A Parametric Texture Model Based on Joint Statistics of Complex Wavelet Coefficients

Javier PortillaEero P. Simoncelli

article2000IJCV2,090 citations

Proposes a parametric texture model and iterative synthesis algorithm based on the joint statistics of complex wavelet coefficients across positions, orientations, and scales to accurately capture human visual texture perception.

Listen

Visual textures represent spatially uniform patterns that are ubiquitous in natural environments, yet mathematically characterizing them in a way that aligns with human perception has historically proven difficult. Early theories hypothesized that matching low-order pixel statistics would yield visually indistinguishable textures, but simple statistical descriptions failed when confronted with human perceptual discrimination. While modern non-parametric methods can generate compelling textures, they lack compact mathematical representations and cannot easily infer texture properties from partial or degraded images. The article develops and evaluates a universal parametric statistical model designed to accurately capture and reproduce visual textures using a fixed, compact set of biologically motivated statistical measurements.

The research evaluates a multi-scale complex wavelet framework (specifically, a complex steerable pyramid) combined with an iterative synthesis-by-analysis algorithm. The approach extracts 710 universal statistical parameters from a single grayscale texture image. These parameters capture lowpass marginal statistics (such as variance, skewness, and kurtosis), raw subband correlations, magnitude correlations across positions, orientations, and scales, and cross-scale relative phase statistics. To synthesize textures matching these parameters, the algorithm iteratively adjusts an initial white Gaussian noise image by projecting it onto the defined statistical constraints across multiple pyramid levels, typically achieving convergence in about 50 iterations.

The findings establish that joint statistical relationships across scales and orientations are essential for successful visual texture representation. First, raw autocorrelation alone proves necessary but insufficient, capturing periodic regularity but failing to represent distinct local visual features. Second, magnitude correlations across scales and orientations successfully bind high-contrast elements into coherent contours, lines, and edges. Third, cross-scale phase statistics are vital for distinguishing lines from edges and correctly reproducing three-dimensional lighting and shadow gradients. Fourth, testing on hundreds of synthetic and photographic textures—including classic counterexamples that defeated earlier models—demonstrates that omitting any single parameter group results in noticeable synthesis failures, whereas the full 710-parameter set reproduces a vast variety of complex, pseudo-periodic, and natural textures.

These results provide a practical and theoretically grounded foundation for visual computing applications. Because the model operates on a fixed, compact parameter set rather than adapting custom filters or storing raw exemplars, it enables significant potential cost and bandwidth savings in image compression through synthetic detail generation. Furthermore, the flexible projection methodology allows straightforward integration into constrained reconstruction tasks, such as seamless texture extrapolation, pattern tiling, and defect restoration (hole filling). However, linear parameter interpolation between different textures results in patchy mixtures rather than smooth perceptual transitions, indicating that the texture parameter space is non-convex.

Organizations evaluating this approach should consider applying the sequential projection framework to image restoration, hole filling, denoising, and compression pipelines where compact representation is required. Future technical initiatives should focus on extending the model to full color channels, refining the parameter space to establish a true perceptual distance metric for seamless texture interpolation, and addressing remaining structural limitations. Specifically, caution is advised when processing textures with multi-oriented intersecting lines, variable line polarities, and closed contours, as the current parameter set does not explicitly track line termination endpoints or complex curvature.

New York University correlation analysis efficient algorithm wavelet parametric texture model complex wavelet coefficients joint statistics

LabForComputationalVision/textureSynth Portilla et al (2000).pdf

Paper: The Design and Use of Steerable Filters, W. Freeman et al. (1991). Reading this foundational work on steerable filters provides the essential mathematical basis for constructing the oriented multiresolution bases used in the source texture model.

Paper: Image Style Transfer Using Convolutional Neural Networks, Leon A. Gatys et al. (2016). This paper builds directly upon earlier statistical texture representations, adapting multi-layer neural network features to achieve high-quality artistic style transfer.

Cover for A Parametric Texture Model Based on Joint Statistics of Complex Wavelet Coefficients

Abstract

We present a universal statistical model for texture images in the context of an overcomplete complex wavelet transform. The model is parameterized by a set of statistics computed on pairs of coefficients corresponding to basis functions at adjacent spatial locations, orientations, and scales. We develop an efficient algorithm for synthesizing random images subject to these constraints, and we use this to test the perceptual validity of the model. In particular, we demonstrate the necessity of subgroups of the parameter set by showing examples of texture synthesis that fail when those parameters are removed from the set. We also demonstrate the power of our model by successfully synthesizing examples drawn from a diverse collection of artificial and natural textures.

Topics

correlation analysis efficient algorithm wavelet wavelet coefficients wavelet packet high resolution image

Keywords

parametric texture model complex wavelet coefficients joint statistics texture statistics visual texture visual discrimination

1 A Framework for Statistical Texture Modeling
1.1 Testing the Julesz Conjecture
1.2 Random Fields from Statistical Constraints
1.3 Sampling via Projection
1.4 Projection onto Constraint Surfaces
1.5 Gradient Projection
2 Texture Model
2.1 Local Linear Basis
2.2 Statistical Constraints
2.3 Summary of Statistical Constraints
3 Implementation and Results
3.1 Sequential Projection, Convergence, and Efficiency
3.2 Synthesis Results
3.3 Extensions
4 Discussion
A Adjustment of Constraints
A.1 Adjusting Marginal Statistics
A.2 Adjusting Subband Auto-correlation
A.3 Adjusting Subband Cross-Correlations
A.4 Adjusting Cross-Correlation with Fixed Subbands
References

Knowls

Knowl 1 — Universal Set of Joint Wavelet Statistical Texture Constraints
model/method
The visual texture model defines a universal, fixed (non-adapted) set of statistical constraints computed on an overcomplete complex multi-scale wavelet representation (the complex steerable pyramid) and raw image pixels. For a decomposition with $N$ pyramid levels (scales), $K$ orientation subbands, and an $M \times M$ spatial autocorrelation neighborhood window (where central non-redundant samples count $\frac{M^2+1}{2}$ ), the complete set consists of 710 parameters (for standard choices $N=4$ , $K=4$ , $M=7$ ):
1. Marginal Statistics (17 parameters for $N=4$ ):
  - Pixel moments and bounds: mean $\mu_1$ , variance $\mu_2$ , skewness $\eta$ , kurtosis $\kappa$ , minimum intensity, and maximum intensity (6 parameters).
  - Highpass residual band variance (1 parameter).
  - Reconstructed lowpass images at each scale: skewness and kurtosis ( $2(N + 1) = 10$ parameters for $N=4$ ).
2. Raw Coefficient Autocorrelation (125 parameters for $N=4, M=7$ ):
  - Central spatial autocorrelation samples of the partially reconstructed lowpass images at each scale and the residual lowpass band: $(N + 1) \frac{M^2+1}{2}$ parameters (with $M=7$ , $5 \times 25 = 125$ ). These capture periodicity, spatial frequency content, and long-range linear regularity.
3. Wavelet Magnitude Statistics (472 parameters for $N=4, K=4, M=7$ ):
  - Spatial autocorrelation of subband complex magnitudes: $N \cdot K \cdot \frac{M^2+1}{2}$ parameters ( $4 \times 4 \times 25 = 400$ ).
  - Cross-correlation of subband complex magnitudes across different orientations at the same scale: $N \frac{K(K-1)}{2}$ parameters ( $4 \times 6 = 24$ ).
  - Cross-correlation of subband complex magnitudes across adjacent scales (fine vs. upsampled coarse): $K^2 (N - 1)$ parameters ( $16 \times 3 = 48$ ).
4. Cross-Scale Phase Statistics (96 parameters for $N=4, K=4$ ):
  - Cross-correlation of the real part of fine-scale complex coefficients with both the real and imaginary components of the phase-doubled coarse-scale complex coefficients: $2 K^2 (N - 1)$ parameters ( $2 \times 16 \times 3 = 96$ ). These capture phase alignment, distinguishing lines from step edges, corners, and asymmetric illumination/shading gradients.
Knowl 2 — Cross-Scale Relative Phase Statistic via Phase Doubling
equation
To represent local structural features such as step edges, lines, and shading gradients, the relative phase between complex wavelet coefficients at adjacent scales is captured. Because the spatial frequency doubles between adjacent pyramid scales (an octave step), the local phase of the fine subband progresses twice as fast spatially as that of the coarse subband. The coarse-scale complex coefficient $c \in \mathbb{C}$ is phase-doubled and normalized in magnitude to yield $\hat{c} = \frac{c^2}{|c|}$ .

The cross-scale phase statistic between the fine coefficient $f = f_r + i f_i$ and the coarse coefficient $c$ at the same spatial location is defined as:

$\phi(f, c) = \frac{c^2 f^*}{|c|} = \hat{c} f^*$

Taking the expectation over the image lattice yields:

$\mathbb{E}(\hat{c} f^*) = \mathbb{E}(f_r \hat{c}_r + f_i \hat{c}_i) + i \mathbb{E}(f_r \hat{c}_i - f_i \hat{c}_r)$

Because $(\hat{c}_r, \hat{c}_i)$ and $(f_r, f_i)$ behave approximately as quadrature pairs (where $\mathbb{E}(f_r \hat{c}_r) \approx \mathbb{E}(f_i \hat{c}_i)$ and $\mathbb{E}(f_r \hat{c}_i) \approx -\mathbb{E}(f_i \hat{c}_r)$ ), the relative phase is captured by the two real expectations:

$\mathbb{E}(\hat{c} f^*) \approx 2\mathbb{E}(f_r \hat{c}_r) + 2i\mathbb{E}(f_r \hat{c}_i)$

where $\mathbb{E}(f_r \hat{c}_r)$ is sensitive to line-like features (even symmetry) and $\mathbb{E}(f_r \hat{c}_i)$ is sensitive to step-edge features (odd symmetry).
Knowl 3 — Iterative Synthesis-by-Analysis Texture Generation Algorithm
algorithm
The texture synthesis algorithm samples an image satisfying the statistical parameter set of a target texture by sequentially projecting an initial white noise image onto the individual constraint surfaces through repeated coarse-to-fine pyramid operations.
Input: Target parameter set estimated from sample image $x_t$ , pyramid scale count $N=4$ , orientation count $K=4$ , autocorrelation window $M=7$ , iteration count $I=50$
Output: Synthesized texture image $x$
Initialize $x$ as Gaussian white noise with mean and variance matching $x_t$
for iter = 1 to $I$ do
     $x_{prev} \leftarrow x$
    Decompose $x$ into complex steerable pyramid with $N$ levels, $K$ orientations, lowpass $L_N$ , and highpass $H_0$
    for scale $n = N$ down to 1 do
        if scale $n == N$ then
            Adjust cross-correlation of subband magnitudes across $K$ orientations
        else
            Upsample and interpolate subbands from scale $n+1$
            Adjust magnitude cross-correlation across $K$ orientations and with scale $n+1$
            Adjust cross-scale relative phase statistics between scale $n$ and scale $n+1$
        end if
        Adjust spatial autocorrelation of subband magnitudes at scale $n$
        Reconstruct lowpass image at scale $n-1$ from adjusted subbands and lowpass at scale $n$
        Adjust central spatial autocorrelation samples of reconstructed lowpass image
        Adjust skewness and kurtosis of reconstructed lowpass image
    end for
    Adjust highpass residual band $H_0$ to match target highpass variance
    Synthesize full image $x \leftarrow \text{reconstructed lowpass}_0 + H_0$
    Adjust pixel mean and variance of $x$
    Adjust pixel skewness of $x$
    Adjust pixel kurtosis of $x$
    Adjust pixel variance and mean of $x$
    Clip pixels of $x$ to lie within target range $[\min(x_t), \max(x_t)]$
     $x \leftarrow x_{prev} + 1.8 \times (x - x_{prev})$
end for
return $x$
The factor of $1.8$ acts as an over-relaxation coefficient to accelerate numerical convergence across the non-convex sequential projections.
Knowl 4 — Complex Steerable Pyramid Decomposition
model/method
The image representation is an overcomplete, shift-invariant, rotation-steerable tight frame. In the polar frequency domain $(r, \theta)$ , the filters are polar-separable.

The recursive lowpass scaling filter $L(r, \theta)$ is given by:

$L(r, \theta) = \begin{cases} 2\cos\left(\frac{\pi}{2}\log_2\left(\frac{4r}{\pi}\right)\right), & \frac{\pi}{4} < r < \frac{\pi}{2} \\ 2, & r \le \frac{\pi}{4} \\ 0, & r \ge \frac{\pi}{2} \end{cases}$

The oriented bandpass analytic filters $B_k(r, \theta)$ for orientations $k \in \{0, \dots, K-1\}$ are defined by $B_k(r, \theta) = H(r) G_k(\theta)$ , with radial component $H(r)$ and directional component $G_k(\theta)$ :

$H(r) = \begin{cases} \cos\left(\frac{\pi}{2}\log_2\left(\frac{2r}{\pi}\right)\right), & \frac{\pi}{4} < r < \frac{\pi}{2} \\ 1, & r \ge \frac{\pi}{2} \\ 0, & r \le \frac{\pi}{4} \end{cases}$

$G_k(\theta) = \begin{cases} \alpha_K \left[\cos\left(\theta - \frac{\pi k}{K}\right)\right]^{K-1}, & \left|\theta - \frac{\pi k}{K}\right| < \frac{\pi}{2} \\ 0, & \text{otherwise} \end{cases}$

where the normalization constant is:

$\alpha_K = 2^{K-1} \frac{(K-1)!}{\sqrt{K[2(K-1)]!}}$

The decomposition is initialized by separating the input into highpass and lowpass components via $H_0(r, \theta) = H(r/2, \theta)$ and $L_0(r, \theta) = \frac{1}{2} L(r/2, \theta)$ . Because $L(r, \theta)$ is bandlimited to $\pi/2$ , subsampling the lowpass branch by a factor of 2 along both spatial axes introduces no aliasing.
Knowl 5 — Subband Magnitude and Cross-Scale Correlation Adjustment
model/method
Adjustment of the $K \times K$ cross-correlation matrix $C = \frac{1}{|L|} X X^T$ of subbands $X$ to match a target correlation matrix $C_t = V_t D_t V_t^T$ (where $C = V D V^T$ is the eigendecomposition of current correlations) is performed by the linear transformation $X' = M X$ , with:

$M = V_t D_t^{1/2} O D^{-1/2} V^T$

where choosing the orthonormal matrix $O = V_t^T V$ minimizes distortion and ensures $M$ becomes symmetric as $C \to C_t$ .

When adjusting subbands $X$ simultaneously with fixed coarse-scale subbands $Y$ (with fixed cross-correlation matrix $E = \frac{1}{|L|} Y Y^T$ , current cross-correlation $B = \frac{1}{|L|} X Y^T$ , and target cross-correlation $B_t$ ), the linear update is $X' = M X + K_m Y$ . The matrix $K_m$ is determined by:

$K_m = (B_t - M B) E^{-1}$

and $M$ satisfies the symmetric quadratic constraint:

$M \left( C - B E^{-1} B^T \right) M^T = C_t - B_t E^{-1} B_t^T$

which is solved via the same eigenvalue factorization method as the unconstrained case.
Knowl 6 — Marginal Moment Adjustment via Gradient Projection
model/method
To enforce target skewness $\eta_t$ or kurtosis $\kappa_t$ on a zero-mean image vector $\vec{x} \in \mathbb{R}^{|L|}$ while minimizing distortion, the vector is adjusted along the gradient direction $\vec{x}' = \vec{x} + \lambda \vec{g}$ .
1. Skewness Adjustment: The gradient direction is $\vec{g} = \vec{x} \odot \vec{x} - \mu_2^{1/2}\eta \vec{x} - \mu_2$ (where $\odot$ denotes element-wise product). The projection step $\lambda$ satisfies:
$\eta_t = \frac{\sum_{n=0}^3 p_n \lambda^n}{\left[\sum_{n=0}^2 q_n \lambda^n\right]^{3/2}}$

Squaring both sides yields a 6th-degree polynomial in $\lambda$ :

$\sum_{n=0}^6 a_n \lambda^n = 0$

where the coefficients $a_n$ are algebraic combinations of sample moments $\mu_2, \dots, \mu_6$ and $\eta_t$ . The solution chosen is the smallest-amplitude root for which $\eta(\lambda)$ has a positive derivative at $\lambda = 0$ .
1. Kurtosis Adjustment: The gradient direction is $\vec{g} = \vec{x} \odot \vec{x} \odot \vec{x} - \alpha \vec{x} - \mu_3$ (where $\alpha = \mu_4 / \mu_2$ ). The parameter $\lambda$ is the root of the 4th-degree polynomial resulting from:
$\kappa_t = \frac{\sum_{n=0}^4 p_n \lambda^n}{\left[\sum_{n=0}^2 q_n \lambda^n\right]^2}$
Knowl 7 — Subband Autocorrelation Adjustment via Zero-Phase Filtering
model/method
Central non-redundant samples $\mathcal{N}$ of the circular autocorrelation $A(n, m) = \frac{1}{|L|} \sum_{i,j} x(i, j) x(|i+n|_N, |j+m|_M)$ are matched to target values $A_t(n, m)$ by applying a zero-phase moving average filter $h_\lambda(i, j)$ with support on a symmetrized neighborhood of $\mathcal{N}$ :

$x'(i, j) = x(i, j) \circledast h_\lambda(i, j)$

An even-symmetric kernel $A_h(n, m)$ with $2|\mathcal{N}| - 1$ non-zero samples is obtained by solving the linear convolution system on $\mathcal{N}$ :

$A_t(n, m) = A(n, m) * A_h(n, m)$

The zero-phase correction filter $h_\lambda(i, j)$ is then computed by taking the inverse discrete Fourier transform of the square root of the absolute magnitude of the DFT of $A_h$ :

$h_\lambda(i, j) \approx \text{DFT}^{-1}\left\{ \sqrt{|\text{DFT}\{A_h(i, j)\}|} \right\}$

This approximate projection achieves spectral accuracy exceeding 70 dB SNR within a few iterations.
Knowl 8 — Definition of Practical Ergodicity on Finite Lattices
definition
A two-dimensional homogeneous random field $X$ on a finite lattice $L \subset \mathbb{Z}^2$ has the property of practical ergodicity with respect to a constraint function $\phi: \mathbb{R}^{|L|} \to \mathbb{R}$ , tolerance $\epsilon > 0$ , and probability $p \in (0, 1]$ , if and only if the spatial average over all circular translations of a single sample image $x(n, m)$ drawn from $X$ , denoted $\overline{\phi(x(n, m))}$ , satisfies:

$P_X\left( \left| \overline{\phi(x(n, m))} - \mathbb{E}(\phi(X)) \right| < \epsilon \right) \ge p$

where the spatial translate average is:

$\overline{\phi(x(n, m))} = \frac{1}{|L|} \sum_{(i,j) \in L} \phi\left(x\left(|n+i|_N, |m+j|_M\right)\right)$

with $|\cdot|_N$ denoting modulo $N$ arithmetic.
Knowl 9 — Non-Convexity of Homogeneous Texture Parameter Space
empirical result
Linear combinations (averaging) of the statistical parameter vectors of two distinct homogeneous textures do not yield a new perceptually homogeneous texture lying intermediate between them. Instead, synthesizing an image from the averaged parameter vector produces a spatially segregated, patchwise mixture consisting of distinct regions of the two initial textures.

This demonstrates that while the space of all image statistics possesses asymptotic convexity (as a linear mixture corresponds to an image split into regions of each texture whose boundary effects vanish as $|L| \to \infty$ ), the subset of parameter vectors that correspond specifically to spatially homogeneous random fields is non-convex.
Knowl 10 — Structural Failure Modes and Limitations of the Texture Model
limitation
The parametric texture model exhibits systematic structural synthesis failures on specific classes of visual patterns:
1. Curved vs. Straight Contours: While the model accurately synthesizes straight contours of a single orientation, textures containing straight line segments at multiple simultaneous orientations cause the model to generate spurious curved contours.
2. Closed Contours and Endpoints: Patterns requiring closed loops (such as randomly oriented ellipses or packed beans) fail to close properly because the parameter set lacks explicit representations for contour terminations (endpoints).
3. Mixed-Polarity Boundaries: For textures composed of flat polygonal regions with sharp step edges where polarity alternates randomly (equal likelihood of positive and negative intensity steps), local subband phase averages out to zero across the image. Consequently, the phase statistic cannot distinguish step edges from line profiles, causing step edges to degrade into bright/dark line artifacts.
4. Non-adapted Filter Set vs. Filter Selection: The model uses a fixed set of orientation bands ( $K=4$ ), which can be less compact than adaptive filter-selection algorithms for textures with very specific non-cardinal dominant orientations.

Coverage note — None was omitted; all key theoretical formulations, representation details, statistical descriptors, projection algorithms, empirical findings, and stated limitations are fully covered.

References

1.C H Anderson and W D Langer. Statistical models of image texture. Technical report, Washington U. Medical School, 1997. Available at ftp://shifter.wustl.edu/pub/.
2.A J Bell and T J Sejnowski. The 'independent components' of natural scenes are edge filters. Vision Research, 37(23):3327–3338, 1997.
3.J R Bergen and E H Adelson. Visual texture segmentation based on energy measures. J. Opt. Soc. Am. A, 3:99, 1986.
4.J R Bergen and M S Landy. Computational modeling of visual texture segregation. In M S Landy and J A Movshon, editors, Computational Models of Visual Processing, pages 253–271. MIT Press, Cambridge, MA, 1991.
5.C A Bouman and M Shapiro. A multiscale random field model for Bayesian image segmentation. IEEE Trans. Image Proc., 3(2), 1994.
6.A C Bovik, M Clark, and W S Geisler. Multichannel texture analysis using localized spatial filters. IEEE Pat. Anal. Mach. Intell., 12(1):55–73, 1990.
7.A C Bovik, M Clark, and W S Geisler. Localized measurements of emergent image frequencies by Gabor wavelets. IEEE Pat. Anal. Mach. Intell., 38:691–712, 1992.
8.P Brodatz. Textures: A Photographic Album for Artists and Designers. Dover, New York, 1966.
9.R W Buccigrossi and E P Simoncelli. Image compression via joint statistical characterization in the wavelet domain. IEEE Trans Image Proc, 8(12):1688–1701, December 1999.
10.J A Cadzow, D M Wilkes, R A Peters II, and X Li. Image texture synthesis–by–analysis using moving–average models. IEEE Trans. Aerospace and Elec. Sys., 29(4):1110–1122, 1993.
11.T M Caelli and B Julesz. Experiments in the visual perception of texture. Biol. Cybernetics, 28:167–175, 1978.
12.D Cano and T H Minh. Texture synthesis using hierarchical linear transforms. Signal Processing, 15:131–148, 1988.
13.P C Chen and T Pavlidis. Segmentation by texture using correlation. IEEE Pat. Anal. Mach. Intell., 5(1):64–69, 1983.
14.G Cross and A Jain. Markov random field texture models. IEEE Trans PAMI, 5:25–39, 1983.
15.I Daubechies. Orthonormal bases of compactly supported wavelets. Comm. on Pure and Appl. Math., 41:909–996, 1988.
16.J G Daugman. Complete discrete 2–D Gabor transforms by neural networks for image analysis and compression. IEEE Trans. Acoust. Speech Signal Proc., 36(7):1169–1179, 1988.
17.J G Daugman and D M Kammen. Pure orientation filtering: A scale-invariant image-processing tool for perception research and data compression. Behavior Research Methods, Instruments, & Computers, 18(6):559–564, 1986.
18.John G. Daugman. Entropy reduction and decorrelation in visual coding by oriented neural receptive fields. IEEE Trans. Biomedical Engineering, 36(1):107–114, 1989.
19.J De Bonet and P Viola. A non-parametric multi-scale statistical model for natural images. In Adv in Neural Info Processing, volume 9. MIT Press, December 1997.
20.H Derin and H Elliott. Modeling and segmentation of noisy and textured images using Gibbs random fields. IEEE Pat. Anal. Mach. Intell., 9(1):39–55, 1987.
21.P Diaconis and D Freedman. On the statistics of vision: the julesz conjecture. J. Math. Psychol, 24:112–118, 1981.
22.A A Efros and T K Leung. Texture synthesis by non-parameteric sampling. In Proc. Int'l Conference on Computer Vision, Corfu, 1999.
23.O D Faugeras and W K Pratt. Decorrelation methods of texture feature extraction. IEEE Pat. Anal. Mach. Intell., 2(4):323–332, 1980.
24.D J Field. Relations between the statistics of natural images and the response properties of cortical cells. J. Opt. Soc. Am. A, 4(12):2379–2394, 1987.
25.J M Francos, A Z Meiri, and B Porat. A unified texture model based on a 2-D Wold-like decomposition. IEEE Trans. Signal Proc., 41(8):2665–2678, 1993.
26.A Gagalowicz. A new method for texture fields synthesis: Some applications to the study of human vision. IEEE Pat. Anal. Mach. Intell., 3(5):520–533, 1981.
27.S Geman and D Geman. Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Pat. Anal. Mach. Intell., 6:721–741, 1984.
28.N Graham. Visual pattern analyzers. Oxford University Press, New York, 1989.
29.M Hassner and J Sklansky. The use of Markov random fields as models of texture. Comp. Graphics Image Proc., 12:357–370, 1980.
30.D Heeger and J Bergen. Pyramid-based texture analysis/synthesis. In Proc. ACM SIGGRAPH, August 1995.
31.Aaron Hertzmann. Painterly rendering with curved brush strokes of multiple sizes. In Proc. ACM SigGraph, pages 453–460, 1998.
32.A N Hirani and T Totsuka. Combining frequency and spatial domain information for fast interactive image noise removal. In ACM SIGGRAPH, pages 269–276, 1996.
33.E T Jaynes. Information theory and statistical mechanics. Phys. Rev., 106:620–630, 1957.
34.E T Jaynes. Where do we stand on maximum entropy? In R. D. Levine and M. Tribus, editors, The Maximal Entropy Formalism. MIT Press, Cambridge, MA, 1978.
35.B Julesz. Visual pattern discrimination. IRE Trans Info Theory, IT-8, 1962.
36.B Julesz, E N Gilbert, L A Shepp, and H L Frisch. Inability of humans to discriminate between visual textures that agree in second-order statistics – revisited. Perception, 2:391–405, 1973.
37.B Julesz, E N Gilbert, and J D Victor. Visual discrimination of textures with identical third-order statistics. Biol. Cybernetics, 31:137–140, 1978.
38.D Kersten. Predictability and redundancy of natural images. J. Opt. Soc. Am. A, 4(12):2395–2400, 1987.
39.H Knutsson and G H Granlund. Texture analysis using two-dimensional quadrature filters. In Workshop on Computer Architecture for Pattern Analysis and Image Database Management, pages 206–213. IEEE Computer Society, 1983.
40.J Malik and P Perona. Preattentive texture discrimination with early vision mechanisms. J. Opt. Soc. Am. A, 7:923–932, May 1990.
41.S G Mallat. A theory for multiresolution signal decomposition: The wavelet representation. IEEE Pat. Anal. Mach. Intell., 11:674–693, July 1989.
42.R Manduchi and J Portilla. Independent component analysis of textures. In Proc. Int'l Conference on Computer Vision, Corfu, 1999.
43.B A Olshausen and D J Field. Natural image statistics and efficient coding. Network: Computation in Neural Systems, 7:333–339, 1996.
44.P Perona and J Malik. Detecting and localizing edges composed of steps, peaks and roofs. In Proc. 3rd Intl. Conf. Computer Vision, Osaka, Japan, 1990.
45.K Popat and R W Picard. Novel cluster-based probability model for texture synthesis, classification, and compression. In Proc SPIE Vis Comm, Cambridge, MA, 1993.
46.K Popat and R W Picard. Cluster-based probability model and its application to image and texture processing. IEEE Trans Im Proc, 6(2):268–284, 1997.
47.M Porat and Y Y Zeevi. Localized texture processing in vision: Analysis and synthesis in Gaborian space. IEEE Trans Biomedical Eng., 36(1):115–129, 1989.
48.J Portilla, R Navarro, O Nestares, and A Tabernero. Texture synthesis-by-analysis based on a multiscale early-vision model. Optical Engineering, 35(8):2403–2417, 1996.
49.J Portilla and E Simoncelli. Texture modeling and synthesis using joint statistics of complex wavelet coefficients. In IEEE Workshop on Statistical and Computational Theories of Vision, Fort Collins, CO, 1999. Available at http://www.cis.ohio-state.edu/~szhu/SCTV99.html.
50.J Portilla and E Simoncelli. Image denoising via adjustment of wavelet coefficient magnitude correlation. In Seventh IEEE Int'l Conf on Image Proc, Vancouver, September 10-13 2000. IEEE Computer Society.
51.W K Pratt, O D Faugeras, and A Gagalowicz. Visual discrimination of stochastic texture fields. IEEE Trans. on Systems Man and Cybernetics, 8:796–804, Nov 1978.
52.T R Reed and H Wechsler. Segmentation of textured images and Gestalt organization using spatial/spatial-frequency representations. IEEE Pat. Anal. Mach. Intell., 12(1):1–12, 1990.
53.D L Ruderman and W Bialek. Statistics of natural images: Scaling in the woods. Phys. Rev. Letters, 73(6):814–817, 1994.
54.E Simoncelli and J Portilla. Texture characterization via joint statistics of wavelet coefficient magnitudes. In Fifth IEEE Int'l Conf on Image Proc, volume I, Chicago, October 4-7 1998. IEEE Computer Society.
55.E P Simoncelli. Statistical models for images: Compression, restoration and synthesis. In 31st Asilomar Conf on Signals, Systems and Computers, pages 673–678, Pacific Grove, CA, November 1997. IEEE Computer Society. Available from http://www.cns.nyu.edu/~eero/publications.html.
56.E P Simoncelli and W T Freeman. The steerable pyramid: A flexible architecture for multi-scale derivative computation. In Second Int'l Conf on Image Proc, volume III, pages 444–447, Washington, DC, October 1995. IEEE Sig Proc Society.
57.E P Simoncelli, W T Freeman, E H Adelson, and D J Heeger. Shiftable multi-scale transforms. IEEE Trans Information Theory, 38(2):587–607, March 1992. Special Issue on Wavelets.
58.A Tabernero, J Portilla, and R Navarro. Duality of log-polar image representations in the space and the spatial-frequency domains. IEEE Trans. on Signal Processing, 47(9):2469–79, September 1999.
59.M R Turner. Texture discrimination by Gabor functions. Biol. Cybern., 55:71–82, 1986.
60.J D Victor. Images, statistics and textures: Implications of triple correlation uniqueness for texture statistics and the Julesz conjecture: Comment. J. Opt. Soc. Am. A, 11(5):1680–1684, May 1994.
61.VisTex: an online collection of visual textures. MIT Media Laboratory, 1995. Available from http://www-white.media.mit.edu/vismod/imagery/VisionTexture/vistex.html.
62.M J Wainwright and E P Simoncelli. Scale mixtures of Gaussians and the statistics of natural images. In S. A. Solla, T. K. Leen, and K.-R. Müller, editors, Adv. Neural Information Processing Systems, volume 12, pages 855–861, Cambridge, MA, May 2000. MIT Press. Presented at Neural Information Processing Systems, Dec 1999.
63.A. B. Watson. Efficiency of a model human image code. J. Opt. Soc. Am. A, 12:2401–2417, 1987.
64.J I Yellott. Images, statistics and textures: Implications of triple correlation uniqueness for texture statistics and the Julesz conjecture. J. Opt. Soc. Am. A, 10(5):777–793, May 1993.
65.D C Youla. Generalized image restoration by the method of alternating orthogonal projections. IEEE Trans. Circuits and Systems, 25:694–702, 1978.
66.D C Youla and H Webb. Image restoration by the method of convex projections. IEEE Trans. Med. Imaging, 1:81–101, 1982.
67.C Zetzsche, B Wegmann, and E Barth. Nonlinear aspects of primary vision: Entropy reduction beyond decorrelation. In Int'l Symposium, Society for Information Display, volume XXIV, pages 933–936, 1993.
68.S Zhu, Y Wu, and D Mumford. Filters, random fields and maximum entropy (FRAME) – towards the unified theory for texture modeling. In IEEE Conf Computer Vision and Pattern Recognition, June 1996.
69.S C Zhu, X Liu, and Y N Wu. Exploring the Julesz ensemble by efficient Markov chain monte carlo. In IEEE Workshop on Statistical and Computational Theories of Vision, Fort Collins, CO, 1999. Available at http://www.cis.ohio-state.edu/~szhu/SCTV99.html.
70.S C Zhu, Y N Wu, and D Mumford. Minimax entropy principle and its application to texture modeling. In Neural Computation, volume 9, pages 1627–1660, 1997.

Citation

MLA

Portilla, J., and E. P. Simoncelli. “A Parametric Texture Model Based on Joint Statistics of Complex Wavelet Coefficients”. International Journal of Computer Vision, vol. 40, no. 1, 2000, pp. 49–70, https://doi.org/10.1023/A:1026553619983.

APA

Portilla, J., & Simoncelli, E. P. (2000). A Parametric Texture Model Based on Joint Statistics of Complex Wavelet Coefficients. International Journal of Computer Vision, 40(1), 49–70. https://doi.org/10.1023/A:1026553619983

Chicago

Portilla, J., and E. P. Simoncelli. 2000. “A Parametric Texture Model Based on Joint Statistics of Complex Wavelet Coefficients”. International Journal of Computer Vision 40 (1): 49–70. https://doi.org/10.1023/A:1026553619983.

Harvard

Portilla, J. and Simoncelli, E.P. (2000) “A Parametric Texture Model Based on Joint Statistics of Complex Wavelet Coefficients”, International Journal of Computer Vision, 40(1), pp. 49–70. Available at: https://doi.org/10.1023/A:1026553619983.

Vancouver

1. Portilla J, Simoncelli EP (2000) A Parametric Texture Model Based on Joint Statistics of Complex Wavelet Coefficients. International Journal of Computer Vision 40:49–70

BibTeX

@article{Portilla_2000, title={A Parametric Texture Model Based on Joint Statistics of Complex Wavelet Coefficients}, volume={40}, ISSN={1573-1405}, url={http://dx.doi.org/10.1023/A:1026553619983}, DOI={10.1023/a:1026553619983}, number={1}, journal={International Journal of Computer Vision}, publisher={Springer Science and Business Media LLC}, author={Portilla, Javier and Simoncelli, Eero P.}, year={2000}, month=Oct, pages={49–70} }

Metadata:Crossref

Source Code

This paper has an official code repository available. Click below to access the source code.

View Repository

Access the Paper

This paper is available from its original source. Click below to access the PDF.

Open PDF

A Parametric Texture Model Based on Joint Statistics of Complex Wavelet Coefficients

Abstract

Topics

Keywords

Table of Contents

Knowls

Knowl 1 — Universal Set of Joint Wavelet Statistical Texture Constraints

Knowl 2 — Cross-Scale Relative Phase Statistic via Phase Doubling

Knowl 3 — Iterative Synthesis-by-Analysis Texture Generation Algorithm

Knowl 4 — Complex Steerable Pyramid Decomposition

Knowl 5 — Subband Magnitude and Cross-Scale Correlation Adjustment

Knowl 6 — Marginal Moment Adjustment via Gradient Projection

Knowl 7 — Subband Autocorrelation Adjustment via Zero-Phase Filtering

Knowl 8 — Definition of Practical Ergodicity on Finite Lattices

Knowl 9 — Non-Convexity of Homogeneous Texture Parameter Space

Knowl 10 — Structural Failure Modes and Limitations of the Texture Model

References

Citation

BibTeX

Source Code

Access the Paper