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Projection-Based Spatially-Adaptive Reconstruction of Block-Transform Compressed Images Yongyi Yang, Nikolas Galatsanos , and Aggelos K. Katsaggelos
Abstract | At the present time block-transform coding is probably the most popular approach for image compression. For this approach the compressed images are decoded using only the transmitted transform data. In this paper, we formulate image decoding as an image recovery problem. According to this approach, the decoded image is reconstructed using not only the transmitted data, but in addition, the prior knowledge that images before compression do not display between-blocks discontinuities. A spatially-adaptive image recovery algorithm is proposed which is based on the theory of projections onto convex sets. This algorithm apart from the data constraint set, uses another new constraint set that enforces between-block smoothness. The novelty of this set is that it captures both the local statistical properties of the image and the human perceptual characteristics. A simpli ed spatially-adaptive recovery algorithm is also proposed and the analysis of its computational complexity is also presented. Numerical experiments are shown that demonstrate that the proposed algorithms work better than both the JPEG deblocking recommendation and our previous projection-based image decoding approach. Keywords | Image compression, blocking artifact, image recovery, artifact visibility, projections onto convex sets.
Block-transform coding is by far the most popular approach for image compression. Evidence of this fact is that JPEG the still image compression standard recommends the use of the block discrete cosine transform (BDCT) , . According to this approach the image in the decoder is reconstructed by taking the inverse transform of the transmitted transform data. In order to reduce the bit-rate and achieve compression before transmission the transform data is quantized. Thus, at high compression ratios the compressed images display coding artifacts. More speci cally, when block-transforms are used the most noticeable artifact is the \blocking artifact". This artifact manifests itself as an arti cial discontinuity between adjacent blocks and is a direct result of the independent processing of the blocks which does not take into account the between-block pixel correlations. Decoding the compressed image by taking the inverse transform of the transmitted quantized data is a very simplistic approach which has been dictated by decoder comY. Yang and N. Galatsanos are with the Department of Electrical and Computer Engineering at the Illinois Institute of Technology, Chicago, IL 60616. A. Katsaggelos is with the Department of Electrical Engineering and Computer Science at Northwestern University, Evanston, IL 60208 This work was supported by the Ping Chun memorial fellowship at the the Department of Electrical and Computer Engineering at the Illinois Institute of Technology.
plexity constraints. However, due to new advances in VLSI technology and the increasing demands for bandwidth these constraints are gradually being relaxed. Therefore, given the current trends of technology, the pursuit of more sophisticated approaches that can the improve the current state-of-art in image compression is a very important problem. In the past, various algorithms have been proposed to improve quality of block-transform compressed images in the decoder without increasing the bit-rate. In  the decoded image is processed using space-invariant lters. In , , the decoded image is processed using space-variant lters. For these approaches the reconstructed blocky image is ltered only. Thus, there is no guarantee that the resulting image will satisfy the transmitted data. In , an image recovery approach is essentially proposed to reconstruct images from the transmitted data. However, the proposed processing is spatially-invariant and the theory of Projections Onto Convex Sets (POCS) ,  was used loosely to justify the convergence of this algorithm. As pointed out in , the convergence of this algorithm can not be justi ed rigorously by the theory of POCS but by the theory of constrained optimization. In the JPEG standard ,  a technique for predicting the AC coecients is recommended in Annex-K.8.2, as an option at the decoder in order to suppress the between block discontinuities of the decoded images. For this approach, the image is assumed to be a quadratic surface and the missing low frequency coecients are predicted to t this surface. However, in areas with sharp intensity transitions, this model is no longer valid and the proposed prediction scheme fails. In , a probabilistic model is assumed and the compressed image is reconstructed using a maximum a posteriori probability (MAP) approach. However, the prior distribution model used for the original image is not spatially-varying. In , , two recovery approaches were proposed to reconstruct the compressed images in the decoder. First, a set theoretic approach based on the theory of POCS, and second, a constrained least-squares approach based on regularization were proposed. However, in both approaches in ,  spatially-invariant prior knowledge was used to reconstruct the compressed image along with the transmitted data. In  another MAP approach based on a compound Gauss Markov image model and mean eld annealing was proposed. However, the spatial adaptivity of this image model is binary and thus somewhat limited. In this paper, a new POCS based recovery algorithm is
proposed to reconstruct in the decoder the compressed image. The novelty of this algorithm is that it uses a spatiallyadaptive smoothness constraint set which captures both the local statistics of the image and the human perceptual characteristics. More speci cally, the rest of this paper is organized as follows: In section II the mathematical background required for the rest of this paper is established and previous relevant work is reviewed. In section III the new spatially-adaptive smoothness convex constraint set is de ned, its projector is found, the mathematical properties of the projector are rigorously established, and nally the recovery algorithm is presented. In section IV, in an eort to make this approach more attractive for practical applications, a simpli ed spatially-adaptive recovery algorithm is proposed. This algorithm is based on a quantized version of the previously de ned smoothness constraint sets. In section V the computational complexity of this algorithm is examined and is compared to the complexity of a traditional JPEG decoder. In section VI numerical experiments are presented which test the proposed algorithms and compare them to previous ones. Finally, in section VII, we present our conclusions. II. Mathematical Background and Our Related Previous Work
Throughout this paper a digital N N image is treated as an N 2 1 vector in the space RN 2 by lexicographic ordering either by rows or columns and as distance measure the l2 norm is used.2 The BDCT is viewed as a linear transformation from RN to RN 2 . Then, for an image f we can write F = B f; (1) where F is the BDCT of f and B is the BDCT matrix. Due to the unitary property of the DCT matrices, the BDCT matrix is also unitary and the inverse transform can be simply expressed by B t where t denotes the transpose of a matrix. Then, the inverse BDCT can be written as f = B t F: (2) The elements of F in Eq. (1) are the transform domain coecients of the image. In a BDCT based coder, each of them is quantized in order to achieve bit-rate reduction for transmission. This quantization operation can be described mathematically by a mapping or an operator from RN 2 to 2 RN . Let Q denote this operator, we have F0 = Q F: (3) Note that Q satis es the idempotent property Q2 = Q: (4) The input output relation of the coder can be modeled by F0 = Q B f : (5) In the receiver only the quantized BDCT coecients F0 are available and the output of a conventional decoder is f 0 = B t F0: (6)
The quantization operator Q in Eq. (5) is nonlinear. Let T denote the concatenation of B and Q. Then, the operator T is2 also nonlinear, and it is a many-to-one mapping from RN to RN 2 . Therefore, Eq. (5) with f unknown may has in general many solutions. Let CT0 denote the set of all such solutions. That is, 4 f f : T f = F0 g: CT0 =
From Eq. (5) we know that the original image at the coder (which is not known at the decoder) is an element of the set CT0 . It is not dicult to see that the blocky image f 0 given in Eq. (6) is also an element of the set CT0 , due to the idempotent property of Q in Eq. (4). The set CT0 is not closed because in general the quantization intervals are not 4 C 0, closed. However, in , , , its closure i.e., CT = t was implicitly used. This subtle dierence was rst noticed in . Let Cs denote the set of images which do not exhibit blocking artifacts. That is, 4 f f : f is smooth in the block boundariesg (8) Cs = T Then, the set C0 = CT Cs contains all the images that satisfy both the received data and also are smooth between blocks. It is therefore clear, that an element from C0 is a better choice than f 0 as the recovered image. The theory of POCS provides the mathematical tools for obtaining an element in C0 , . Therefore, in what follows we shall brie y review the results from the theory of POCS which are necessary for the development of the rest of this paper. For more details and some recent developments in set theoretic estimation the interested reader is referred to , , . The main result from POCS theory that we shall use in the rest of this paper is the following: Given m closed 4 Tm C nonempty convex sets Ci ; i = 1; 2; :::;m, and C0 = i=1 i the iteration fk+1 = Pm Pm?1 P1 fk ;
k = 0; 1; 2; :::
where Pi the projector or projection operator onto Ci de ned by k f ? Pif k = gmin k f ? g k; (10) 2C i
where g is called the projection of f will converge to a point of C0 for any initial f0 , . It is worth pointing out that if the set C0 contains more than one point, as it usually does, then the solution is not unique and is in uenced by the order of the projections, and the starting point f0 of the algorithm. The key to applying this theory to image recovery problems lies in expressing every known property of the unknown image by a closed convex set. Then, the POCS theory yields automatically a recovery algorithm. This approach is very exible in incorporating prior knowledge into the recovery process, provided that it can be described in the form of convex constraint sets.
The de nition of the set Cs in Eq. (8) is only qualitative. A set that can be used in the POCS context requires a rigorous mathematical de nition. In , , the smoothness constraint set Cs was de ned by 4 ff : kQ f k E g; (11) Cs = where E is a scalar upper bound that de nes the size of this set; Q is a linear operator de ned by writing an N N image f in its column vector form as: f = f f1; f2 ; :::; fN g; (12) where fi denotes the ith column of the image. Then, Q f gives the dierence between adjacent columns at the block boundaries of f . For example, for the case of N = 512 and 8 8 blocks, 2 3 f8 ? f 9 66 f16 ? f17 77 6 77 : Q f = 66 (13) 77 : : 64 5 : f504
The norm of Q f
kQ f k =
"X 63 i=1
k f8i ? f8i+1 k2
III. Projection-Based Spatially-Adaptive Image Recovery
In most images between-pixel correlation varies spatially. More speci cally, in smooth areas , strong correlations exist among the local neighboring pixels, whereas in texture or edge areas the local correlations are weaker. A number of image models have been proposed to capture these properties for image recovery problems, see for example , , . The set de ned in Eq. (13) treats the block variations of the entire image equally, and consequently the processing in Eq. (15) does not adapt to the local statistics of the image. In addition, it is also well known that the noise visibility in images is not space-invariant and depends on the characteristics of the region under observation. For example, noise in smooth regions is more visible than that in texture or edge areas . The blocking artifacts can be viewed as noise with a regular pattern. Therefore, from both points of view, delity to the true statistical nature of real images, and human perceptual characteristics, a spatially-adaptive smoothness constraint set would be very helpful in recovering high-quality images. A. A Spatially-Adaptive Smoothness Constraint Set
is a measure of the total intensity variation between the boundary columns of adjacent blocks. The projection onto Cs was computed in . For an image f 2= Cs in column form f f1; f2; :::; fN g, its projection ~f = Ps f onto set Cs is represented in column form also by f~f1; ~f2; :::; ~fN g. For a 512 512 image and 8 8 blocks ~fi = fi+1 + (fi ? fi+1); ~fi+1 = fi ? (fi ? fi+1 )
(15) ~fi = fi,
for i = 8 k hand k = 1; i 2; :::;63; otherwise; E 1 + 1 . where = 2 kQf k In a similar fashion, a set Cs0 which captures the intensity variations between the rows of the block boundaries was also de ned. When the quantizer Q is known the projector PT onto the data set in Eq. (7) is easy to nd , . Using these convex sets a POCS based recovery algorithm can be de ned to reconstruct in the decoder the compressed images. The experimental results showed that the this approach worked well . The recovered images are, both visually and objectively using a distance metric, better than the images obtained by traditional decoders. However, according to this approach an entire block boundary column/row vector fi is treated uniformly. Thus, the local properties of the image along this column/row are not explicitly used during the recovery process. In what follows we present a spatially-adaptive recovery approach that explicitly uses the local image properties to impose smoothness constraints along the columns/rows of the reconstructed image.
From the de nition in Eq. (13), for a 512512 image and 8 8 blocks, Q f is a (512 63) 1 vector. Q f captures the variations between all the pixels at the neighboring vertical block boundaries. Let W be a (512 63) (512 63) diagonal matrix of the form 2 w 0 0 0 3 1 66 .. 77 . 7 66 0 w2 0 7 . . (16) 66 0 0 . . 0 .. 77 ; 64 .. .. 7 . 0 5 . . 0 .. 0 0 w51263 where the wi 's, i = 1; 2; ; (512 63), are weights which capture the local properties of the image. Then, the vector WQ f is a weighted version of the vertical between block variations. De ne the set 4 ff : kW Q f k E g: Cw =
It is straightforward to show that the set CW is both convex and closed. Before describing how the weights wi0 s are obtained we will examine in detail the projection operation onto the set Cw . For an image f 2= Cw represented in column vector form as in Eq. (12), the projection onto Cw will be also given in column form. Let Pw be the projector onto Cw . Then, we can write ~f = Pw f = f ~f1; ~f2 ; :::; ~f512 g: (18)
2 3 f8 66 f16 77 6 : 77 x0 = 6 66 : 77 ; 4 : 5 f504
2 ~ 3 f8 66 ~f16 77 6 : 77 u=6 66 : 77 ; 4 : 5 ~f504
3 2 f9 66 f17 77 6 : 77 y0 = 6 66 : 77 4 : 5 f505
2 ~ 3 f9 66 ~f17 77 6 : 77 v=6 66 : 77 : 4 : 5 ~f505
Using the notation in Eqs. (23) and (24), Eq. (21) can be rewritten as ~fi = 1 [ I + (I + 2wkt wk )?1] fi + (19) 2 1 [ I ? (I + 2wt w )?1 ] f i+1 (26) k k 2 ~fi+1
(20) for i = 8 k and k = 1; 2; :::; 63; otherwise; ~f = f . i i The spatially adaptive nature of the projector Pw can be seen by rewriting Eq. (27) as
We show in Appendix I that the projection ~f is given by u
[ I + (I + 2W t W)?1 ] x0+ 1 [ I ? (I + 2W tW)?1 ] y0 2
= 12 [ I ? (I + 2wkt wk )?1] fi + 1 [ I + (I + 2wt w )?1 ] f i+1 (27) k k 2
= fi+2fi+1 + (I + 2wkt wk )?1 fi ?2fi+1 = fi+2fi+1 ? (I + 2wkt wk )?1 fi ?2fi+1 :
From Eq. (24), matrix (I + 2wkt wk )?1 can be written as (21) 2 3 1 v = 12 [ I ? (I + 2W t W)?1 ] x0+ 0 0 0 1+2(w1k )2 1 [ I + (I + 2W tW)?1 ] y0; 77 66 .. 2 1 0 0 . 77 66 1+2(w2k )2 .. 77 : (29) 66 . ~fi = fi: for i 6= 8 k or 8 k + 1; k = 1; 2; ; 63, . . 0 . 0 0 77 66 .. .. 5 where the scalar in Eq. (21) is the positive root of the 4 0 . . 0 ... nonlinear equation 0 0 1+2(1w512 k )2
k W Q ~f k = E:
(22) From Eqs. (28) and (29) it is clear that in areas where the weighting factors wi 's (i.e. wjk 's) are large, the dierence In order to better understand the nature of the projector between the neighboring pixels of the projected image is Pw , we rewrite W in a block form as reduced more than in areas where wi0 s are small. It is informative to investigate the following two extreme cases: 2 w 0 0 0 3 1 when wi = 1, the projected pixels are simply the average .. 77 66 of the two neighboring pixels; when wi = 0, the projected 66 0 w2 0 .. 77 pixels remain unchanged. (23) 66 0 0 . . . 0 .. 77 ; Comparing the projector onto Cw in Eq. (28) with the 64 .. .. 7 . projector onto Cs in Eq. (15), the spatially-adaptive nature . . 0 .. 0 5 of the new constraint set becomes clear. All pixels along 0 0 w63 a column in the projector Ps in Eq. (15) are processed without taking into account the local properties where wk is a 512 512 diagonal matrix corresponding to uniformly of the image. However, the penalty for the more sophistithe kth block column boundary. For k = 1; 2; ; 63. wk cated model used in Cw is that the projection can not be is written in the form expressed in closed form. The parameter must be found 2 wk 0 0 0 3 by solving numerically the nonlinear equation in Eq. (22). 1 Finally, we like to point out that a similar in form spatially. 7 66 . adaptive smoothness constraint has also been used in reg66 0 w2k 0 .. 777 .. 77 : (24) ularized image restoration , , . wk = 6 0 0 . . . 0 66 . . 7 B. On the properties of the constant in the projector Pw 4 .. .. 0 . . . 0 5 k 0 0 w512 The de nition of the projector in Eq. (27) requires the computation of the parameter from Eq. (22). Therefore, Note that the factors in wk are related to those in W by in what follows the properties of the roots of Eq. (22) and the relation the numerical techniques used to nd them are examined. Using the notation in Eq. (20), we have wjk = w512k+j; (25) for j = 1; 2; ; 512.
Qf = x0 ? y0 and Q~f = u ? v:
From Eq. (21), we have u ? v = (I + 2W t W)?1 (x0 ? y0 ); that is, Q ~f = (I + 2W tW)?1 Q f : Eq. (22) can be rewritten as k W (I + 2W tW)?1 Q f k = E: De ne 3 2 d1 6 d2 77 d=6 64 .. 75 = Q f : . d51263 Then, Eq. (33) can be rewritten as k W (I + 2W tW)?1 d k = E; or N2 X wi2 d2i = E 2: (1 + 2w2 )2 i
(31) (32) (33) (34) (35) (36)
Eq. (36) can be expanded further into a 2N 2 th order equation of . However, before this equation can be used two questions are in order: First, does it have any solutions ? Second, if it has several solutions which one corresponds to the projector in Eq. (27) ? These questions are answered by the following theorem: Theorem 1: For all wi 's real in Eq. (36), the following holds: 1. Eq. (36) has one and only one positive root in . 2. Eq. (36) has at least one negative root . 3. The negative roots of Eq. (36) are all greater than the positive root in magnitude. 4. Only the positive root corresponds to the projector in Eq. (27). The proof of Theorem 1 is given in Appendix II. The above theorem establishes the existence of a positive root for the projector in Eq. (27). In order though to nd this root numerical methods have to be used. For example, Newton's method can be used. Since there are more than one solutions to Eq. (36), a legitimate question is: How can we guarantee that the numerical iterations will converge to the correct root? This question is answered by the following theorem: Theorem 2: Let n 2 2 X i di ? E2: (37) g() = (1 +w2w 2 2 ) i i=1 Then, the iterations generated by Newton's method: g k = 0; 1; 2; (38) k+1 = k ? g(0 k ) ; k )
with 0 = 0 will always converge to the positive root of Eq. (36); Furthermore, k+1 > k and jk+1 ? j < jk ? j where is the true solution. The proof of Theorem 2 is given in Appendix III. A similar equation has been previously studied in  in the context of image restoration. However, the properties of its roots were not rigorously established.
C. On the Choice of the Weights in W
From the previous discussion, it is clear that the weights wi 's should be chosen based on the local statistics of the image and the human perceptual properties. The pixel intensity at location (i; j) can be treated as a random variable with mean i;j and variance i;j . The mean serves as a measure of the local brightness, and the variance is a measure of the local detail at the pixel location (i; j). From the nature of both Pw and the visibility of the blocking artifacts, the weights wi 's should be a decreasing function of i;j . An example of such a function is w(i;j ) = 1 +1 ; (39) i;j where 1 is added in the denominator to avoid mathematical diculties when i;j = 0. A range compressed form of this function is given by (40) w(i;j ) = ln 1 + 1 +1 : i;j In our experiments, we noticed that the blocking artifact is more visible in bright rather than in dark areas of the image. A function which captures this property is pi;j w(i;j ) = ln 1 + 1 + : (41) i;j Several forms of the weighting function were previously suggested, , . Since all of them were based on the human perceptual characteristics, we shall refer to them as the visibility functions of the blocking artifact (VFBA). The above de nitions are only examples of VFBA's and illustrate how the human perceptual characteristics can be incorporated in a spatially-adaptive smoothness constraint set. A study of the exact form of VFBA's is beyond the scope of this paper. The appropriate form of VFBA depends also on the medium through which the images are presented to the users, e.g., hard copy or CRT monitor. D. The Recovery Algorithm
The constraint set de ned in Eq. (17) captures the weighted variations between the columns at the block boundaries. From the projection in Eq. (27) we see that this constraint set results only in the direct processing of the columns at the block boundaries. This processing will yield new intensity discontinuities between the columns at the block boundaries and their neighboring columns inside the blocks. To avoid this undesired eect, constraint sets that capture the variations between the columns at block boundaries and their neighboring columns inside the blocks can be introduced in a similar fashion. This type of sets can be de ned also for the columns inside the blocks. The set Cw captures only the image smoothness (horizontally) between block columns only. The smoothness constraint set Cw0 captures the between block row smoothness. Similarly, constraint sets capturing also o-block boundary between row smoothness can also be de ned.
Besides the sets de ned previously, the set CT in Eq. (7), is also used. Another valuable set is the set that captures the information about the range of the pixel intensity of an image. This set is de ned by 4 ff : 0 f 255; 1 i; j N; g: Cp = (42) i;j The projectors of sets CT and Cp , PT and Pp , respectively, are well known, see for example , , and they are not discussed in this paper. Using the previous sets the POCS theory yields the following recovery algorithm: Set f0 = f 0. For k = 1; 2; , compute fk from fk = Pw Pw0 PT Pp fk?1; (43) where Pw ; Pw0 ; PT and Pp denote the projections onto the sets Cw ; Cw0 ; CT and Cp respectively. Continue this iteration until kfk ? fk?1k is less than some prescribed bound. IV. A Simplified Spatially-Adaptive Algorithm
The algorithm in Eq. (43) requires the projections Pw and Pw0 which are not given in closed forms. As mentioned previously they require the numerical solution of Eq. (22) to nd the projection constant . From an application point of view, a simpler algorithm would be preferable. Therefore, a valid question is: can we avoid the numerical computations to nd the projection in Eq. (27) and still maintain the adaptive nature of the recovery algorithm? The weighting factors wi's in W have a continuous range of values. A natural simpli cation is to quantize the continuous range of the values of these weights to a xed number of values. Assuming that W has only M quantized levels for its weighting factors which are denoted by q1; q2; ; qM . Then, for each i, i = 1; 2; ; M, de ne a set Ii using the following two rules: 1. Assign all the pixels in block boundary pixels whose corresponding weighting have values equal to qi to set Ii . 2. Assign all pixels o the block boundaries to the same set to which its closest boundary pixel belongs to. For this segmentation rule we can write M M [ \ Ii = the entire image : (44) Ii = ;; and
The idea of using the operators Ii to select the pixels in the set Ii is not new. It was used before in the context of image restoration in . By the de nition of the Ii 's, we have WQf = q1QI1 f + q2QI2f + + qM QIM f ;
k WQf k2 = q12 k QI1 f k2 + q22 k QI2 f k2 + + qM2 k QIM f k2: (49) Note that the set CW de ned in Eq. (17) puts an upper bound on the total variation kWQf k. From Eq. (49) it is clear that this constraint can be enforced approximately by putting constraints on each term kWQIi f k; i = 1; 2; ; M, i.e., k QI1f k E1 k QI2f k E2 (50) .. . k QIM f k EM ; where Ei; i = 1; 2; ; M are constants. Consequently, we have the following constraint sets Cw1 = f f : k QI1 f k E1 g Cw2 = f f : k QI2 f k E2 g (51) .. . CwM = f f : k QIM f k EM g:
Note that for each i, Ii f is only a segment of the entire image and the set Cwi constrains this segment of the image only. The projection Pi onto the set Cwi is given by Eq. (15) with Ii f as the image. From Eq. (46), we see that in essence the image is partitioned into dierent segments based on the value of the VFBA. The image pixels within a segment are treated uniformly since they have nearly the same local statistical properties. Therefore, the spatially-adaptive nature of the algorithm is preserved and the computational complexity in the new approach is almost the same as that in  that uses the projection of Eq. (15). The previously de ned sets capture only the smoothness between the columns at block boundaries. It is straightfor0 that capture the smoothness beward to de ne sets Cwi tween the rows of the block boundaries and sets that capi=1 i=1 ture column/row o-block-boundary smoothness. Based De ne Ii to be the operator which selects only the pixels on these sets a POCS recovery algorithm can be de ned in a similar fashion to Eq. (43). in the set Ii and sets the rest to 0. Then, we have V. Computational Cost of the Simplified I = I 1 + I2 + + I M ; (45) Algorithm where I is the identity operator. Hence In this section the computational cost of the simpli ed algorithm presented in section IV is compared to the cost f = I1 f + I 2 f + + I M f ; (46) of a traditional decoder. The complexity cost is measured in terms of numbers of real additions and multiplications. and therefore, The computational cost of the segmentation process is not WQ f = WQI1 f + WQI2f + + WQIM f : (47) included in the following calculations.
cost of this recovery algorithm is: (3 5) 10 real multiplications per pixel and (3 5) 26 real additions per pixel. For a traditional decoder, an inverse BDCT is required, this takes about 3 real multiplications and 8 real additions. Therefore, the proposed algorithm requires 10 17 times the computations of a conventional decoder without any postprocessing. The computations in this algorithm are dominated by the BDCT transforms. Therefore, a hardware implementation for the BDCT could speed up this algorithm significantly. Furthermore, the multiplication operation in Eq. (50) is only a scaling operation of a vector which is very amenable to parallel implementation. Finally, the total ~fi = fi+1 +(fi ?fi+1 ); ~fi+1 = fi ?(fi ?fi+1): (52) cost of the proposed recovery process is approximately the same as that of a 2-D 512 512 Fast Fourier Transform For N N images and K K blocks, Eq. (52) requires (FFT). N real multiplications and 3N real additions. There are ( NK ? 1) such block boundaries, hence the total cost VI. Experiments is: N( NK ? 1) real multiplications and 3N( NK ? 1) real In this section experiments are presented in order to additions. In addition to this, the computation of in test the proposed recovery algorithms and compare them Eq. (52) requires approximately N( KN ? 1) multiplica- to previous approaches. The 512 512 \Lena" image is tions and 2N( KN ? 1) additions. used as a test image. This image was compressed using a The above calculations are only for the smooth- JPEG based coder-decoder with quantization table shown ness constraint applied only at the block bound- in Fig. 1 which yields a bit-rate of .24 bpp . Each entry aries. Assume that a total of (m ? 1) o-block- in the quantization table denotes the quantizer step-size for boundary smoothness constraints are also enforced the corresponding DCT coecient, . For presentation on the columns inside the blocks. Then, the total purposes the center 256 256 section of this image is shown computational cost for the smoothness constraints is: in Fig. 2. The same section of the processed images will be 2mN( KN ? 1) real multiplications and 5mN( NK ? 1) real also shown in what follows. additions. The maximum value for m is K. As an objective measure of the distance between a re2. The projectors (P10 PM0 ) are of the same nature as constructed image g and its original image f , we used the (P1 PM ), and their computational costs are identi- peak-signal-to-noise-ratio (PSNR). For N N images cal. with [0; 255] gray-level range PSNR is de ned in dB by 3. The projector Pp is simply a thresholding operation. N 2 2552 Every pixel outside the intensity range is thresholded (53) PSNR = 10 log 10 kg ? f k2 : to the correct value. No additions and multiplications are required for this projection. 4. The projector PT is also a thresholding operation ex- The PSNR of the blocky image in Fig. 2 is 29:579 dB. cept that it is performed in the BDCT domain. ThereThe weights in Eq. (16) are computed based on Eq. fore, both forward and inverse BDCT are required (41) using the transmitted transform data. The weight to convert the data from the spatial domain to the matrices for the column and row smoothness constraint transform domain and vice versa. The computational sets are W and W 0 , respectively. Thus, both means cost of BDCT is dependent on the speci c implemen- and 0 corresponding to W and W 0 , respectively, and both tations. In the algorithm reported in , for each variances and 0 must be estimated. The weights wij K K DCT transform, K 2 log2 K real multiplication and wij0 used for the implementation of our algorithm were and K(3K log2 K ? K +1) real additions are required. constant within a 8 8 region surrounding a vertical or For an N N image there are ( NK )2 blocks and the total horizontal block boundary, respectively. number of real multiplications is N 2 log2 K and the toSince the DC coecient in each block is the average (to a 2 N 2 2 constant) of the pixel intensity within this block the mean tal number of real additions is (3N log2 K ? N + K ). was estimated as follows: Consider the vertical boundary Therefore, the total number of operations in each iteraN 2 l and let DC ; DC tion are: 2[2mN( K ? 1)]+2[N log2 K] real multiplications L R denote the DC coecients of its left 2 and right blocks, respectively. Then, the estimate of the N N 2 2 and 2[5mN( K ?1)+2[3N log2 K ?N + K ] real additions. mean used for the computation of the weights in W is l For example, when N = 512, K = 8 and for the worst given by case m = K, the computations required for each iteration R: are: 10 real multiplications per pixel (6 out of which is for (54) ^l = DC2L +8DC 2 BDCT) and 26 real additions per pixel (16 out of which is for BDCT). As we will see in section VI, it only takes 3 5 A similar equation is used to estimate the mean 0l0 used iterations for this algorithm converge. Therefore, the total for the computation of the weights in W 0 . However, in this Since the analyzed algorithm is iterative the cost of each iteration will be rst examined. Each iteration of the simpli ed algorithm in section IV is a concatenation of projection operators. Thus, the computational cost per-iteration is obtained by adding the computational cost of each projector. 1. The projectors (P1 PM ) for each of the constraint sets in Eq. (51) are only de ned for one segment of the entire image. Thus, the total cost for in (P1 PM ) is the same as for the projection in Eq. (15). Hence, we will count the operations required in Eq. (15). This equation can be rewritten as
case the DC coecients of the blocks above and below this Then, wi ; i = 1; 2; ; 63 512, are classi ed according to the following rule: vertical boundary l0 are used. The variance l at the vertical block boundary l is esti if wi < w ? s=2, then assign it to class 1; mated by if wi > w + s=2, then assign it to class 3; r otherwise, assign it to class 2. ^l = V AC2L+ 8V2ACR : (55) The resulting weight maps obtained by quantizing the where V ACL and V ACR are the sums of the squared AC weight maps in Figs. 3 (a) and (b) are shown in Figs. 5 (a) coecients in the rst column of the blocks left and right and (b), respectively. As one can see the maps in Figs. 5 to the boundary l, respectively. In a similar fashion 0 l0 is (a) and (b) are very close to those in Figs. 3 (a) and (b), reestimated using the sums of the squared AC coecients in spectively. After the image is segmented, the upper bounds the rst row of the blocks above and below the horizontal Ei's in Eq. (51) have to be determined. This is accomboundary l0 . This choice of estimators for the variances plished by computing Sk in Eq. (57) with wk = I for the captures the local properties of the image in the direction pixels of a single segment. Then Eq. (58) is used to compute the bounds Ei for that segment. This can be done that smoothing is performed. After proper scaling, the weight map for the \Lena" im- by the scheme used to determine E. Then, the recovery age corresponding to W and W 0 are shown in Fig.s 3 (a) algorithm can be applied and the reconstructed image is and (b), respectively. In the bright areas of these maps the shown in Fig. 6. The corresponding PSNR is 30:385 dB. For comparison purposes, we show results from the preblocking artifact is more visible than in the dark areas. viously proposed algorithms. In Fig. 7 we show the reFor the implementation of the algorithm in Eq. (43), constructed image using the POCS algorithm in . The the upper bound E in Eq. (17) has to be determined. It corresponding PSNR is 30.300 dB. In Fig. 8 we show the was estimated from the received data as follows: Write the 0 reconstructed image using the JPEG AC prediction recomblocky image f in Eq. (6) in its column vector form mendation in Annex-K.8.2. The corresponding PSNR is 0 0 0 0 f = f f1; f2; :::; f512 g: (56) 29.525 dB. All the previously shown images were obtained after 5 iterations of the respective algorithms. To illustrate the conFor k = 0; 1; 2; ; 7, de ne vergence properties of the proposed algorithms, the average # 21 "X 63 dierence per-pixel between-iterates given by k fk ?Nf2k?1 k 0 0 4 2 (57) and the PSNR of the recovered images are plotted versus Sk = k wk (f 8i+k ? f 8i+k+1) k ; i=1 the number of iterations in Figs. 10 and 9, respectively. proposed algorithms were tested in a number of other where wk is de ned in Eq. (24). Note that S0 = kW Q f 0 k. The experiments using both higher and lower compression raIn our experiments, we found that S0 is about 10 100 tios. In all cases proposed approaches outperformed times larger than the rest of the Sk 's. We determined E the approaches in the ,  using both the PSNR metric by and subjective visual quality. However, due to space con7 X 1 straints we can not include images of our other experiments (58) E = 7 Sk : in this paper. Instead, in Table 1 we furnish the PSNR k=1 results using the quantization tables in . For completeA simpler approach is to chose E = 10S0100 . The bound ness, the results obtained using the quantization table in E 0 can be determined similarly. The algorithm in Eq. (43) Fig. 1 are also included in this table. is implemented and the center part of the reconstructed image is shown in Fig. 4. The corresponding PSNR is VII. Summary and Conclusions 30:426 dB. A new spatially-adaptive image recovery algorithm was To implement the algorithm in section IV, the image pixels are classi ed according to their corresponding weighting proposed to decode block-transform compressed images. factors. In this experiment, we found that M = 3 levels The main advantage of this approach is that the decoded yields satisfactory results. The image is segmented into 3 image is reconstructed using both the transmitted data and areas: one with high, one with medium and one with low the prior knowledge that prior to compression it does not values of the weights. The segmentation scheme used is have discontinuities between-blocks. This prior knowledge complements the information conveyed by the transmitted given bellow: data. Thus, at the same bit-rate, the reconstructed imDe ne 63512 ages by this approach are of higher quality than those of X wi w = 63 1512 traditional decoders which use the transmitted data only. i=1 The new spatially-adaptive smoothness constraint set that we proposed is based both on the perceptual properties of and 12 # "63X humans and on the local image statistics. Therefore, it re512 1 2 sulted in better images, both visually and objectively using s = 63 512 (wi ? w) : a distance metric, than the previous POCS space-invariant i=1
algorithm in . The penalty for the improvement in image quality is an increase in the complexity of the decoder. In a forth-coming paper we propose a new improved POCS based recovery algorithm that explicitly incorporates in the smoothness constraints the image edge structure , . This algorithm can correct both ringing and blocking artifacts, therefore, can be used to decode both BDCT and subband/wavelet transform compressed images. Furthermore, POCS image recovery can be extended to the video decoding problem also. In this case convex constraints that capture the between-frame image relations are also included .
where ~f = f~f1; ~f2; :::; ~fN g 2 Cw . By the de nition of Qf given in Eq. (13), the relation kWQ ~fk E only puts constraint columns at the block boundaries of ~f. Therefore, for all columns not at block boundaries, ~fi = fi. Also, with the notations in Eq. (20), we have Q~f = u ? v. The projection in Eq. (21) follows from Lemma 1. II. Proof of Theorem 1
First, consider the following Lemma Lemma 2: For any real numbers x < 0; y > 0 and jxj >
y, the following holds,
j 1 +x x j > 1 +y y Acknowledgements The rst two authors acknowledge Professor Henry Stark Proof: Consider the following two cases: for his in uence in using POCS-based algorithms to solve Case 1: x < ?1. We have engineering problems and the reviewers who helped improve the quality of this presentation. j 1 +x x j = j1 ? 1 +1 x j = 1 + j 1 +1 x j > 1 > 1 +y y : Appendix I. Derivation of the projector onto Cw
Case 2: x > ?1. Then 0 < x + 1 < 1. We have First, consider the following Lemma j 1 +x x j = 1 j+xjx > jxj > y > 1 +y y : Lemma 1: Let x0 ; ; y0; u and v be vectors in the space Rn with the Euclidean norm k k. Let also W be an arbitrary n n matrix. Then under the constraint that Proof of Lemma 2 is completed. k W(u ? v) k E, the functional Proof of Theorem 2: (u; v) = k u ? x0 k2 + k v ? y0 k2 1. Let n 2 2 X i di is minimized when g() = (1 +w2w ? E 2: 2 2 ) i i=1 1 [ I + (I + 2W t W)?1 ] x + u = 0 2 Then g(0) = k WQf k ? E 0; [ I ? (I + 2W tW)?1 ] y0 1 t ?1 since f 2= Cw . v = 2 [ I ? (I + 2W W) ] x0 + And also [ I + (I + 2W tW)?1 ] y0; lim g = ?E 0: !+1 () where is to be solved by k W(u ? v) k = E. Proof: Form the Lagrange auxiliary function Since g() is continuous for all 0, there must be an + 2 (0; 1) such that g(+ ) = 0. Note also that g() J (u; v) = (u; v) + ( k W(u ? v) k2 ? E 2 ): is strictly decreasing for all 0. The uniqueness of + follows. Taking the gradients of J (u; v) with respect to vectors u 2. Let and v and setting them to 0, yields 2 = min fwi2; i = 1; 2; ; ng wmin (u ? x0 ) + ( W t W(u ? v) ) = 0; And let and max = ? 2w12 : (v ? y0 ) + ( W t W(v ? u) ) = 0: min Therefore Then g() is continuous for all 2 (?1; max ). Note that t u + v = x0 + y0 and (I + 2W W)(u ? v) = x0 ? y0 : lim g = ?E 0 and !lim g = +1: Lemma 1 follows immediately. !?1 () max () Using Lemma 1, the projector Pw can be derived as follows: Writing the image f in its column vector form, the Therefore, there must be an ? 2 (?1; max ) such projection of f onto the set Cw is the vector ~f in Cw that that g(? ) = 0. minimizes the distance function g() may possibly have some other negative roots. 3. For 0, we have N X (I?1) k f ? ~f k2 = k fi ? ~fi k2; (1 ? 2wi2 )2 < (1 + 2wi2 )2; i=1
for each wi . Therefore, for > 0, g(?) > g() . Let + denote the positive root of Eq. (36). Since g() > 0 for all 2 (0; + ), there won't be any root for g() in the interval [?+ ; 0). Therefore, all the negative roots have a larger magnitude than the positive root. 4. Form Eq. (21), the projection ~f of the image f is a function of . Let D() = k ~f ? f k2 : Then from Eq. (21), we have D() = k u ? x0 k2 + k v ? y0 k2 : (II?1) From Eq. (21), we also have u + v = x0 + y0 and u ? v = (I +2W tW)?1 (x0 ? y0 ): Therefore, u ? x0 = y0 ? v: And Eq. (II-1) can be rewritten as D() = 2k u ? x0 k2 = 2k 12 [(u ? x0 ) + (u ? x0 )] k2 = 21 k (u ? x0 ) + (y0 ? v) k2 = 21 k (u ? v) + (x0 ? y0 ) k2 = 12 k [(I + 2W t W)?1 ? I](x0 ? y0 ) k2 = 21 k [(I + 2W t W)?1 ? I]Qf k2: From Eq. (34), we have D() = 21 k [(I + 2W t W)?1 ? I]d k2 n 2 X i )2 d2: ( 1 +2w = 2w2 i
which is continuous for 0. And also g(0 ) is an increasing function of and g(0 ) < 0 for all 0. If k 0 is such that g(k) > 0, then from the Newton's iteration g k+1 = k ? g(0 k) : k )
we have k+1 > k . On the other hand, g(k+1 ) =
Z k+1 k
g(0 ) d + g(k ) :
For 2 (k ; k+1], g(0 ) > g(0 k ) It follows that Z k+1 g(k+1 ) > g(0 k ) d + g(k) k
= g(0 k ) (k+1 ? k ) + g(k ) = 0: (III-1) Therefore, g(k+1 ) > 0 also. Hence, for 0 = 0, the Newton's iteration k+1 = k ? gg(0 k) ; (k ) will generate a sequence 0 = 0 < 1 < 2 < which will converge to the positive root + , since g(k ) > 0 = g(+) guarantees that k < + for all k = 0; 1; 2; . The proof to theorem 2 is completed.
References  Committee Draft ISO/IEC CD 10918-1, \Title: Digital Compression and Coding of Continuous-tone Still Images, Part 1: Requirements and Guidelines," March 15, 1991.  W. B. Pennebaker and J. L. Mitchell, JPEG still image data compression standard, Van Nostrand Reinhold, New York, 1992. i i=1  H. C. Reeves and J. S. Lim, \Reduction of blocking eects im image coding," Optical Eng., Vol. 23, No. 1, pp.34-37, Jan/Feb 1984. Let ? denote a negative root of Eq. (36) and + be Ramamurthi and A. Gersho, \Nonlinear space-variant postits positive root. Then from 3 we have j? j > + .  B. processing of block coded images," IEEE Trans. on Acoust., Hence from Lemma 2 we have Speech and Signal Processing, Vol. 34, No. 5, pp. 1258-1267, October 1986. 2 2 ? wi )2 > ( 2+ wi )2;  K. Sauer, \Enhancement of low bit-rate coded images using ( 1 +22 edge detection and estimation", Computer Vision Graphics and 1 + 2+ wi2 ? wi2 Image Processing: Graphical Models and Image Processing, Vol.53,No.1, pp.52-62., January 1991. for each i. Therefore, we have  R. Rosenholtz and A. Zakhor, \Iterative Procedures for reduction of blocking eects in transform image coding", IEEE Trans D(? ) > D(+ ) : on Circuits and Systems for Video Tech., Vol. 2, No. 1, pp. 91-94, March 1992. G. Gubin, B. T. Polyak, and E. V, Raid, \The method of proThe theorem follows since ? is an arbitrary negative  L. jections for nding the common point of convex sets", U.S.S.R. root. Computational Mathematics and Mathematical Physics 7 (6) 124 (1967). III. Proof of theorem 2  D. C. Youla, \Generalized image restoration by the method of alternating orthogonal projections," IEEE Trans. Circuits and Note that the function g() has derivative Systems., Vol. 25, No. 9, pp. 694-702, September 1978.  S. J. Reeves and S. L. Eddins, \Comments on iterative procen 4 2 X dures for reduction of blocking eects in transform image codw d i i 0 ing", IEEE Trans. on Circuit and Systems for Video Tech. Vol. g() = ?4 (1 + 2w2 )3 : 3, No. 6, pp. 439-440, December 1993. i i=1
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Yongyi Yang was born in Shandong, China
in 1964. He received the BSEE and MSEE degrees from Northern Jiaotong University, Beijing, China in 1985 and 1988 respectively. He then continued his education in the United States where he received his MS degree in Applied Mathematics and a Ph.D degree in Electrical Engineering from Illinois Institute of Technology, Chicago, IL in 1992 and 1994 respectively. Currently he is a lecturer at the department of Electrical and Computer Engineering at the Illinois Institute of Technology. His research interests are in the general area of signal processing and more speci cally in signal recovery and image compression.
Nikolas P. Galatsanos was born in AthensGreece in 1958. He received his Diploma degree in Electrical Engineering from the National Technical University of Athens, Athens, Greece, in 1982. He received the M.S. and the Ph.D. degrees both in Electrical and Computer Engineering from the University of WisconsinMadison, in 1984 and 1989, respectively. Since August of 1989, he has been on the faculty of the Department of Electrical and Computer Engineering at the Illinois Institute of Technology, Chicago IL, where he is currently an Assistant Professor. His current research interests include image processing and multidimensional signal processing and more speci cally recovery, and compression of single and multi-channel/frame images. Dr. Galatsanos presently serves as an associate editor for the IEEE Transactions on Image Processing. Aggelos K. Katsaggelos received the Diploma degree in electrical and mechanical engineering from the Aristotelian University of Thessaloniki, Thessaloniki, Greece, in 1979 and the M.S. and Ph.D. degrees both in electrical engineering from the Georgia Institute of Technology, Atlanta, Georgia, in 1981 and 1985, respectively. .sp In 1985 he joined the Department of Electrical Engineering and Computer Science at Northwestern University where he is currently an associate professor. During the 1986-1987 academic year he was an assistant professor at Polytechnic University, Department of Electrical Engineering and Computer Science. His current research interests include image recovery, processing of moving images (motion estimation, compression) and computational vision. Dr. Katsaggelos is an Ameritech Fellow and a member of the Associate Sta, Department of Medicine, at Evanston Hospital. He is a senior member of IEEE, and also a member of SPIE, the Steering Committees of the IEEE Transactions on Medical Imaging and the IEEE Transactions on Image Processing, the IEEE Technical Committees on Visual Signal Processing and Communications and on Image and Multi-Dimensional Signal Processing, the Technical Chamber of Commerce of Greece and Sigma Xi. He is an Associate editor for the IEEE Transactions on Signal Processing, an area editor for the Journal Graphical Models and Image Processing, and also the editor of the Springer-Verlag book Digital Image Restoration (1991).
The PSNR of the reconstructed images from different algorithms at different compression bit-rates
Reconstruction Compression Bit-Rate Algorithm .43 bpp .24 bpp .15 bpp JPEG Blocky 32.36 29.58 26.44 AC Prediction 32.30 29.52 26.53 POCS Nonadaptive 32.80 30.32 27.52 POCS Adaptive 32.81 30.43 27.58 POCS Segmented 32.79 30.38 27.50
Fig. 1. The quantization table used for JPEG based compression Fig. 2. Blocky \Lena" from JPEG compression at .24 bpp
Fig. 3. The weight maps of the blocky \Lena": (a) horizontal; (b) vertical. Fig. 4. Reconstructed image using algorithm in Eq. (43).
Fig. 5. The segmented map of the blocky \Lena": (a) horizontal; (b) vertical. Fig. 6. Reconstructedimage using the simpli ed algorithm in section (IV) Fig. 7. Reconstructed image using the POCS algorithm in  . Fig. 8. Reconstructed image using JPEG recommended AC prediction. Fig. 9. PSNR versus the number of iterations Fig. 10. Dierence Per Pixel Between Successive Iterates (DPPBSI) versus the number of iterations