scene geometric complexity and relaxation convergence rates.
Downhill propagation (Bichsel & Pentland) provides moderate speed (6.1 to 27.6 seconds) due to fast convergence in under 10 iterations.
Linear (Tsai & Shah, Pentland) and local (Lee & Rosenfeld) methods are the fastest (0.9 to 18.4 seconds), with execution times governed primarily by image resolution rather than surface structure.

Knowl 11 — Systematic Failure Modes of Shape from Shading Algorithms
limitation
Comparative evaluation of SFS techniques on synthetic and real images identifies several fundamental limitations:
1. Synthetic-to-Real Generalization Failure: While algorithms achieve measurable reconstruction on ideal Lambertian synthetic models, performance degrades severely on real images (e.g., Lenna, Pepper, Vase). Error metrics on synthetic data do not predict performance on real data due to non-Lambertian reflectance, albedo variations, and non-collimated lighting.
2. Oversmoothing of High-Frequency Surface Detail: Iterative minimization methods (Lee & Kuo, Zheng & Chellappa) recover macro surface shape effectively but progressive relaxation smooths out fine facial features and sharp surface details.
3. Sensitivity to Singular Point Inaccuracy: Propagation methods (Bichsel & Pentland) depend critically on singular points for depth initialization; noise or specularity in real images produces incorrect singular points, causing catastrophic reconstruction failures.
4. Nonlinear Breakdown and Shadow Artifacts: Linear approaches (Pentland, Tsai & Shah) suffer from frequency doubling and severe distortion when quadratic reflectance terms dominate, and fail completely in self-shadowed regions.
5. Singularities Under Direct Illumination: When illumination aligns directly with the optical axis $(0, 0, 1)$ , update terms in algorithms such as Zheng & Chellappa and Lee & Kuo become singular or vanish, requiring artificial light direction perturbations.

Coverage note — Brief survey mentions of historical non-implemented algorithms (e.g., Horn 1970, Ikeuchi & Horn 1981, Brooks & Horn 1985, Frankot & Chellappa 1988) and background reflectance models (Phong, Torrance-Sparrow, Nayar et al.) were omitted as standalone knowls because they represent prior literature rather than the contributed implementations and experimental evaluations of this study.

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Citation

MLA

Ruo Zhang, et al. “Shape-from-shading: A Survey”. IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 21, no. 8, 1999, pp. 690–706, https://doi.org/10.1109/34.784284.

APA

Ruo Zhang, Ping-Sing Tsai, Cryer, J. E., & Shah, M. (1999). Shape-from-shading: a survey. IEEE Transactions on Pattern Analysis and Machine Intelligence, 21(8), 690–706. https://doi.org/10.1109/34.784284

Chicago

Ruo Zhang, Ping-Sing Tsai, J. E. Cryer, and M. Shah. 1999. “Shape-from-shading: A Survey”. IEEE Transactions on Pattern Analysis and Machine Intelligence 21 (8): 690–706. https://doi.org/10.1109/34.784284.

Harvard

Ruo Zhang et al. (1999) “Shape-from-shading: a survey”, IEEE Transactions on Pattern Analysis and Machine Intelligence, 21(8), pp. 690–706. Available at: https://doi.org/10.1109/34.784284.

Vancouver

1. Ruo Zhang, Ping-Sing Tsai, Cryer JE, Shah M (1999) Shape-from-shading: a survey. IEEE Transactions on Pattern Analysis and Machine Intelligence 21:690–706

BibTeX

@article{Ruo_Zhang_1999, title={Shape-from-shading: a survey}, volume={21}, ISSN={0162-8828}, url={http://dx.doi.org/10.1109/34.784284}, DOI={10.1109/34.784284}, number={8}, journal={IEEE Transactions on Pattern Analysis and Machine Intelligence}, publisher={Institute of Electrical and Electronics Engineers (IEEE)}, author={Ruo Zhang and Ping-Sing Tsai and Cryer, J.E. and Shah, M.}, year={1999}, pages={690–706} }

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Shape from Shading: A Survey

Abstract

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Table of Contents

Knowls

Knowl 1 — Taxonomy of Shape from Shading Approaches

Knowl 2 — Variational Energy Constraints for Shape from Shading

Knowl 3 — Zheng and Chellappa Variational SFS Method

Knowl 4 — Lee and Kuo Triangular Element Multigrid SFS Method

Knowl 5 — Bichsel and Pentland Minimum Downhill Propagation SFS Algorithm

Knowl 6 — Lee and Rosenfeld Local Spherical Shape from Shading Method

Knowl 7 — Pentland Linear Closed-Form Fourier Shape from Shading Method

Knowl 8 — Tsai and Shah Discrete Linear Shape from Shading Algorithm