COMPLEXITY–REGULARIZED DENOISING OF ... - Semantic Scholar

COMPLEXITY–REGULARIZED DENOISING OF POISSON–CORRUPTED DATA Juan Liu and Pierre Moulin U. of Illinois, Beckman Inst. and ECE Dept 405 N. Mathews Ave., Urbana, IL 61801 Email: {j-liuf, moulin}@ifp.uiuc.edu ABSTRACT In this paper, we apply the complexity–regularization principle to Poisson imaging. We formulate a natural distortion measure in image space, and present a connection between complexity–regularized estimation and rate–distortion theory. For computational tractability, we apply constrained coders such as JPEG or SPIHT to solve the optimization problem approximately. Also, we design a simple predictive coder which lends itself well to our optimization problem.

2. RELATIVE ENTROPY LOSS AND INDEX OF RESOLVABILITY

1. INTRODUCTION In a number of imaging applications where images are acquired by photon–counting devices, such as in MRI or astronomical imaging applications, data are corrupted by Poisson noise. Given the true image f , the observations y obey the conditional probability N p(y|f ) = ΠN i=1 pi (yi |fi ) = Πi=1

fiyi −fi e , yi !

(1)

where N is the number of data samples in the image. Examples of image estimation from Poisson–contaminated data can be found in [1, 2, 3]. In this paper, we use a theoretically motivated complexity– regularized approach to estimate the unknown image (Poisson intensity field) f from the observations y. Complexity is used as the regularization penalty, following the view that typical images have low complexity in a data compression sense. Complexity–regularized estimators take the general form fˆ(y) = arg min [− ln p(y|f ) + µL(f )] , f ∈Γ

penalty term penalizes unlikely estimates in a more flexible way. Moreover, it admits Rissanen’s minimum description length (MDL) [4] principle as a special case with µ = 1. Theoretical justification and practical applications of complexity regularization can be found in our previous work [5, 6, 7]. We also note related work by Nowak and Figueiredo [2] which uses the MDL principle to estimate piecewise homogeneous Poisson fields.

(2)

where Γ is a discrete set of candidate images, informally referred to as a codebook, and µ is a regularization parameter. Complexity is measured by a codelength L(f ) associated with each f ∈ Γ. With a properly designed codebook, unlikely images are assigned long codewords and hence are strongly penalized by the criterion (2). Compared to the more standard L2 , L1 and Besov penalties, the complexity

To characterize the performance of the estimator, we use relative entropy loss d(f ∗ , f ) =

1 D(p(y|f ∗ )||p(y|f )) N

between the true image f ∗ and the estimate f , where D(p||q) = R p(y) ln p(y) q(y) dy is the relative entropy between densities Y p and q [8]. Relative–entropy loss is the natural choice to characterize the performance of penalized likelihood estimators. It inherits the basic properties of relative entropy. In particular, d(f ∗ , f ) is nonnegative but is not symmetric, hence is not a metric. Relative-entropy loss is also invariant under application of an invertible transform to the observations. In the denoising of additive white Gaussian noise, d(f ∗ , f ) is simply the mean–squared error scaled by a constant [6, 7]. For our Poisson model (1),  N  1 X ∗ fi∗ ∗ ∗ − fi + fi . f ln (3) d(f , f ) = N i=1 i fi In this case, d(f ∗ , f ) is convex both in f ∗ and f . We demonstrate in [6] that a complexity–regularized estimator guarantees a certain level of performance in terms of an index of resolvability that is defined as   L(f ) Rµ (f ∗ ) = min d(f ∗ , f ) + µ . (4) f ∈Γ N The index of resolvability (4) describes how well the image f ∗ can be approximated in the relative–entropy sense

by a moderately-complex element of the codebook Γ. In [6], we show that the expected relative–entropy loss is upper bounded by a linear function of Rµ (f ∗ ). The criterion (4) can be thought of as a rate–distortion criterion, in which distortion is measured by d(f ∗ , f ) and rate by L(f ). We are interested in computing both the complexityregularized estimator fˆ in (2) and the index of resolvability (4). Let f˜ be the image that achieves best resolvability. For our Poisson imaging model, both optimization problems can be solved using the same algorithm. To see this, observe that the negative log-likelihood can be written in the form X [−yi ln fi + ln(yi !) + fi ] − ln p(y|f ) = i

yi − yi + fi ] fi

=

X

[yi ln

+

X

[−yi ln yi + yi − ln(yi !)]

i

i

4. PREDICTIVE CODERS

where the first term in the right-hand side is N times the relative entropy loss d(y, f ), and the second term is independent of f . Hence fˆ = arg minf ∈Γ E(y, f ), where E(y, f ) = d(y, f ) + µ

in. One may think of q as a sieve parameter [10]: the covering of F is coarse for large values of q, but gradually improves as q → 0. The minimization over f ∈ Γ can be written as a double minimization: minq minf ∈Γq . For each value of q, the inner minimization produces fˆq represented by a pair (L(fˆq ), d(y, fˆq )) in rate-distortion coordinates. The collection of such points defines the operational rate-distortion curve of the coder, for the given input y. The outer minimization (over q) amounts to finding the point at slope −1/µ on the operational rate-distortion curve. One can use practical coders such as a JPEG coder [9] and the state-of-the-art SPIHT wavelet coder [11] to find suboptimal solutions to the optimization problems (2) and (4). The suboptimality is due to the fact that these coders are designed for the l 2 norm distortion, not for the relative– entropy loss (3).

L(f ) . N

(5)

This criterion is of the same mathematical form as (4). The function that achieves best resolvability is given by f˜ = arg minf ∈Γ E(f ∗ , f ). The solution to such minimization problems may be viewed as the solution to an operational rate-distortion optimization problem, with µ in (5) playing the role of a Lagrange multiplier. Given a codebook Γ, fˆ and f˜ are obtained by optimally compressing an input image (respectively y and f ∗ ). The optimization problem for an arbitrarily complex codebook Γ is a high-dimensional discrete optimization problem (the dimensionality is N , and the number of codewords is usually even larger) whose solution is beyond reach. To make complexity-regularized imaging practically useful, one needs to design efficient approximate optimization algorithms. This can be done for some structured codebooks, as discussed below. 3. CONSTRAINED CODERS Our approach is to constrain the coder to have one single free parameter (or perhaps a few parameters) that can be adjusted to provide the desired tradeoff between rate L(f ) and distortion d(y, f ). The idea is analogous to the design of JPEG coders [9], where the so-called quality factor determines the scaling of the default quantization matrix applied to the discrete-cosine-transform coefficients. Formally, we consider a family of codebooks Γq indexed by a quantizer step size q, and define the actual codebook (supercodebook) as Γ = ∪q Γq ∪ {q}. We denote as F the space f lives

While predictive coding techniques have long been displaced by transform-based coding techniques as the method of choice for practical high-performance image compression, fairly good compression performance can still be obtained by constructing good prediction rules and suitably quantizing and encoding prediction errors. In this paper, we investigate predictive coding as it lends itself quite well to optimization of our particular rate-distortion cost function. We consider a predictive coder and decoder operating in a closed–loop mode [12] shown in Fig. 1. Here raster scanning is used, x is the input image to be compressed, and x˜ is the reconstructed (decompressed) image. The quantizer Q for prediction errors is uniform with step size q. In Fig. 1, G is the prediction filter, x ˆ = G˜ x is the recursively predicted image, e = x − x ˆ is the prediction error, and e˜ = Q(e) = x ˜−x ˆ is the quantized prediction error image. The image e˜ is encoded using a variable-length code. Two simple choices of the prediction filter G are studied in our experiment. The first one uses the value of the left neighbor of the current pixel as the predicted value. The second prediction filter is a 3–point median filter whose support set is made of the left, upper, and upper-right neighbors of the current pixel. This is the same filter that is used for predicting the value of motion vectors in the H.263 video coding standard [13]. For lossless encoding of the quantized prediction errors e˜ we use Shannon codes [8], which have ideal length − log p(˜ ei ). The distribution of the quantized prediction errors is modeled as a discretized exponential distribution P (˜ ei = jq) = Ae−α|j| , where j ∈ Z, and the parameters A and α can be estimated using maximum likelihood estimation from the clean image f ∗ . This defines the codebook Γq . With such a predictive coder, it is possible to recur-

x

+

e

~ e

Q

^x

G

+

~ x

+

~ e

~ x ^x

G

(b)

(a)

Fig. 1. A lossy predictive coding scheme: (a) encoder; (b) decoder. sively optimize the rate–distortion cost function E(x, x˜) = x) ˜, where x is the input imd(x, x˜) + µ L(˜ N with respect to x age, and x˜ is the compressed image. There is a one-to-one mapping between e˜ and x ˜, as implemented by the decoder. The admissible set of quantized prediction errors e˜i lies in a discrete grid {jq, j ∈ Z}. The original problem of finding the best x ˜i can thus be reduced to a one–dimensional optimization over a discrete grid. Further computational savings in the search for the optimal j can be obtained by restricting the search to a small subset of quantization levels in the vicinity of the prediction error at that location. In our experiment, we use four levels, as indicated in Fig. 2. The optimization algorithm is simple but greedy. At each pixel location i, the algorithm minimizes the contribution of the current pixel, without evaluating the consequences of this choice on future pixel values to be encoded.

e= x i - x^ i

q

e

Fig. 2. Greedy algorithm for quantization of prediction errors.

5. NUMERICAL RESULTS We have implemented the predictive coding algorithms described above and apply the denoising methods to Poisson– corrupted images, including a simple CAD tank image T62 of size 128×128 and a Moon surface image of size 256×256. Table 1 reports the estimation results. We used the MDL estimator, corresponding to µ = 1. The denoised T62 images are displayed in Fig. 3c,d. The two predictive coders studied respectively use a left-neighbor predictor and a twodimensional median-filter predictor, as described in Sec. 4. We also present results using a JPEG coder [9] and the SPIHT coder [11] in Tables 1 and in Fig. 3e,f. The conventional

quantization strategies used by these coders lead to suboptimal solutions over the set Γ. Although the performance bounds derived in [6] do not apply to the MDL case, we still found a good agreement between estimation performance as measured by the relativeentropy loss d(f, fˆ), and compression performance as measured by the index of resolvability Rµ (f ∗ ). Similar agreement was found on other images we tested; these results are also consistent with the more comprehensive study we undertook for the denoising of additive white Gaussian noise [5, 7]. Observe that the predictive coders outperform the JPEG and SPIHT coders on the T62 image both in terms of denoising and compression capabilities under our rate-distortion cost function. The choice of a median-filter predictor is particularly well adapted to the T62 image. In particular, the use of JPEG and SPIHT coders results in characteristic blocking and ringing artifacts, respectively. However, for the more complex Moon image, the JPEG and SPIHT coders produce somewhat better results. We reemphasize that the results quoted for the predictive coders are obtained using greedy optimization techniques, and that a number of simplifications have been made regarding the choice of the rate-distortion optimization algorithm and the predictors and lossless codes used. In contrast, the JPEG and SPIHT coders use vastly more sophisticated compression techniques, but these techniques are not aimed at optimizing the particular rate-distortion cost function of interest here. 6. REFERENCES [1] T. J. Hebert and R. Leahy, “Statistic–based MAP image– reconstruction from Poisson data using Gibbs priors,” IEEE Trans. on Signal Process., vol. 40, pp. 2290–2303, Sept. 1992. [2] R. D. Nowak and M. A. T. Figueiredo, “Unsupervised progressive parsing of Poisson fields using minimum description length criteria,” in Proc. of ICIP’99, (Kobe, Japan), pp. II. 26–30, Oct. 1999. [3] K. E. Timmermann and R. D. Nowak, “Multiscale modeling and estimation of Poisson processes with application to photon-limited imaging,” IEEE Trans. on Info. Theory, vol. 45, pp. 846–862, Apr. 1999. [4] J. Rissanen, Stochastic Complexity in Statistical Inquiry. Singapore: World Scientific, 1989.

ˆ (in bpp) L(f) ˆ d(f ∗ , f) Rµ (f ∗ )

OptLeft 0.34 0.22 0.26

T62 OptMedian 0.47 0.09 0.36

JPEG 0.61 0.49 0.80

SPIHT 0.72 0.31 0.67

OptLeft 1.05 0.59 0.81

Moon surface OptMedian JPEG 0.77 0.30 0.46 0.38 0.66 0.42

SPIHT 0.24 0.30 0.35

Table 1. Estimation results for T62–tank CAD image and Moon surface image. Each of the columns corresponds to a different coder. OptLeft corresponds to a predictive coder using a left predictor, and OptMedian corresponds to a predictive coder using a 3–point (left, upper and upper–right) median predictor.

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 3. Denoising results for T62–tank image: (a) original image f ∗ ; (b) data y, corrupted by Poisson noise; (c) estimate obtained

using a predictive coder with a left neighbor predictor; (d) using a predictive coder with a median predictor; (e) using a JPEG coder; (f) using a SPIHT coder. [5] J. Liu and P. Moulin, “Complexity–regularized image denoising,” in Proc. of ICIP’97, (Santa Barbara, CA), pp. II. 370–373, Oct. 1997. [6] P. Moulin and J. Liu, “Statistical imaging and complexity regularization,” IEEE Trans. on Info. Theory, vol. 46, pp. 1762–1777, Aug. 2000. [7] J. Liu and P. Moulin, “Complexity-regularized image denoising,” IEEE Trans. on Image Processing, submitted, 1999. [8] T. M. Cover and J. A. Thomas, Elements of Information Theory. New York, NY: John Wiley and Sons, Inc., 1991. [9] “JPEG software codec.” Portable Research Video Group, Stanford University, 1997. Available via anonymous ftp from havefun.stanford.edu:pub/jpeg/JPEGv1.1.tar.Z.

[10] U. Grenander, Abstract Inference. New York: Wiley, 1981. [11] A. Said and W. A. Pearlman, “A new, fast, and efficient image codec based on set partitioning in hierarchical trees,” IEEE Trans. Circuits Syst. Video Technol., vol. 6, pp. 243–250, June 1996. [12] A. Gersho and R. M. Gray, Vector Quantization and Signal Compression. Norwell: Kluwer Academic Press, 1992. [13] “Video coding for low bitrate communication.” ITU-T recommendation H.263 version 2, Feb. 1998.