Deblurring Text Images via L0 -Regularized Intensity and Gradient Prior

Jinshan Pan† , Zhe Hu‡ , Zhixun Su† , Ming-Hsuan Yang‡

† School of Mathematical Sciences, Dalian University of Technology
‡
Electrical Engineering and Computer Science, University of California at Merced
jspan@mail.dlut.edu.cn, zhu@ucmerced.edu, zxsu@dlut.edu.cn, mhyang@ucmerced.edu

Abstract

We propose a simple yet effective L0 -regularized prior

based on intensity and gradient for text image deblurring.
The proposed image prior is motivated by observing distinct
properties of text images. Based on this prior, we develop (a) Blurred image
an efficient optimization method to generate reliable inter-
mediate results for kernel estimation. The proposed method
does not require any complex filtering strategies to select
salient edges which are critical to the state-of-the-art de-
blurring algorithms. We discuss the relationship with other
deblurring algorithms based on edge selection and provide (b) Cho et al. [2]
insight on how to select salient edges in a more principled
way. In the final latent image restoration step, we develop
a simple method to remove artifacts and render better de-
blurred images. Experimental results demonstrate that the
proposed algorithm performs favorably against the state-
(c) Ours
of-the-art text image deblurring methods. In addition, we
show that the proposed method can be effectively applied to Figure 1. A challenging blurred text image. (best viewed on high-
deblur low-illumination images. resolution display).

direct method that exploits sparse characteristics of natu-

ral images is proposed for deblurring natural and document
1. Introduction images [14]. Nevertheless, the blur kernel is not explic-
The recent years have witnessed significant advances in itly estimated from an input image and the computational
single image deblurring [8]. Much success of the state-of- load for learning an over-complete dictionary for deblur-
the-art algorithms [6, 16, 3, 18, 12, 10, 20] can be attributed ring is significant. Li and Lii [13] propose a joint estima-
to the use of learned prior from natural images and the se- tion method to estimate blur kernels from two-tone images.
lection of salient edges for kernel estimation. Although nu- However, this method is only applied to two-tone images
merous methods [6, 16, 12, 10, 20] have been proposed for and is less effective for text images with complex back-
motion deblurring, these priors are less effective for text im- grounds. Cho et al. [2] develop a method to incorporate
ages due to the contents of interest being mainly two-toned text-specific properties (i.e., sharp contrast between text and
(black and white) which do not follow the heavy-tailed gra- background, uniform gradient within text, and background
dient statistics of natural images. gradient following natural image statistics) for deblurring.
Text image deblurring has attracted considerable atten- While this algorithm achieves the state-of-the-art deblurring
tion in recent years due to its wide range of applications. results, the kernel estimation process is complicated and the
In [1], Chen et al. propose a new prior based on the im- performance depends largely on whether the stroke width
age intensity rather than the heavy-tailed gradient prior of transform (SWT) [5] separates an image into text and non-
natural scenes. However, this method is developed specif- text regions well or not. If the characters in a text image are
ically for document images (i.e., binary text images) and small and connected, it is unlikely to perform well. Figure 1
is unlikely to work well for cluttered images with text. A shows one example, where blurred characters are connected

1
due to large camera motion, and the deblurred result from 0.1 0.8

0.7

the state-of-the-art algorithm [2]. 0.08

0.6

0.5
0.06
In this paper, we propose a novel L0 -regularized in- 0.4

0.04 0.3
tensity and gradient prior for text image deblurring, and 0.2
0.02

present an efficient optimization algorithm based on the 0

0.1

0
0 50 100 150 200 250 300 −150 −100 −50 0 50 100 150
half-quadratic splitting method. The splitting method guar-
(a) (b) (c)
antees that each sub-problem has a closed-form solution and 0.1 0.4

ensures fast convergence. We present analysis on the re- 0.08

0.35

0.3

lationship with other methods based on salient edges, and 0.06 0.25

0.2

show that the proposed algorithm generates reliable inter- 0.04 0.15

0.1
mediate results for kernel estimation without any ad-hoc 0.02
0.05

selection processes. Compared to the state-of-the-art meth- 0

0 50 100 150 200 250 300
0
−150 −100 −50 0 50 100 150

ods [1, 2], the proposed algorithm is simple and easy to im- (d) (e) (f)
plement as it requires no additional operations (e.g., adap- Figure 2. Statistics of text images. (a) clean text image. (b) his-
tive segmentation [1], smoothing intermediate latent im- togram of pixel intensities from (a). (c) histogram of horizontal
ages, or SWT [2]). In the latent image restoration step, we gradients from (a). (d) blurred image. (e) histogram of pixel in-
present a simple method to deal with artifacts. Furthermore, tensities from (d). (f) histogram of horizontal gradients from (d).
we show that the proposed algorithm can also be applied to
gradient histograms of one clean text image and one blurred
effectively process natural blurred images containing text
image. The nonzero values of blurred image gradients are
and low illumination images which are not handled well by
denser than those of clear text image gradients. Thus we use
most state-of-the-art deblurring methods.
L0 -regularized prior, Pt (∇x), to model image gradients.
2. Text Deblurring via L0 -Regularized Prior The image prior for text image deblurring is defined as

In this section, we present a novel L0 -regularized prior P (x) = σPt (x) + Pt (∇x), (2)
of intensity and gradient for text image deblurring. where σ is a weight. Although P (x) is developed based
2.1. L0 -Regularized Intensity and Gradient Prior on the assumption that background regions of a text image
are uniform, we show that this prior can also be applied to
The proposed L0 intensity and gradient prior is based
image deblurring with complex backgrounds.
on the observation that text characters and background re-
gions usually have near uniform intensity values in clean 2.2. Text Deblurring via L0 -Regularized Prior
images without blurs. Figure 2(b) illustrates that the pixel The proposed prior P (x) is used as a regularization term
intensities of a clean text image (Figure 2(a)) center around for text deblurring,
two values and the distribution has two peaks (near 0 and
255). In other words, the pixel values of text images are min kx ∗ k − yk22 + γkkk22 + λP (x), (3)
very sparse if we only consider zero peaks. For a blurred x,k

text image, the histogram of pixel intensity is different from

where x and y denote the latent and blurred images, respec-
that of a clean image. Figure 2(e) shows the histogram of
tively; k is a blur kernel with the convolution operator * and
pixel intensity (from a blurred image in (d)) where it can-
L2 regularized term kkk22 ; and γ and λ are the weights.
not be modeled well by narrow peaks. Most importantly, it
does not contain the zero peak. Namely, the pixel values of 3. Deblurring Text Images
blurred text images are more dense. This intensity property
is generic for text images and used as one regularization We obtain the solution for (3) by alternatively solving
term in our formulation. For an image x, we define
min kx ∗ k − yk22 + λP (x), (4)
x
Pt (x) = kxk0 , (1)
and
where kxk0 counts the number of nonzero values of x. With min kx ∗ k − yk22 + γkkk22 . (5)
k
this criterion on pixel intensity, clean and blurred images These two sub-problems are solved as follows.
can be differentiated.
Gradient priors are widely used for image deblurring as 3.1. Estimating x with k
they have been shown to be effective in suppressing arti- Due to the L0 regularization term in (4), minimiz-
facts. As the intensity values of a clean text image are ing (4) is commonly regarded as computationally in-
close to two-tone, the pixel gradients are likely to have a tractable. Based on the half-quadratic splitting L0 mini-
few nonzero values. Figure 2(c) and (f) show the horizontal mization method [19], we propose a new method to solve
it using an efficient alternating minimization method. We Algorithm 1 Solving (6)
introduce auxiliary variables u and g = (gh , gv )> corre- Input: Blur image y and blur kernel k.
sponding to x and ∇x respectively, and rewrite the objective x ← y, β ← 2λσ.
function as repeat
min kx∗k−yk22 +βkx−uk22 +µk∇x−gk22 +λ(σkuk0 +kgk0 ), solve for u using (10).
x,u,g
(6)
µ ← 2λ.
where σ is the weight defined in (2). When β and µ are repeat
close to ∞, the solution of (6) approaches that of (4). With solve for g using (11).
this formulation, (6) can be efficiently solved through alter- solve for x using (8).
natively minimizing x, u, and g independently by fixing the µ ← 2µ.
other variables. until µ > µmax
β ← 2β.
The values of u and g are initialized to be zeros. In each
until β > βmax
iteration, the solution of x is obtained by solving
Output: Intermediate latent image x.
min kx ∗ k − yk22 + βkx − uk22 + µk∇x − gk22 , (7)
x
Algorithm 2 Blur kernel estimation algorithm
and the closed-form solution for this least squares mini- Input: Blur image y.
mization problem is initialize k with the results from the coarser level.
! for i = 1 → 5 do
−1 F(k)F(y) + βF(u) + µFG solve for x using Algorithm 1.
x=F , (8)
F(k)F(k) + β + µF(∇)F(∇) solve for k using (12).
λ ← max{λ/1.1, 1e−4 }.
where F(·) and F −1 (·) denote the Fast Fourier Trans- end for
form (FFT) and inverse FFT, respectively; the F(·) is the Output: Blur kernel k and intermediate latent image x.
complex conjugate operator; and FG = F(∇h )F(gh ) +
F(∇v )F(gv ) where ∇h and ∇v denote the horizontal and After obtaining k, we set the negative elements to 0, and
vertical differential operators, respectively. normalize it so that the sum of its elements is 1.
Given x, we compute u and g separately by Similar to the state-of-the-art methods, the proposed ker-
nel estimation process is carried out in a coarse-to-fine man-
min βkx − uk22 + λσkuk0 , ner using an image pyramid [3]. Algorithm 2 shows the
u
(9) main steps for kernel estimation algorithm on one pyramid
min µk∇x − gk22 + λkgk0 .
g level.
Note that (9) is a pixel-wise minimization problem, thus,
3.3. Removing Artifacts
the solutions of u and g are obtained based on [19],
Although latent text images can be estimated from (4)
x, |x|2 > λσ

u= β , (10)
as shown in Figure 3(c), this formulation is less effective
0, otherwise, for scenes with complex backgrounds or fine texture details.
We note that non-blind deblurring methods with Laplacian
and priors [9] have been shown to preserve fine details. How-
|∇x|2 > µλ ,

∇x,
g= (11) ever, significant artifacts are likely to be included with this
0, otherwise. prior as shown in Figure 3(b). In contrast, the proposed al-
The main steps for solving (6) are summarized in Algo- gorithm with L0 -regularized prior produces fewer fine de-
rithm 1. tails and ringing artifacts as shown in Figure 3(c).
3.2. Estimating k with x The proposed algorithm can be further enhanced to
deblur text and natural images with fine details by the
With the given x, (5) is a least squares minimization following approach similar to the ringing suppression
problem from which a closed-form solution can be com- method [16]. First, we estimate latent images Il (See Fig-
puted by FFT. As the solution directly from (5) based on ure 3(b)) by using the method with Laplacian prior [9]. Sec-
intensity values is not accurate [3, 12], we estimate the blur ond, we estimate latent images I0 (See Figure 3(c)) using
kernel k in the gradient space by the proposed algorithm via (4) but only with the gradient
min k∇x ∗ k − ∇yk22 + γkkk22 (12) information Pt (∇x) at this stage (i.e., setting σ of (6) to 0).
k
Similar to [16], we then compute a difference map between
and the solution can be efficiently computed by FFTs [3]. these two estimated images and remove artifacts with bi-
 (a) (b ) (c ) (d )

(a) (b) (c) (d) (e)

Figure 3. Non-blind deconvolution examples. (a) blurred images
and our estimated kernels. (b) results by [9] with Laplacian prior.
(c) results by (4) where σ = 0. (d) ringing suppression results (e)
by [16]. (e) our results.

lateral filtering. Finally, we subtract the filtered difference

map from Il . The result in Figure 3(e) shows that this ap-
proach works well for text and natural images, and performs x u x u
favorably against the ringing suppression method [16]. (f)

4. Analysis of the Proposed Algorithm

In this section, we provide more insight and analysis on
how the proposed algorithm performs on text deblurring.
We also demonstrate the importance of intensity prior for (g)
text deblurring and discuss its relationship with other meth-
ods based on edge selection. Furthermore, we show that the
proposed algorithm can be applied to deblur natural images.
4.1. Effectiveness of the L0 -Regularized Prior
(h)
The intensity prior Pt (x) can be regarded as a regular-
ization of segments based on image pixels from (10). In the
proposed algorithm, the threshold value of (10) is decreas-
ing with the effect of selecting segments in a coarse-to-fine
manner. We note (10) is the critical step adopted in [1] for
text deblurring. (i)
The text deblurring method [2] has a similar step
((7) in [2]) to the sub-problem of u in (9). The main dif-
ference is that the threshold value in [2] is determined by
SWT [5]. In addition, this text deblurring algorithm uses
the sparse gradient minimization method [19] to remove
(j)
ringing artifacts from the intermediate latent images, and
Co arse to fine
thus has more computational loads. The effectiveness of
this method is limited by SWT in terms of speed and accu- Figure 4. An example presented in [2]. (a) blurred image and ker-
racy. Figure 4 shows one example for which the method [2] nel. (b) results of [2]. (c) our results without using Pt (x) in the
does not perform well. The reason is that SWT is not effec- kernel estimation. (d) our final results. (e) intermediate results
tive in detecting text when the blurred characters are con- of [2]. (f) our intermediate results (including x and u). (g) inter-
mediate salient edges of [20]. (h) intermediate salient edges using
nected. The images in Figure 4(e) show the intermediate
only Pt (∇x). (i) intermediate results using only Pt (x). (j) our
results generated by SWT in [2]. intermediate salient edges, i.e., g in (11).
The success of recent deblurring methods hinges on in-
termediate estimations of the latent image explicitly [3, 18] like the filter-based edge selection methods [3, 18], the pro-
or implicitly [6, 16, 10]. The proposed method is distin- posed algorithm computes intermediate estimations itera-
guished from existing methods as it does not involve ad-hoc tively by solving a few optimization problems in a way sim-
edge selection (e.g., spatial filtering [2, 3, 18], or edge re- ilar to [20]. By using (10) and (11) in the proposed algo-
weighting [16, 10]) for kernel estimation. Instead of find- rithm, pixels with small intensity values or tiny structures
ing one good threshold to remove subtle image structures can be removed while salient edges are retained. Further-
more, our method exploits the gradient prior with Pt (∇x). (e.g., Figure 4(j)). Figure 6(b) shows an intermediate latent
If σ of (6) is set to 0, then the proposed algorithm is similar image x from a natural image. The result demonstrates that
to the recent methods based on L0 gradient priors [20, 15] the use of prior P (x) can also preserve salient edges and
which achieve the state-of-the-art results for deblurring nat- removes tiny details in natural images, thereby facilitating
ural images. Thus, the proposed algorithm is likely to per- kernel estimation in natural images.
form well for natural image deblurring. On the other hand,
these two methods [20, 15] (L0Deblur for short) do not per- 4.4. Deblurring Saturated Images
form well for text images. Figure 4(g) shows intermediate Estimating motion kernels from blurred images with re-
salient edges extracted by [20]. As no sharp edges are ex- gions of saturated pixels has been known as a difficult prob-
tracted, the blur kernel is not estimated well by this method. lem. Although some non-blind deblurring methods [4, 17]
We note that image deblurring using only intensity prior have been proposed, it remains challenging to develop ef-
Pt (x) is less effective (See Figure 4(i)) as the intensity prior fective blind deblurring algorithms. Saturated regions usu-
does not guarantee the sparsity properties of text image gra- ally appear sparsely in clear images and these areas are
dients. On the other hand, image deblurring with only gra- much larger (e.g., blobs or streaks) in the blurred images.
dient prior Pt (∇x) is not effective (See Figure 4(h)) as no Figure 7(b) shows two examples of saturated images from
salient edges are extracted. Figure 7(a). As the L0 norm used in the proposed algo-
4.2. Convergence of the Proposed Algorithm rithm is similar to an adaptive hard threshold strategy, we
use the binary images for illustration. Figure 7(c) and (d)
Our kernel estimation algorithm is mainly based on the show the corresponding binary images of clear and blurred
alternating minimization method which ensures that each saturated images where there are more nonzero elements in
sub-problem has a closed-form solution. Thus, it has the the blurred binary images than those in the clear binary im-
fast convergence property. Figure 5(c) shows kernel simi- ages. As the L0 norm in Pt (x) minimizes the number of
larity [7] with respect to iterations. With more iterations, nonzero coefficients, the proposed deblurring algorithm fa-
the quality of kernel estimates becomes higher. vors solutions with fewer blobs or streaks in the latent clear
1

images. We present results from challenging examples in

Kernel Similarity

0.8
Section 5.2.
0.6

0.4
0 10 20 30 40
Iterations

(a) Input and kernel (b) Our results (c) Kernel similarity plot
Figure 5. Convergence of the proposed algorithm.
4.3. Deblurring Non-document Text Images
Although the prior P (x) is developed based on the as-
sumptions that text images have uniform backgrounds, the (a) (b) (c) (d)
proposed method can be applied to text deblurring in non- Figure 7. Saturated images. (a) Clear images with saturated areas
document text images as shown in Figure 6. and kernel. (b) Blurred images with saturated areas. (c) Binary
images of (a). (d) Binary images of (b). (c) and (d) are obtained
from (a) and (b) with the same threshold value.

5. Experimental results
(a) Blurred image (b) Intermediate result x (c) Our results
Figure 6. An example with complex background regions. The size We present experimental evaluations of the proposed al-
of estimated kernel is 99 × 99 pixels. gorithm against the state-of-the-art methods for text deblur-
ring and results for saturated images in this section. All the
In the proposed Algorithm 1, the solution u from (10) experiments are carried out on a desktop computer with an
contains large intensity values and g from (11) contains Intel Xeon processor and 12 GB RAM. The execution time
large gradient values. That is, u and g contain main struc- for a 255×255 image is around 50 seconds on MATLAB. In
tures of x. Thus, the intermediate result from (7) is likely all the experiments, we set λ = 4e−3 , γ = 2, and σ = 1, re-
to inherit the properties of u and g. Compared to the nat- spectively. We empirically set βmax = 23 and µmax = 1e5
ural image deblurring method [20], our intermediate latent in Algorithm 1. More experimental results can be found
image restoration step introduces the intensity prior Pt (x). in the supplementary document, and the MATLAB code
This prior can help preserve more salient edges in the inter- and datasets are available at http://eng.ucmerced.
mediate latent image rather than destroy the salient edges edu/people/zhu/cvpr14_textdeblur.
 Blurred image Cho and Lee Xu and Jia Krishnan et al. Levin et al. Xu et al. Zhong et al. Ours
40

35

30
Average PSNR Values

25

20

15

10

0
im01 im02 im03 im04 im05 im06 im07 im08 im09 im10 im11 im12 im13 im14 im15
Image index
Figure 8. Quantitative comparison on the dataset. The numbers below the horizontal axis denote the image index. Our method performs
the best.
5.1. Document Images 5.2. Non-document Images
Synthetic images: We provide the example from [2] as Non-document text images: We present an example in
shown in Figure 4 for comparison. Table 1 shows the SSIM Figure 10 where the complex image contains rich text and
values and the kernel similarity values of the recovered im- cluttered background regions. The state-of-the-art natural
ages and estimated kernels by some state-of-the-art meth- image deblurring methods [3, 18, 21, 20, 6] do not perform
ods. Overall, the proposed algorithm performs well in terms well in this image. Although the text deblurring method [2]
of both metrics. In addition, we build a dataset containing handles this image well, the estimated kernel contains a cer-
15 ground truth document images and 8 kernels from [11]. tain amount of noise and the deblurred result contains some
For each sharp image, we compute the average PSNR on the unnatural colors as a result of the SWT process. In con-
blurred images from different kernels and compare among trast, the proposed algorithm generates the deblurred image
different methods [16, 3, 18, 10, 12, 20, 21] in Figure 8. The (clear text, sharp edges, and natural color) and blur kernel
details about this dataset can be found in the supplementary well. Figure 10(h) and (i) show the results using the pro-
document. posed algorithm without Pt (x) and Pt (∇x), respectively.
The results in (h)-(i) show that sharp images cannot be ob-
Real images: We evaluate the proposed algorithm and other tained by using only the gradient prior or intensity prior,
methods using real images. For fair comparison with [2], which indicates that the proposed prior P (x) plays a criti-
we use an example from [2] and show the deblurred re- cal role in text image deblurring.
sults in Figure 9. The natural image deblurring methods
do not perform well on text images. The deblurred result Low-illumination images: It is known that most state-
of [1] contains some ringing artifacts and some details are of-the-art deblurring methods are less effective in process-
missing. Although the state-of-the-art method by Cho et ing blurred images with saturated regions [4] which often
al. [2] performs well, the motion blur is not fully removed appear in low-illumination scenes. As discussed in Sec-
as shown in the red box in Figure 9(g). In addition, the tion 4.4, the proposed algorithm is likely to handle this
deblurred results contain unnatural colors as a result of the problem to a certain extent.
SWT process. Compared with [2], the proposed algorithm Figure 11 shows a real captured image which contains
generates a sharper and visually more pleasant deblurred several saturated regions (red boxes in (a)). We com-
image. We note that the L0Deblur [20] does not estimate pare the proposed algorithm with the state-of-the-art meth-
the blur kernel or deblurs the image well which also demon- ods [3, 18, 10, 21, 20]. As the priors of the state-of-the-art
strates the importance of Pt (x) of the proposed prior P (x). methods are developed to exploit salient edges for motion
 Table 1. Quantitative comparison using the example shown in Figure 4(a).
[3] [18] [12] [20] [21] [2] Ours without Pt (x) Ours without Pt (∇x) Ours
SSIM of images 0.6457 0.6269 0.5611 0.4867 0.6190 0.5526 0.5812 0.7473 0.8659
Kernel similarity 0.5200 0.5200 0.4170 0.6407 0.4938 0.6456 0.6303 0.7133 0.9140

(a) Blurred image (b) Cho and Lee [3] (c) Xu and Jia [18] (d) L0Deblur [20]

(e) Zhong et al. [21] (f) Chen et al. [1] (g) Cho et al. [2] (h) Our results
Figure 9. A real blurred image from [2]. The part in the red box in (g) contains some blur and unnatural colors.

deblurring, these algorithms do not perform well for images age deblurring.
containing numerous saturated regions. While the recent
method [21] is developed to handle large Gaussian noise, it Acknowledgements We thank Hojin Cho for generating
is less effective for saturated images. Although the saturated the deblurred results of his method [2]. Jinshan Pan and
areas (e.g., the highlighted blobs, streaks, and the characters Zhixun Su are supported by the NSFC (Nos. 61300086,
in Figure 11(a)) are large due to motion blur, as discussed 61173103 and 91230103), the China Postdoctoral Science
in Section 4.4, the L0 -regularized prior P (x) favors a clean Foundation, and National Science and Technology Ma-
image with few blobs and streaks. Thus, the proposed al- jor Project (2013ZX04005021). Zhe Hu and Ming-Hsuan
gorithm is able to estimate the blur kernel well due to the Yang are supported partly by the NSF CAREER Grant (No.
proposed prior P (x). The recovered image shown in Fig- 1149783) and NSF IIS Grant (No. 1152576).
ure 11(g) is sharper and clearer and characters can be rec-
ognized. We note that while our method is able to estimate References
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 (a) Blurred image (b) Cho and Lee [3] (c) Xu and Jia [18] (d) Krishnan et al. [10] (e) L0Deblur [20]

(f) Zhong et al. [21] (g) Cho et al. [2] (h) Ours without Pt (x) (i) Ours without Pt (∇x) (j) Our results
Figure 10. A blurred image with rich text. Our method performs well at kernel estimation and image recovery.

(a) Blurred image (b) Cho and Lee [3] (c) Xu and Jia [18] (d) Krishnan et al. [10]

(e) L0Deblur [20] (f) Zhong et al. [21] (g) Our results (h) Our kernel + [17]
Figure 11. A real blurred image with numerous saturated regions. The red boxes in (a) enclose some saturated pixels (e.g., the highlighted
blobs, streaks, and the characters).

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