Blind Deblurring of THz Time-Domain Images Based on Low-Rank

Representation

Marina Ljubenovi

1 a

, M

ario A. T. Figueiredo

2 b

and Arianna Traviglia

1 c

Center for Cultural Heritage Technology, Istituto Italiano di Tecnologia, Venice, Italy

Instituto de Telecomunicac¸

oes, Instituto Superior T

ecnico, Lisbon, Portugal

ﬁ

Keywords:

THz TDS, Deblurring, Denoising, Low-Rank.

Abstract:

Terahertz (THz) time-domain imaging holds immense potential for material characterization, capturing three-

dimensional data across spatial and temporal dimensions. Despite its capabilities, the technology faces hurdles

such as frequency-dependent beam-shape effects and noise. This paper proposes a novel, dual-stage frame-

work for improving THz image resolution beyond the wavelength limit. Our method combats blur at lower

frequencies and noise at higher frequencies. The ﬁrst stage entails selective deblurring of lower-frequency

bands, addressing beam-related blurring, while the second stage involves denoising the entire THz hyperspec-

tral cube through dimensionality reduction, exploiting its low-rank structure. The synergy of these advanced

techniques—beam shaping, noise removal, and low-rank representation—forms a comprehensive approach

to enhance THz time-domain images. We present promising preliminary results, showcasing signiﬁcant im-

provements across all frequency bands, which is crucial as samples may display varying features across the

THz spectrum. Our ongoing work is extending this methodology to complex scenarios such as analyzing mul-

tilayered structures in closed ancient manuscripts. This approach paves the way for broader application and

reﬁnement of THz imaging in diverse research ﬁelds.

1 INTRODUCTION

Terahertz (THz) radiation occupies a region between

the microwave and infrared portions of the elec-

tromagnetic spectrum, approximately spanning from

0.1 to 10 THz. This radiation can penetrate vari-

ous non-conducting materials, which makes it suit-

able for imaging diverse objects. THz waves have

longer wavelengths than visible and infrared light,

which traditionally limits their spatial resolution due

to diffraction. The advent of THz time-domain imag-

ing has marked a signiﬁcant milestone in the ﬁeld

of spectroscopy and imaging, offering unparalleled

non-destructive analysis and resolution capabilities in

a diverse array of applications ranging from secu-

rity screening and biomedical imaging to cultural her-

itage (Kemp et al., 2003; Darmo et al., 2004; Fuku-

naga, 2012). The unique interaction of THz radiation

with different materials allows for the extraction of

material-speciﬁc signatures, which are imperative for

https://orcid.org/0000-0002-4404-3630

https://orcid.org/0000-0002-0970-7745

https://orcid.org/0000-0002-4508-1540

accurate imaging and analysis (Hashimoto and Tri-

pathi, 2022).

In time-domain THz imaging in the reﬂection ge-

ometry, short pulses of THz radiation are directed

onto an object. Some of the radiation is reﬂected,

transmitted, or scattered depending on the object’s

properties. A detector measures the time delay and

intensity of the returning pulses, creating a time-

resolved proﬁle of the radiation, often referred to as

a waveform, for each pixel of the image. The struc-

ture of a THz time-domain image is inherently three-

dimensional (3D), with two spatial dimensions and

one temporal dimension (x, y, t). For each pixel (x,

y), there is a corresponding time-domain waveform

(t) that contains information about the THz signal’s

interaction with the object at that point. This wave-

form can provide a wealth of information, including

phase and amplitude data that are sensitive to the ma-

terial’s characteristics such as thickness, density, and

chemical composition. In the frequency domain, THz

time-domain images have the shape of 3D hyperspec-

tral (HS) cubes where each band corresponds to dif-

ferent frequencies.

However, despite its considerable potential, the

672

Ljubenovi

c, M., Figueiredo, M. and Traviglia, A.

Blind Deblurring of THz Time-Domain Images Based on Low-Rank Representation.

DOI: 10.5220/0012436800003660

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 19th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISIGRAPP 2024) - Volume 3: VISAPP, pages

672-679

ISBN: 978-989-758-679-8; ISSN: 2184-4321

practical deployment of THz imaging, particularly

in the 0.35–6 THz range, has been challenged by

a variety of factors, including frequency-dependent

beam-shape effects and the inherent noise present in

these images (Ljubenovi

c et al., 2020). Additionally,

the resolution, being wavelength-dependent, is con-

strained by the THz wavelength, which typically pre-

vents resolving features smaller than the wavelength

itself. Figure 1 illustrates beam shape and noise ef-

fects on different bands of THz time-domain images.

1.1 Related Work

In reﬂection geometry, the shape of the THz beam

plays a pivotal role in the quality of the acquired im-

ages. The spatial distribution of the beam, which

can be signiﬁcantly affected by diffraction and scat-

tering, often leads to images with less than optimal

resolution. To circumvent these limitations, advanced

methodologies have been developed to manipulate

and reﬁne the beam proﬁle, leading to improved im-

age resolution and contrast (Recur et al., 2012; Pod-

zorov et al., 2010; Popescu and Hellicar, 2010). Nev-

ertheless, these methods are mostly focused on single-

frequency THz images and noiseless scenarios.

In recent times, there has been an emergence of

methods leveraging neural networks for the restora-

tion of THz images, including super-resolution (Li

et al., 2017; Long et al., 2019), deblurring (Ljuben-

ovi

c et al., 2020), and noise reduction (Dutta et al.,

2022). However, these techniques often rely on syn-

thetic datasets for training due to the scarcity of real-

world training samples. Consequently, while they

exhibit promise, their performance tends to be lim-

ited when applied to real THz image data. Noise

removal techniques are of paramount importance, as

THz time-domain images are particularly susceptible

to noise stemming from a variety of sources including

environmental conditions and system-induced arte-

facts.

HS data is naturally low-rank because the spectral

bands often have a high correlation; only a few dis-

tinct materials are present in the scene, which reﬂect

similar spectral signatures (Nascimento and Bioucas-

Dias, 2007). The idea is that while the dataset is

high-dimensional, the intrinsic dimensionality—due

to redundancies and correlations in the spectral do-

main—is much lower. For denoising, the low-rank

property has been exploited by assuming that the

clean HS image data resides in a low-dimensional

subspace. Techniques such as robust principal com-

ponent analysis (RPCA) and its variants can separate

the low-rank clean image from sparse noise (Cand

et al., 2011). These methods assume that the noise

is sparse and uncorrelated, allowing it to be sepa-

rated from the low-rank data matrix. Several methods

by Zhuang et al. are based on a similar assumption

providing state-of-the-art results for denoising remote

sensing HS data (Zhuang and Bioucas-Dias, 2018;

Zhuang et al., 2021). Additionally, some of these

methods exploit a plug-and-play framework where

the core idea is to decouple the inversion process (i.e.,

restoration) from the denoising process (Venkatakr-

ishnan et al., 2013).

Furthermore, image restoration techniques based

on low-rank representation have emerged as a pow-

erful tool to exceed the diffraction limit imposed by

the THz wavelength (Ljubenovi

c et al., 2022). These

techniques exploit the inherent sparsity of the image

data in the THz spectral range, enabling the recon-

struction of high-resolution and sharp images from

low-resolution counterparts. By disentangling the

true signal from the noise, low-rank representation

methods provide a robust framework for the restora-

tion of THz time-domain images without the need for

hardware improvements.

In this paper, we aim to integrate these advanced

techniques—beam shaping in reﬂection geometry,

noise removal, low-rank representation, and plug-

and-play-based restoration—to present a comprehen-

sive framework that pushes the boundaries of THz

time-domain imaging. Our approach leverages the

synergies between these methods, fostering develop-

ments that could improve the quality and applicability

of THz imaging across multiple disciplines.

2 PROPOSED METHOD

We observed that bands associated with lower fre-

quencies exhibit stronger blur due to the larger beam

waist sizes at these frequencies. To address this issue,

our approach involves a sequential two-step method:

initially, we apply a targeted deblurring process to the

lower-frequency bands. Subsequently, we perform

denoising of the entire THz HS cube by applying di-

mensionality reduction and a plug-and-play approach,

taking advantage of its inherent low-rank structure.

The rationale underpinning our proposed method

is based on the insight that dimensionality reduction

applied to the original data cube tends to retain certain

blurring artifacts within the subspace components (as

seen in Figure 2). Conversely, when this reduction is

carried out on data that has already been deblurred,

there is a notable preservation of ﬁner details (as il-

lustrated in Figure 3).

Assuming additive noise, an observation model

for THz HS image with b spectral bands where each

Blind Deblurring of THz Time-Domain Images Based on Low-Rank Representation

673

Figure 1: Bands corresponding to lower, medium, and higher frequencies.

Figure 2: Subspace components when dimensionality re-

duction is performed on the original data cube.

Figure 3: Subspace components when dimensionality re-

duction is performed after deblurring.

band has n pixels is

Y = HX + N, (1)

where Y ∈ R

b×n

and X ∈ R

b×n

represent an observed

(blurred and noisy) and underlying (clean) THz HS

image respectively, and N ∈ R

b×n

stands for Gaussian

i.i.d. noise. The matrix H = bkdiag(H

, H

, ..., H

) ∈

bn×bn

, representing the unknown blurring degrada-

tion, is a block diagonal matrix, where each block

corresponds to a 2D cyclic convolution related to

the point spread function (PSF) of the corresponding

band.

2.1 Band-by-Band Deblurring

We propose deblurring each separate band up to some

(predeﬁned) frequency limit. The vectorised observed

and underlying images corresponding to each band

are related through the observation model:

= H

+ n

, (2)

with i = 1, 2, ..., m, and m < b. Here, the selection

of the value for m is empirically determined based on

the characteristics of the THz system utilized for data

acquisition (TOPTICA TeraFlash Pro). Speciﬁcally,

the deblurring process is conducted until the spectral

bands exhibit minimal noise (in our case up to a fre-

quency of 2 THz). The unknown blur is assumed to

be spatially invariant, but frequency-dependent. By

assuming the unknown underlying image x

and un-

known blur H

, the model in equation (2) is ill-posed

thus requesting regularization.

To perform blind deblurring of each band, the fol-

lowing optimization problem is formulated:

= argmin

||y

−Hx

+αφ(x

)+βψ(h

), (3)

where h

is a vectorised form of blurring ﬁlter H

The ﬁrst term of the objective functions is a data ﬁ-

delity term, premised on the assumption of additive

white Gaussian noise. The terms φ and ψ act as regu-

larizers that encode prior knowledge about the image

being recovered and the blurring kernel respectively,

each weighted by their respective regularization pa-

rameters, α and β.

Given that simultaneously estimating the latent

image x and the blurring kernel h leads to an ill-posed

optimization problem, our strategy is to ﬁrst estimate

the blurring kernel followed by a non-blind deconvo-

lution to restore the image.

Kernel Estimation. Should we employ a consistent

THz time-domain system and maintain identical set-

tings for data acquisition, we can enhance the accu-

racy of the blurring kernel estimation by utilizing a

straightforward sample, such as a circular object. This

reﬁned kernel estimate can subsequently be applied to

deblur more intricate samples.

The PSF corresponding to each lower-frequency

band is estimated by solving

= argmin

||y

− X

i, j

+ β||h

, (4)

where X

i, j

∈ R

bn×bn

is the matrix representing the

convolution of one band x

and the kernel correspond-

ing to that component, h

. Blurring kernel estimation

from (4) is a convex optimization problem and has a

closed-form solution.

To estimate the kernels, we use a simple shape ob-

ject, i.e., a circular hole in a metallic surface. From

this experiment, we conclude that beyond a certain

frequency (e.g., 2 THz), the estimate becomes unreli-

able most likely due to noise.

VISAPP 2024 - 19th International Conference on Computer Vision Theory and Applications

674

Image Estimation. To estimate the underlying im-

age, we solve

= argmin

||y

− H

+ αφ(x

). (5)

Here, we use the regularizer formulated as a sum

of two so-called l

-norms on image intensities and

gradients with the following form

φ(x

) = λ||x

+ ||∇x

, (6)

with λ controlling the relative contribution of each

component.

We utilize a half-quadratic splitting method for the

optimization to recover the sharp image. For an in-

depth explanation of this approach, please refer to the

work by Xu et al. and the citations therein (Xu et al.,

2011).

Our image restoration method builds upon the

framework initially presented in (Pan et al., 2014),

which was speciﬁcally developed for images contain-

ing text. Furthermore, it has been demonstrated that

the scope of this method extends beyond textual im-

agery, proving its efﬁcacy in deblurring a broader

range of natural images as well.

2.2 Low-Rank Representation

It has been demonstrated that THz HS images can be

projected onto a lower dimensional subspace with al-

most no loss of useful information (Ljubenovi

c et al.,

2020). The number of subspace components strongly

depends on the material composition of a sample and

important features of the sample that should be pre-

served (edges, textural features, etc.). Thus, after we

perform deblurring of bands corresponding to lower

frequencies, we remove the remaining noise by ex-

ploiting a low-rank property of a THz HS cube.

Under the assumption that the columns of X in

equation (1) live in a p-dimensional subspace S

, with

p ≪ b, spanned by the columns of subspace basis

E = [e

, ..., e

] ∈ R

b×p

, the underlying image is rep-

resented as X = EZ. Here, Z ∈ R

p×n

holds the repre-

sentation coefﬁcients of X in S

. The above assump-

tion is enabled by the discovery that E may be learned

directly from Y by applying a singular value de-

composition (SVD) or subspace identiﬁcation meth-

ods such as HySime (Nascimento and Bioucas-Dias,

2007).

By projecting the original THz HS image onto a

lower dimensional subspace, we remove a bulk of

noise. The remaining noise can be further removed by

applying a so-called plug-and-play approach and an

off-the-shelf denoiser to each separate subspace com-

ponent z

, for k = 1, 2, ..., p, as introduced in (Zhuang

and Bioucas-Dias, 2018). In this work, we use two

off-the-shelf denoisers, namely BM3D (Dabov et al.,

2007) and DnCNN (Zhang et al., 2017).

3 EXPERIMENTAL RESULTS

The results obtained by the proposed approach

are compared to several state-of-the-art approaches

for denoising HS images: FastHyDe (Zhuang and

Bioucas-Dias, 2018), FastHyMix (Zhuang and Ng,

2023), FastSuDeep (Zhuang et al., 2021), and SSTV

(Aggarwal and Majumdar, 2016). We also compared

with a method for deblurring HS images, PCA + TV

(Liao et al., 2013), and a method for joint THz image

deblurring and denoising (Ljubenovi

c et al., 2020).

In all experiments, the number of subspace com-

ponents is set to 10, α = 0.001, β = 0.1, and the num-

ber of iterations for kernel estimation is set to 30.

Figure 4 showcases the restoration results for the

1-cent coin using various methods. The denoising

techniques for HS imagery, FastHyDe, FastHyMix,

and FastSuDeep, leverage the low-rank property of

HS data akin to our approach. However, these meth-

ods are originally designed for image denoising and

speciﬁcally optimized for HS remote sensing im-

agery. In a similar vein, both the PCA + TV de-

blurring approach and the SSTV denoising technique

are devised for remote sensing data where a uniform

blurring effect across all bands is presumed. A joint

deblurring and denoising method is proposed specif-

ically for THz time-domain images where the blur

removal is performed after dimensionality reduction.

Due to the careful design of the proposed approach,

we are able to better preserve ﬁne details from the

bands corresponding to lower frequencies and simul-

taneously remove noise from the higher frequency

bands.

Additionally, we demonstrate the outcomes

achieved on a sample possessing ﬁne details, such as

a pendant with intricate engravings (Figure 5). The

results are displayed for three selected frequencies: a

lower frequency (0.35 THz) predominantly affected

by blurring artifacts and two higher frequencies (4

and 5.85 THz) where the original bands are heavily

corrupted by strong noise.

The evaluation of our method on both the 1-cent

coin and the engraved pendant samples illustrates its

robustness across the entire frequency spectrum as-

sessed. It effectively enhances the clarity of bands

at lower frequencies while simultaneously reducing

noise in the higher-frequency bands. It is important to

note, however, that some methods we tested, such as

FastHyDe, FastHyMix, FastSuDeep, and SSTV, are

Blind Deblurring of THz Time-Domain Images Based on Low-Rank Representation

675

Figure 4: Results obtained on the 1 cent coin. The ﬁrst row shows raw bands on seven different frequencies (from low to high).

Rows two to seven show the results obtained by six different methods for denoising and deblurring of HS cubes (FastHyDe,

FastHyMix, FastSuDeep, PCA + TV, SSTV, Joint Deblurring & Denoising). The ﬁnal row shows the results obtained by the

proposed method.

designed primarily for noise reduction and do not ad-

dress the blurring present in lower-frequency bands.

While PCA + TV is aimed at blur and noise removal,

it falls short when confronting the strong noise lev-

els typical in THz HS images. Although the tested

joint deblurring and denoising method can mitigate

both noise and blur, our proposed technique exhibits a

more reﬁned restoration of small details, particularly

in lower frequencies up to 2 THz.

To compare the results of the proposed approach

by using two different off-the-shelf denoisers (BM3D

and DnCNN) within the plug-and-play approach ex-

plained in Subsection 2.2, we performed an experi-

ment with the 1-cent coin (Figure 6). All other vari-

ables set in the experiments are the same. The per-

formance of the two (plugged) denoising algorithms,

BM3D and DnCNN, was found to be closely compa-

rable. DnCNN may be given a marginal preference

due to its ability to run effectively without the need

for manual adjustment of input parameters.

To evaluate the effectiveness of our method on

THz time-domain imagery that reveals layered com-

VISAPP 2024 - 19th International Conference on Computer Vision Theory and Applications

676

Figure 5: Results obtained on a silver pendant. The ﬁrst row shows the results for the lower frequency (0.35 THz), and the

middle and last rows show the results for the higher frequencies (4 THz and 5.85 THz respectively).

Figure 6: Results obtained with two different off-the-shelf denoisers plugged into the plug-and-play approach used for addi-

tional denoising of subspace components.

Figure 7: Results obtained with the proposed approach on a sample representing a ”closed book”.

positions, such as multiple layers of paper, we applied

our technique to data presented in (Redo-Sanchez

et al., 2016). This dataset comprises a stack of 9

pages, each imprinted with a single character (T, H,

Z, L, A, B, C, C, G). Only the top page is directly

observable, while the subsequent eight are concealed.

The data collection was conducted using a FICO THz

time-domain system from Zomega Terahertz Corpo-

ration, which has a bandwidth capability of up to 2

THz.

Figure 7 displays the results obtained with our

proposed methodology on a sample referred to as the

”closed book”. Given that the THz scanning system

utilized for data acquisition differs from our system,

a minor modiﬁcation to our technique is needed, in-

cluding the direct estimation of PSFs for various fre-

quency bands from the data itself. The ﬁndings con-

ﬁrm the resilience of our method, even when applied

to datasets characterized by distinctly unique blur and

noise proﬁles. Our approach successfully restores

frequency-speciﬁc bands and unveils some of the con-

cealed text, all without the need for any preliminary

preprocessing steps. We also present selected sub-

space components, eigen-images, in Figure 8. No-

Blind Deblurring of THz Time-Domain Images Based on Low-Rank Representation

677

Figure 8: Selected eigen images of ”closed book” showing

visibility of letters obscured by blank pages and/or another

leters.

tably, it is possible to discern letters from deeper lay-

ers; for instance, the letters ’L’ and ’A’ on the fourth

and ﬁfth pages, respectively, despite being masked by

other (blank) pages and letters.

4 CONCLUSION

We introduced a new methodology for the simulta-

neous deblurring and denoising of THz time-domain

images. Addressing the challenge of pronounced

blurring at lower frequencies and signiﬁcant noise

at higher frequencies, we developed a two-pronged

process: 1) a selective band-by-band deblurring for

the lower frequency bands, and 2) a projection of

the HS data cube onto a lower-dimensional sub-

space to effectively mitigate noise. The initial re-

sults are encouraging, demonstrating robust perfor-

mance across the frequency spectrum. This is par-

ticularly noteworthy given that various samples may

exhibit distinct characteristics at different frequen-

cies within the THz range. Moving forward, our ef-

forts are directed towards evaluating our approach on

complex, multi-layered samples, including sealed an-

cient manuscripts, to validate further and reﬁne our

method.

ACKNOWLEDGEMENTS

This project has received funding from the European

Union’s Horizon 2020 research and innovation pro-

gramme under grant agreement No. 101026453. The

authors would like to thank Alessia Artesani and Ste-

fano Bonetti for their support in data acquisition.

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