# Enhancing Kinship Verification through Multiscale Retinex and Combined Deep-
Shallow features

El Ouanas Belabbaci1, Mohammed Khammari2, Ammar Chouchane3,4, Mohcene
Bessaoudi4, Abdelmalik Ouamane4, Yassine Himeur5, Shadi Atallal5 and Wathiq
Mansoor5

1 LIMED Laboratory, Faculty of Technology , University of Bejaia, 06000
Bejaia, Algeria  2 LIMED Laboratory, Faculty of Exact Sciences, University of
Bejaia, 06000 Bejaia, Algeria<EMAIL_ADDRESS><EMAIL_ADDRESS> 3University Center of Barika. Amdoukal
Road, Barika, 05001, Algeria  4dept. Electrical Engineering LI3C, University
of Mohamed Khider Biskra, Algeria<EMAIL_ADDRESS>;
<EMAIL_ADDRESS> 5College of Engineering and Information
Technology University of Dubai Dubai UAE<EMAIL_ADDRESS>
###### Abstract

The challenge of kinship verification from facial images represents a cutting-
edge and formidable frontier in the realms of pattern recognition and computer
vision. This area of study holds a myriad of potential applications, spanning
from image annotation and forensic analysis to social media research. Our
research stands out by integrating a preprocessing method named Multiscale
Retinex (MSR), which elevates image quality and amplifies contrast, ultimately
bolstering the end results. Strategically, our methodology capitalizes on the
harmonious blend of deep and shallow texture descriptors, merging them
proficiently at the score level through the Logistic Regression (LR) method.
To elucidate, we employ the Local Phase Quantization (LPQ) descriptor to
extract shallow texture characteristics. For deep feature extraction, we turn
to the prowess of the VGG16 model, which is pre-trained on a convolutional
neural network (CNN). The robustness and efficacy of our method have been put
to the test through meticulous experiments on three rigorous kinship datasets,
namely: Cornell Kin Face, UB Kin Face, and TS Kin Face.

###### Index Terms:

Kinship Verification, CNN, Deep Features, Shallow Features, MSR, LR Fusion.

##  I Introduction

Smart cities represent the next phase in urban development, harnessing the
power of digital technology, the Internet of Things (IoT), and data analytics
to enhance urban life on numerous fronts [1] Face recognition and biometrics
play a pivotal role in the evolution and functionality of smart cities . As
urban environments become increasingly interconnected and data-driven, the
need for efficient, secure, and personalized services becomes paramount [2,
3]. Face recognition serves as an advanced tool that aids in public safety,
streamlining traffic and crowd management, and even enhancing personalized
user experiences in public transportation or retail settings [4].
Concurrently, biometric systems, which go beyond just facial features, offer
an added layer of security, ensuring that services are accessed only by
authorized individuals. Whether it is for efficient service delivery,
security, or fostering a seamless urban experience, face recognition and
biometrics are instrumental in realizing the full potential of smart cities,
making them safer, more efficient, and responsive to their citizens’ needs
[5].

Kinship verification and face recognition are both sub-domains of facial image
analysis but serve different primary objectives, albeit with intertwined
techniques and methodologies [4, 6, 7]. Kinship verification through facial
images aims to ascertain the biological kinship between two individuals by
examining their facial characteristics [8, 9, 10]. By verifying kinship ties,
cities can ensure that rights related to cultural practices, lands, or
hereditary roles are correctly passed to genuine relatives, preserving
traditions and heritage.

This image-based verification adds a unique layer to facial analysis by
emphasizing the recognition of shared familial traits. This not only adds
depth to the challenge of facial image analysis but also broadens its scope
[11, 12]. Recognizing kinship is arduous due to the subtle interplay of facial
attributes, which include identity, age, gender, ethnicity, and expression [7,
13]. Furthermore, kinship identification has wide-ranging applications. It can
be harnessed to organize photos, build family trees, support forensic
inquiries, tag images, and aid in locating lost or sought-after individuals
[14, 7, 15]. Though DNA has been the traditional touchstone for verifying
kinship, automated facial image algorithms can offer both cost-effective and
rapid solutions [9, 16, 17].

In this research, we strive to harness the complexities and nuances of these
factors to craft a reliable kinship verification system, proficient in
overcoming the challenges delineated. To achieve this, we present several
innovative contributions and rigorously evaluate our methodology on three
renowned datasets: Cornell Kin Face, UB Kin Face, and TS Kin Face. Our primary
contributions are outlined as follows:

  * •

We introduce an advanced preprocessing method termed Multiscale Retinex (MSR).
This technique significantly enhances color restoration and overall image
quality. Our meticulous experimental evaluation reveals significant
improvements in kinship verification outcomes directly linked to the
deployment of the MSR approach.

  * •

For subspace projection and dimensionality reduction, we employ the robust
TXQDA+WCCN algorithm, which emphasizes multidimensional data representation.

  * •

In a bid to refine our feature extraction process, we implement score-level
fusion using Logistic Regression (LR). This fusion strategy pairs the shallow
texture features of LPQ with the profound features sourced from the VGG16
model. By leveraging the synergies between these attributes, we achieve
superior kinship verification performance.

The remainder of this article is structured as follows: In Section 2, we
discuss related work from three perspectives: Shallow features, CNNs, and
Multilinear Subspace Learning for Kinship Verification. Section 3 details our
methodology, introducing the preprocessing method (MSR) and the integration of
deep and shallow texture features. Section 4 delves into the experimental
setup used in our study and discusses the results derived from these
experiments. Lastly, in Section 5, we offer concluding remarks that
encapsulate the primary findings and contributions of our research.

##  II Related Works

Over recent years, a multitude of shallow texture models and algorithms for
kinship verification have been introduced. These can be broadly categorized
into two main streams. The first encompasses methods employing established
feature descriptors such as HOG [16], [18], SIFT [19], LBP [19], and D-CBFD as
suggested by [20]. Typically, these techniques lean on low-level facial
features or combinations thereof for kinship verification. The second stream
zeroes in on creating straightforward yet distinctive metrics to ascertain if
two facial images possess a kinship link. Noteworthy contributions in this
realm include the NRML as proposed by Lu et al. [19], PDFL by Yan et al. [21],
and TSL [8], [17].

Recently, the Convolutional Neural Network (CNN) has also carved a niche for
itself in kinship verification. For example, Li et al. [22] put forth the
SMCNN, leveraging two identical CNNs supervised by a similarity metric-based
loss function. Another innovative technique, termed CNN-points, was introduced
by Zhang et al. [23]. Despite these methodologies showcasing encouraging
results, advancements in this domain are somewhat stymied. This is, in part,
due to data paucity and the still-evolving understanding of deep convolutional
networks.

Multilinear Subspace Learning (MSL) stands as a potent machine learning
technique, adept at discerning discriminant features from an array of feature
extraction methods, each operating at distinct scales [24, 25, 26]. It is
designed to uncover hidden patterns in expansive datasets, making it valuable
for discerning relationships amongst various variables. The integration of MSL
with tensor data has cemented its stature as a formidable approach for kinship
verification endeavors. Among the most pivotal algorithms bolstering kinship
verification, [10] showcased the Multilinear Side-information-based
Discriminant Analysis (MSIDA). MSIDA projects the input region tensor into a
novel multilinear subspace. This enhances the separation between samples of
different classes while minimizing the distance within samples of the same
class. Another notable algorithm in this spectrum is the Tensor Cross-View
Quadratic Analysis (TXQDA) [13]. TXQDA not only retains the intrinsic data
structure and augments the spacing between samples but also adeptly navigates
the pitfalls of limited sample sizes, all the while mitigating computational
overheads.

##  III Methodology

This section elucidates the architecture of the face kinship verification
system posited in our research, as depicted in Fig.1. The structure
encompasses four pivotal components: (A) Face Pre-processing, (B) Feature
Extraction, (C) Multilinear Subspace Learning, and (D) Matching and Fusion. We
delve into a detailed discourse of each phase in the ensuing sections.

Figure 1: Pipeline diagram of the proposed kinship verification and feature
extraction using LPQ and VGG16.

###  III-A Pre-processing

In the data pre-processing stage, we employ the MTCNN method [27] to detect
facial regions within images. Following this, the MSR algorithm [28] is
leveraged for image enhancement. The MSR algorithm amplifies the dynamic range
of images while preserving their color accuracy. Fig. 2 ilustrates an example
of test image processing :(a) original images;(b) MSR images.

Input: Image RGB

Output: Image MSR

Initialization:

//Input Processing

multiscaleRetinex(image)

Split the input image into its RGB channels

for each color channel do

$L\leftarrow\log(\text{image})$ Convert to logarithmic domain

for each scale do

$F_{s}\leftarrow\text{SurroundFunction}(L)$ $R_{s}\leftarrow L-F_{s}$
$W_{s}\leftarrow R_{s}\times w_{s}$ Weight the result according to scale
endfor

$R\leftarrow\exp\left(\sum_{s}R_{s}w_{s}\right)$ Sum all the weighted single
scale results and convert back to the linear domain

endfor

Merge all the color channels

Algorithm 1 Description of the proposed ViT Algorithm. Figure 2: Example of
test image processing :(a) original images;(b) MSR images.

short for Visual Geometry Group with 16 layers, is a pretrained convolutional
neural network (CNN) that is widely acclaimed for its exceptional accuracy in
image recognition tasks. It comprises 13 convolutional layers and 3 fully
connected layers, specifically the "fc6," "fc7," and "fc8" layers [29]. For
shallow feature extraction, we employ the Local Phase Quantization (LPQ)
descriptor [30], a well-regarded local texture descriptor. To optimize the
verification rate, features are extracted at various scales by adjusting
certain parameters, notably the window size $R$.

###  III-B Multilinear subspace learning using TXQDA+WCCN

In the offline training phase, the TXQDA+WCCN technique involves projecting
the training tensors X and Z into a novel discriminant subspace [15]. This
projection reduces the dimensions of both tensors, resulting in new dimensions
$\acute{\mathrm{I}}_{1}$x$\acute{\mathrm{I}_{2}}$ for mode-1 and mode-2,
respectively, where $\acute{\mathrm{I}}_{1}$x$\acute{\mathrm{I}_{2}}$ ${<<}$
$\mathrm{I}_{1}$x$\mathrm{I}_{2}$. Notably, the dimension of mode-3 remains
consistent, representing the individuals in the dataset. This method aims to
reduce the high dimensionality of the higher-order tensor, thereby producing a
new feature representation that augments inter-class distinctions while
diminishing high intra-class variability.

###  III-C Matching and Logistic Regression Fusion

Upon projection of the facial image data using the TXQDA+WCCN algorithm, the
matching process is executed by computing the cosine distance between two
vectors in the discriminant subspace [31, 32, 33]. To combine the scores
derived from both deep and shallow texture features, we employ a robust
technique known as Logistic Regression (LR) [34]. The choice of this fusion
technique is influenced by its proven efficacy in prior fusion studies [15,
35]. It enables us to harness the advantages of both feature types, leading to
enhanced performance in our facial image-matching system.

##  IV Experiments

In this section, we undertake a set of experiments to gauge the efficacy of
the proposed kinship verification system. We subject our system to tests using
three distinct datasets: Cornell Kin Face, UB Kin Face, and TS Kin Face. The
experimental results, gleaned from these datasets, are delineated in Tables I
through VI.

###  IV-A Benchmark Datasets

Cornell Kin Face [16]: The dataset being referred to consists of a total of
286 facial images, specifically related to 143 pairs of subjects. The facial
images in this dataset depict subjects with a frontal pose and a neutral
expression.

UB Kin Face [17]: This dataset consists of 600 images of 400 people, divided
into 200 groups of child-young parent (c-yp) and 200 groups of child-old
parent (c-op). this dataset is regarded as the first of its kind, presenting a
novel approach to the kinship verification problem, as it includes both young
and old face images of parents.

TS Kin Face [36]: The Tri-subject kinship face dataset consists of images
belonging to the child, mother, and father. The dataset comprises 513 images
in the Father, Mother, and Son group, as well as 502 images in the Father,
Mother, and Daughter group.

###  IV-B Parameter Settings

In our experiments, we utilize the 5-fold cross-validation protocol [36], [20]
to evaluate the performance of our approach. This protocol ensures that our
results can be directly compared to the state of the art in the field. Prior
to analysis, all face images in the datasets undergo pre-processing,
specifically the detection of the facial region using the MTCNN method.
Additionally, we employ MSR techniques to enhance the quality of the images.
Subsequently, we extract two distinct types of features, shallow Texture
feature and Deep feature. For shallow texture features, we utilize the LPQ
descriptor on the facial image, with a window size of R = 3, 4, 5, 6, 7, 8,
and 9. The facial image is subsequently partitioned into 12 blocks. Each block
is summed to create a histogram consisting of 256 bins. These individual
histograms are then concatenated, resulting in a final feature vector with a
size of (1 × 3072). For deep features, we extract them from the face image
with a size of 224 × 224 × 3. We utilize four layers from the VGG16 network,
specifically fc6,relu6, fc7, and relu7. The resulting features from these
layers are concatenated to form a feature matrix with a size of (4 × 4096).

###  IV-C Result analysis and discussion

The experimental results for the Cornell Kin Face, UB Kin Face, and TS Kin
Face datasets from our study are presented in Tables I-VI. Tables I, III, and
V show the mean accuracy when inputting the original images into our system,
with and without a histogram. Additionally, these tables illustrate the
outcomes when utilizing LPQ descriptors, both with and without preprocessing.
Tables II, IV, and VI display the mean accuracy of the LPQ descriptor across
its 7 scales (R= 3, 4, 5, 6, 7, 8, and 9). These tables also highlight the
performance on the fc6, relu6, fc7, and relu7 layers of the pre-trained VGG16
model. Furthermore, they provide details on the average accuracy achieved by
fusing the scores from the top-performing LPQ and VGG16 outcomes using LR
fusion. Figs. 3, 4 and 5 depict the ROC curves, showcasing the best results
our methodology secured across the three datasets.

TABLE I: The mean accuracy using LPQ with and without (histograms and MSR) on the Cornell Kin Face dataset. Settings | Mean Acc (%)
---|---
Without histogram | 54.51
With histogram | 59.34
Without MSR | 76.58
With MSR | 94.16
TABLE II: The mean accuracy using LPQ on Cornell Kin Face dataset with different scales and the LR fusion with VGG16. Method | Scales | Mean Acc (%)
---|---|---
| R=3 | 94.16
| R=4 | 92.72
LPQ | R=5 | 93.43
| R=6 | 93.39
| R=7 | 92.06
| R=8 | 93.09
| R=9 | 92.78
VGG16 | fc6,relu7, fc7 and relu7 | 91.02
LR Fusion | LPQ (R=3) + VGG16 | 95.18
Figure 3: ROC curves of the Shallow texture and Deep features on Cornell Dataset. TABLE III:  The mean accuracy using LPQ with and without (histograms and MSR) on the UB Kin Face dataset. Settings | c-yp | c-op | Mean
---|---|---|---
|  |  | Acc (%)
Without histogram | 55.28 | 56.00 | 55.64
With histogram | 60.75 | 60.58 | 60.67
Without MSR | 88.17 | 88.55 | 88.36
With MSR | 89.42 | 90.78 | 90.10
TABLE IV:  The mean accuracy using LPQ on UB Kin Face dataset with different scales and the LR fusion with VGG16. Method | Scales | c-yp | c-op | Mean
---|---|---|---|---
|  |  |  | Acc (%)
| R=3 | 89.42 | 90.78 | 90.10
| R=4 | 88.92 | 88.29 | 88.61
LPQ | R=5 | 84.23 | 86.55 | 85.39
| R=6 | 83.41 | 85.60 | 84.51
| R=7 | 82.43 | 83.36 | 82.90
| R=8 | 80.18 | 82.08 | 81.13
| R=9 | 80.93 | 83.08 | 82.01
VGG16 | fc6,relu6 | 88.39 | 86.21 | 87.30
| fc7, and relu7 |  |  |
LR Fusion | LPQ (R=3) | 91.16 | 91.52 | 91.34
| \+ VGG16 |  |  |

(a)

(b)

Figure 4: ROC curves of the Shallow texture and Deep features on UB Kin Face dataset: (a) c-yp set,(b)c-op set. TABLE V:  The mean accuracy using LPQ with and without (histograms and MSR) on the TS Kin Face dataset. Settings | FS | FD | MS | MD | Mean
---|---|---|---|---|---
|  |  |  |  | Acc (%)
Without histogram | 54.46 | 54.26 | 53.76 | 52.97 | 53.86
With histogram | 67.09 | 64.13 | 66.91 | 66.97 | 66.27
Without MSR | 79.41 | 77.03 | 81.68 | 81.98 | 80.03
With MSR | 85.14 | 87.13 | 86.83 | 88.02 | 86.78
TABLE VI:  The mean accuracy using LPQ on the TS Kin Face dataset with different scales and the LR fusion with VGG16. Method | Scales | FS | FD | MS | MD | Mean
---|---|---|---|---|---|---
|  |  |  |  |  | Acc (%)
| R=3 | 85.94 | 86.93 | 87.23 | 88.22 | 87.08
| R=4 | 87.43 | 86.13 | 87.81 | 86.92 | 87.07
LPQ | R=5 | 86.83 | 86.24 | 86.83 | 86.53 | 86.60
| R=6 | 87.41 | 86.03 | 87.91 | 86.52 | 86.97
| R=7 | 85.14 | 85.01 | 87.97 | 86.26 | 86.09
| R=8 | 84.93 | 82.89 | 84.91 | 85.13 | 84.46
| R=9 | 85.36 | 82.34 | 83.56 | 84.50 | 83.94
VGG16 | fc6,relu6 | 77.38 | 78.12 | 79.32 | 79.70 | 78.63
| fc7 and relu7 |  |  |  |  |
LR | LPQ (R=3) | 90.30 | 91.49 | 93.17 | 92.28 | 91.81
Fusion | \+ VGG16 |  |  |  |  |

(a)

(b)

(c)

(d)

Figure 5: ROC curves of the Shallow texture and Deep features on TS Kin Face
dataset: (a)F–S set, (b) F–D set, (c) M–S set, and (d) M–D.

###  IV-D Discussion

Based on experiments with our proposed approach, which leverages fusion from
two types of features (Deep and Shallow texture), across three datasets
(Cornell Kin Face, UB Kin Face, and TS Kin Face), we draw the following
conclusions:

  * •

Integrating the image histogram markedly improves the precision of metric
evaluations. The inclusion of the histogram enhanced our system’s accuracy by
4.83% for the Cornell dataset, 5.03% for the UB dataset, and 12.41% for the TS
dataset. Additionally, the benefits of employing the MSR-based preprocessing
technique were evident. This preprocessing step elevated the accuracy rates by
15.43%, 1.74%, and 6.75% on the Cornell, UB, and TS Kin Face datasets,
respectively.

  * •

Our findings underscore the superior performance of LR fusion compared to
utilizing individual feature types. In this study, we employed score-level
fusion that amalgamates scores generated by the CNN-based VGG16 and the LPQ
descriptor. Leveraging the LR fusion method, we realized remarkable accuracy
rates: 95.18% for the Cornell Kin Face dataset, 91.34% for the UB Kin Face
dataset, and 92.81% for the TS Kin Face dataset. Detailed outcomes can be
found in Tables II, IV, and VI.

###  IV-E Comparison against the state of the art

The effectiveness of our proposed method, which involves fusing LPQ and VGG16
scores using the LR fusing technique, is compared to more modern methods in
Table VII for the Cornell Kin Face, UB Kin Face, and TS Kin Face datasets. The
comparison clearly shows that our proposed technique outperforms the recent
state-of-the-art methods on the three datasets.

TABLE VII: Performance comparison of kinship verification state of the art on Cornell Kin Face, UB Kin Face, and TS Kin Face datasets Algorithm | year | Cornell dataset | UB dataset | TS dataset
---|---|---|---|---
MSIDA [10] | 2019 | 86.87 | 83.34 | 85.18
FMRE2 [37] | 2021 | 84.16 | 85.03 | 90.85
AdvKin [9] | 2021 | 81.40 | 75.00 | -
BC2DA [38] | 2022 | 83.07 | 83.30 | 83.55
TXQEDA [4] | 2022 | 93.77 | - | 90.68
MLDPL [39] | 2023 | - | 87.90 | 92.40
Proposed | 2023 | 95.18 | 91.34 | 91.81

##  V Conclusion

This study presents a kinship verification system that leverages a novel and
efficient facial description method. This method harnesses the power of
Logistic Regression (LR) fusion between deep and shallow texture features.
Additionally, the system is enhanced with the integration of Multiscale
Retinex (MSR), addressing challenges related to contrast, lighting, and noise.
This enhancement boosts image quality, ultimately leading to superior
performance. Employing tensor subspace learning, our method showcases notable
results. The system’s efficacy is further amplified by applying LR fusion at
the score level of LPQ combined with a pre-trained VGG16. Our results suggest
that deep and handcrafted texture attributes synergize effectively at the
score level, with the fusion substantially elevating kinship verification
accuracy.

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# The stable rank of $\mathbb{Z}[x]$ is $3$

Luc Guyot

###### Abstract.

Grunewald, Mennicke and Vaserstein proved with [GMV94, Proposition 1.9] that
the Bass stable rank of $\mathbb{Z}[x]$, the ring of the univariate
polynomials over $\mathbb{Z}$, is $3$. This note addresses minor errors found
in their proof. Using their method, we show in addition that the unimodular
row $(1+x,12,x^{2}+16)$ is not stable.

###### Key words and phrases:

Bass stable rank; univariate polynomial ring; special Whitehead group;

Rings are assumed to be commutative and unital. Let $R$ be a ring. A row
$(r_{1},\dots,r_{n})\in R^{n}$ is _unimodular_ if $\sum_{i}Rr_{i}=R$. We
denote by $\operatorname{Um}_{n}(R)$ the set of unimodular rows of size $n$. A
row $(r_{1},\dots,r_{n+1})\in\operatorname{Um}_{n+1}(R)$ ($n>0$) is _stable_
if there is $(s_{1},\dots,s_{n})\in R^{n}$ such that

$(r_{1}+s_{1}r_{n+1},\dots,r_{n}+s_{n}r_{n+1})$

belongs to $\operatorname{Um}_{n}(R)$. An integer $n>0$ lies in the _stable
range_ of $R$ if every row in $\operatorname{Um}_{n+1}(R)$ is stable. If $n$
lies in the stable range of $R$, then so does $k$ for every $k>n$ [MR87, Lemma
11.3.3]. The _Bass stable rank $\operatorname{sr}(R)$ of $R$_ is the least
integer in the stable range of $R$. Grunewald, Mennicke and Vaserstein proved
the following:

######  Theorem A.

([GMV94, Proposition 1.9]) $\operatorname{sr}(\mathbb{Z}[x])=3$.

The authors claimed that the row $(21+4x,12,x^{2}+20)$ is unimodular but not
stable. This would be enough to establish Theorem A since
$\operatorname{sr}(\mathbb{Z}[x])\leq 3$. Indeed, we have
$\operatorname{sr}(\mathbb{Z}[x])\leq\dim_{\text{Krull}}(\mathbb{Z}[x])+1$
[MR87, Corollary 6.7.4] and $\dim_{\text{Krull}}(\mathbb{Z}[x])=2$ [MR87,
Proposition 6.5.4]. However, it is unknown if the previous row is stable:
because of a typographical error one should read $(21+2x,12,x^{2}+20)$ instead
of $(21+4x,12,x^{2}+20)$. Besides [GMV94, Proof of Proposition 1.9, page 191]
uses the false statement “$\operatorname{SK}_{1}(\mathcal{O},2\mathcal{O})\neq
1$” where $\mathcal{O}=\mathbb{Z}+\mathbb{Z}\sqrt{-5}$.

In Section 2, we present a proof of Theorem A which addresses these errors.
Our proof follows closely the lines of the original. It consists in exhibiting
a quotient $R$ of $\mathbb{Z}[x]$ such that $\operatorname{SK}_{1}(R)\neq
1$.This suffices to show that $\operatorname{sr}(\mathbb{Z}[x])\neq 2$.
Indeed, if $\operatorname{sr}(\mathbb{Z}[x])=2$, then the natural map
$\operatorname{SK}_{1}(\mathbb{Z}[x])\rightarrow\operatorname{SK}_{1}(R)$
would be surjective (Corollary 1.6 below), which is impossible as
$\operatorname{SK}_{1}(\mathbb{Z}[x])=1$ [BHS64, Theorem 1].

In Section 3, we show in addition

######  Proposition B.

The unimodular row $(1+x,12,x^{2}+16)$ of $\mathbb{Z}[x]$ is not stable.

#### Ackowledgements.

This note originates from stimulating discussions with Tsit Yuen Lam regarding
the results of [GMV94] and related topics. I am extremely thankful to him for
his continuous encouragements and his kind support.

##  1\. Bass’s stable rank and surjective homorphisms

In this section, we introduce notation and definitions. We prove subsequently
Corollary 1.6, which is a simple, but important argument in the proof of
Theorem A

###  1.1. Bass’s stable rank

The Bass stable rank can be characterized in terms of a lifting property for
unimodular rows of quotient rings.

######  Proposition 1.1.

Let $R$ be a commutative and unital ring. Then the following are equivalent.

  * $(i)$

$\operatorname{sr}(R)\leq n$.

  * $(ii)$

For every ideal $I\subseteq R$, the map
$\operatorname{Um}_{n}(R)\rightarrow\operatorname{Um}_{n}(R/I)$ sending
$(r_{1},\dots,r_{n})$ to $(r_{1}+I,\dots,r_{n}+I)$ is surjective.

The proof is straightforward, hence left to the reader. Specializing $n$ to
$1$, we obtain:

######  Proposition 1.2.

([EO67, Lemma 6.1]) Let $R$ be a commutative and unital ring. Then the
following are equivalent.

  * $(i)$

$\operatorname{sr}(R)\leq 1$.

  * $(ii)$

For every ideal $I\subseteq R$, the natural map
$R^{\times}\rightarrow(R/I)^{\times}$ is surjective.

The condition $\operatorname{sr}(R)\leq 1$ received special attention in
[Vas84]. We shall see with Proposition 1.5 below that the condition
$\operatorname{sr}(R)\leq 2$ can also be interpreted in terms of surjective
group homomorphisms. We denote by $\operatorname{Jac}(R)$, the _Jacobson
radical of $R$_, that is, the intersection of the maximal ideals of $R$.

######  Proposition 1.3.

[MR87, Proposition 11.3.11 and Lemma 11.4.6] Let $I$ be an ideal of $R$. Then
we have $\operatorname{sr}(R/I)\leq\operatorname{sr}(R)$ and equality holds if
$I\subseteq\operatorname{Jac}(R)$.

###  1.2. Rings of stable rank at most $2$

######  Lemma 1.4.

([GMV94, Lemma 6.2], [EO67, Corollary 8.3]) Let $R$ be a ring. Let
$(a,b,c)\in\operatorname{Um}_{3}(R)$ and denote by $\overline{r}$ the image of
$r\in R$ in $R/Rc$. Then the following are equivalent:

  * $(i)$

The row $(a,b,c)$ is stable.

  * $(ii)$

Every matrix $\begin{pmatrix}\overline{a}&\overline{b}\\\
*&*\end{pmatrix}\in\operatorname{SL}_{2}(R/Rc)$ has a lift in
$\operatorname{SL}_{2}(R)$.

###### Proof.

$(i)\Rightarrow(ii)$. Let $\begin{pmatrix}\overline{a}&\overline{b}\\\
\overline{d}&\overline{e}\end{pmatrix}\in\operatorname{SL}_{2}(R/Rc)$. Since
$(a,b,c)$ is stable we can find $(r,s)\in\operatorname{Um}_{2}(R)$ such that
$\overline{r}=\overline{a}$ and $\overline{s}=\overline{b}$. Let $u.v\in R$ be
such that $\begin{pmatrix}r&s\\\ u&v\end{pmatrix}\in\operatorname{SL}_{2}(R)$.
Let $A=\begin{pmatrix}\overline{a}&\overline{b}\\\
*&*\end{pmatrix}\in\operatorname{SL}_{2}(\overline{R})$. Set
$\overline{E}\Doteq\begin{pmatrix}\overline{a}&\overline{b}\\\
\overline{u}&\overline{v}\end{pmatrix}A^{-1}$. Then
$\overline{E}=\begin{pmatrix}1&0\\\ \overline{w}&1\end{pmatrix}$ for some
$w\in R$. Then $\begin{pmatrix}1&0\\\
w&1\end{pmatrix}^{-1}\begin{pmatrix}r&s\\\ u&v\end{pmatrix}$ is a lift of $A$.

$(ii)\Rightarrow(i)$. By assumption, we can find
$(r,s)\in\operatorname{Um}_{2}(R)$ such that $r=a+\lambda c$ and $s=b+\mu c$
for some $\lambda,\mu\in R$. It follows immediately that $(a,b,c)$ is stable.
∎

######  Proposition 1.5.

Let $R$ be a ring. Then the following are equivalent:

  * $(i)$

$\operatorname{sr}(R)\leq 2$.

  * $(ii)$

The natural map
$\operatorname{SL}_{2}(R)\rightarrow\operatorname{SL}_{2}(\overline{R})$ is
surjective for every quotient $\overline{R}$ of $R$.

  * $(iii)$

The natural map
$\operatorname{SL}_{n}(R)\rightarrow\operatorname{SL}_{n}(\overline{R})$ is
surjective for every quotient $\overline{R}$ of $R$ and every $n\geq 2$.

We denote by $\operatorname{E}_{n}(R)$ the subgroup of
$\operatorname{SL}_{n}(R)$ generated by the _elementary matrices_ , i.e., the
matrices which differ from the identity matrix by a single off-diagonal entry.

###### Proof of Proposition 1.5.

Clearly, we have $(iii)\Rightarrow(ii)$. The implication $(ii)\Rightarrow(i)$
is given by Lemma 1.4. Hence it only remains to show that
$(i)\Rightarrow(iii)$.

Assume that $(i)$ holds and let $n\geq 2$,
$A\in\operatorname{SL}_{n}(\overline{R})$ where $\overline{R}=R/I$ for some
ideal $I$ of $R$. If $n=2$, then it follows from Lemma 1.4 that $A$ has a lift
in $\operatorname{SL}_{2}(R)$. Indeed, write $A=\begin{pmatrix}a+I&b+I\\\
d+I&e+I\end{pmatrix}\in\operatorname{SL}_{2}(\overline{R})$. As
$(a+I,b+I)\in\operatorname{Um}_{2}(\overline{R})$, there is $c\in R$ such that
$(a,b,c)\in\operatorname{Um}_{3}(R)$ and
$\widetilde{A}=\begin{pmatrix}a+Rc&b+Rc\\\
d+Rc&e+Rc\end{pmatrix}\in\operatorname{SL}_{2}(R/Rc)$. By hypothesis, we have
$\operatorname{sr}(R)\leq 2$, so that Lemma 1.4 applies and provides us with a
lift of $\widetilde{A}$ in $\operatorname{SL}_{2}(R)$ which is also a lift of
$A$.

We can now assume that $n>2$ and proceed by induction on $n$. Since
$\operatorname{sr}(\overline{R})\leq 2$ by Proposition 1.3, and because the
first row of $A$ is unimodular, we can find
$E,E^{\prime}\in\operatorname{E}_{n}(\overline{R})$ such that
$A=E\begin{pmatrix}1&0\\\ 0&A^{\prime}\end{pmatrix}E^{\prime}$ with
$A^{\prime}\in\operatorname{SL}_{n-1}(\overline{R})$. By induction hypothesis,
the matrix $A^{\prime}$ has a lift in $\operatorname{SL}_{n-1}(R)$. Clearly,
the matrices $E$ and $E^{\prime}$ lift to $\operatorname{SL}_{n}(R)$. Thus $A$
has a lift in $\operatorname{SL}_{n}(R)$. ∎

The next result refers to the special Whitehead group
$\operatorname{SK}_{1}(R)$ of a ring $R$. We define this group as follows. Let
$\operatorname{E}(R)\Doteq\bigcup_{n}\operatorname{E}_{n}(R)$ and
$\operatorname{SL}(R)\Doteq\bigcup_{n}\operatorname{SL}_{n}(R)$ be the
ascending unions for which the embeddings
$\operatorname{E}_{n}(R)\rightarrow\operatorname{E}_{n+1}(R)$ and
$\operatorname{SL}_{n}(R)\rightarrow\operatorname{SL}_{n+1}(R)$ are defined
through $A\mapsto\begin{pmatrix}A&0\\\ 0&1\end{pmatrix}$. Then
$\operatorname{E}(R)$ is a normal subgroup of $\operatorname{SL}(R)$ [Mag02,
Whitehead Lemma 9.7] and the _special Whitehead group_
$\operatorname{SK}_{1}(R)$ of $R$ is the quotient group
$\operatorname{SL}(R)/\operatorname{E}(R)$.

######  Corollary 1.6.

Let $R$ be a ring satisfying $\operatorname{sr}(R)\leq 2$. Let $\overline{R}$
be a quotient of $R$. Then the natural map
$\operatorname{SK}_{1}(R)\rightarrow\operatorname{SK}_{1}(\overline{R})$ is
surjective.

$\square$

Following [Coh66], we call a ring $R$ a _$\operatorname{GE}_{2}$ -ring_ if
$\operatorname{SL}_{2}(R)=\operatorname{E}_{2}(R)$.

The next corollaries are of independent interest.

######  Corollary 1.7.

Let $R$ be a $\operatorname{GE}_{2}$-ring satisfying $\operatorname{sr}(R)\leq
2$. Then every quotient of $R$ is a $\operatorname{GE}_{2}$-ring.

$\square$

######  Remark 1.8.

Corollary 1.7 provides a straightforward proof of [Guy18, Theorem A]. However
[Guy18, Lemma 3], which implies [Guy18, Theorem A], doesn’t follow from
Corollary 1.7.

######  Corollary 1.9.

[McG08, Theorem 3.6] Let $R$ be a ring such that any proper quotient of
$R/\mathcal{J}(R)$ has stable rank $1$. Then $\operatorname{sr}(R)\leq 2$.

###### Proof.

It is easy to check that rings of stable rank $1$ are
$\operatorname{GE}_{2}$-rings. The result follows by combining Proposition 1.5
with Proposition 1.3. ∎

###  1.3. The groups $\operatorname{SK}_{1}(R,I)$, $\operatorname{K}_{2}(R)$
and the relative sequence

In this section, we define the relative special Whitehead group
$\operatorname{SK}_{1}(R,I)$ for $I$ an ideal of a ring $R$ and the group
$\operatorname{K}_{2}(R)$ from algebraic K-theory. We describe then an exact
sequence involving these groups called the _relative sequence_. This sequence
will come in handy for the $K$-theoretical computations of Section 2.

Let $R$ be a ring. Let $n\geq 2$ and denote by $I_{n}$ the $n\times n$
identify matrix. For $1\leq i,j\leq n$, let $\epsilon_{ij}$ be the $n\times n$
matrix whose $(i,j)$ entry is $1$ and whose other entries are zero. For $a\in
R$ and $i\neq j$, we set $e_{ij}(a)=I_{n}+a\epsilon_{ij}$ and call any such
matrix an _elementary matrix_.

Recall that $\operatorname{E}_{n}(R)$ is the subgroup of
$\operatorname{SL}_{n}(R)$ generated by the elementary matrices. Let $I$ be an
ideal of $R$. We denote by $\operatorname{E}_{n}(R,I)$ the normal subgroup of
$\operatorname{E}_{n}(R)$ which is normally generated by the matrices
$e_{ij}(a)$ with $a\in I$, $1\leq i\neq j\leq n$. We denote by
$\operatorname{SL}_{n}(R,I)$ the kernel of the natural map
$\operatorname{SL}_{n}(R)\rightarrow\operatorname{SL}_{n}(R/I)$.

The _relative special Whitehead group $\operatorname{SK}_{1}(R,I)$_ is defined
as follows. Let
$\operatorname{E}(R,I)\Doteq\bigcup_{n}\operatorname{E}_{n}(R,I)$ and
$\operatorname{SL}(R,I)\Doteq\bigcup_{n}\operatorname{SL}_{n}(R,I)$ be the
ascending unions for which the embeddings
$\operatorname{E}_{n}(R,I)\rightarrow\operatorname{E}_{n+1}(R,I)$ and
$\operatorname{SL}_{n}(R,I)\rightarrow\operatorname{SL}_{n+1}(R,I)$ are
defined through $A\mapsto\begin{pmatrix}A&0\\\ 0&1\end{pmatrix}$. Then
$\operatorname{E}(R,I)$ is a normal subgroup of $\operatorname{SL}(R,I)$
[Mag02, Relative Whitehead Lemma 11.1] and
$\operatorname{SK}_{1}(R,I)\Doteq\operatorname{SL}(R,I)/\operatorname{E}(R,I)$.

We now turn to the definition of $\operatorname{K}_{2}(R)$. The _Steinberg
group_ $\operatorname{St}_{n}(R)$ is the group with generators $x_{ij}(r)$,
with $i\neq j$, $1\leq i,j\leq n$ and $r\in R$ subject to the defining
relations:

(1) $\begin{split}x_{ij}(r)x_{ij}(r)=x_{ij}(r+s),\\\
[x_{ij}(r),x_{kl}(s)]=\left\\{\begin{array}[]{c}1,\text{if }i\neq l,j\neq
k,\\\ x_{il},\text{ if }i\neq l,j=k.\end{array}\right.\end{split}$

Removing the restrictions $i\leq n$ and $j\leq n$ on the generators
$x_{ij}(r)$, the same presentation defines the _Steinberg group_
$\operatorname{St}(R)$. Because the elementary matrices $e_{ij}(r)$ obey the
above standard relations and generate $\operatorname{E}(R)$, the map
$x_{ij}(r)\mapsto e_{ij}(r)$ induces a surjective group homomorphism
$\operatorname{St}(R)\twoheadrightarrow\operatorname{E}(R)$. The kernel of
this homomorphism is $\operatorname{K}_{2}(R)$.

Finally, we introduce the so-called _relative sequence_ :

(2)
$\operatorname{K}_{2}(R/I)\xrightarrow{\partial_{1}}\operatorname{SK}_{1}(R,I)\rightarrow\operatorname{SK}_{1}(R)\rightarrow\operatorname{SK}_{1}(R/I)$

where the second and third arrows are induced respectively by the inclusion
$\operatorname{SL}(R,I)\subseteq\operatorname{SL}(R)$ and the natural map
$\operatorname{SL}(R)\rightarrow\operatorname{SL}(R/I)$. For the connecting
homomorphism $\partial_{1}$, we refer the reader to [Mag02, Theorem 13.20 and
Example 13.22] where this sequence is shown to be exact.

##  2\. Proof of Theorem A

We shall establish

######  Proposition 2.1.

$\operatorname{SK}_{1}(\mathbb{Z}+4i\mathbb{Z})\neq 1$ where $i=\sqrt{-1}$.

As outlined in the introduction, Theorem A immediately follows from the
combination of Proposition 2.1 with Corollary 1.6 and

######  Theorem 2.2.

[BHS64, Theorem 1] $\operatorname{SK}_{1}(\mathbb{Z}[x])=1$.

Our proof of Proposition 2.1 relies on

######  Lemma 2.3.

Let $I$ be an ideal of a ring $R$ such that $R/I\simeq\mathbb{Z}/n\mathbb{Z}$
for some $n\geq 0$. Then the natural map

(3) $\operatorname{SK}_{1}(R,I)\rightarrow\operatorname{SK}_{1}(R)$

induced by the inclusion $\operatorname{SL}(R,I)\subseteq\operatorname{SL}(R)$
is an isomorphism.

###### Proof.

In the exact sequence (2) introduced in Section 1.3

$\operatorname{K}_{2}(R/I)\xrightarrow{\partial_{1}}\operatorname{SK}_{1}(R,I)\rightarrow\operatorname{SK}_{1}(R)\rightarrow\operatorname{SK}_{1}(R/I)$

the last term, namely $\operatorname{SK}_{1}(R/I)$, is trivial since $R/I$ is
Artinian and hence of stable rank $1$ [Bas64, Corollary 10.5]. In addition,
the image of $\operatorname{K}_{2}(R/I)$ in $\operatorname{SK}_{1}(R,I)$ is
also trivial. Indeed $\operatorname{K}_{2}(R/I)$ is generated by the Steinberg
symbol $\\{-1+I,-1+I\\}_{R/I}$ because $R/I\simeq\mathbf{Z}/n\mathbf{Z}$
[Mag02, Exercises 12.B.6 and 13A.9]. As $\\{-1,-1\\}_{R}$ is a lift of the
previous symbol in $\operatorname{K}_{2}(R)$ our claim follows from the
definition of $\partial_{1}$ [Mag02, Theorem 13.20], which completes the
proof. ∎

###### Proof of Proposition 2.1.

Let $S=\mathbb{Z}[i],I=4S$ and $R=\mathbb{Z}+I$. By Lemma 2.3 we have
$\operatorname{SK}_{1}(R)\simeq\operatorname{SK}_{1}(R,I)$. The inclusion
$R\subset S$ induces a surjective group homomorphism

(4) $\operatorname{SK}_{1}(R,I)\twoheadrightarrow\operatorname{SK}_{1}(S,I).$

As $\operatorname{SK}_{1}(S,I)\simeq\mathbb{Z}/2\mathbb{Z}$ by the Bass-
Milnor-Serre Theorem [Mag02, Theorem 11.33] (Theorem 3.1 below), we conclude
that $\operatorname{SK}_{1}(R)\neq 1$. ∎

######  Remark 2.4.

As an alternative to Proposition 2.1, one can show that
$\operatorname{SK}_{1}(\mathbb{Z}+3\zeta_{3}\mathbb{Z})\neq 1$ where
$\zeta_{3}=e^{\frac{2i\pi}{3}}$. This is a direct consequence of [Swa71, Lemma
3.2 and subsequent remark] and Lemma 2.3.

##  3\. Proof of Proposition B

We shall prove that the unimodular row

$(1+x,12,x^{2}+16)\in\operatorname{Um}_{3}(\mathbb{Z}[x])$

is not stable. We apply the strategy devised in [GMV94, Proof of Proposition
1.9]: we look for an explicit matrix in $\operatorname{SL}_{2}(R)$ that
defines a non-trivial element of $\operatorname{SK}_{1}(R)$ for a suitable
quotient $R$ of $\mathbb{Z}[x]$. To do so, we resort to the Bass-Milnor-Serre
Theorem [BMS67, Theorems 3.6 and 4.1] and its description of
$\operatorname{SK}_{1}(S)$ for $S$ the ring of integers of a totally imaginary
number field, in terms of power residue symbols. The following definitions are
required to state the latter theorem.

Let $m>0$ be a rational integer. We denote by $\mu_{m}$ the group of $m$-th
roots of unity in the field of complex numbers. Let $S$ be the ring of
integers of a number field and suppose that $S$ contains $\mu_{m}$ for some
$m>0$. For $b\in S$ and $\mathfrak{a}$ an ideal of $S$ such that
$\mathfrak{a}+Sbm=S$, define the _$m$ -th power residue symbol_

$\left(\frac{b}{\mathfrak{a}}\right)_{m}\Doteq\prod_{\mathfrak{p}|\mathfrak{a}}\left(\frac{b}{\mathfrak{p}}\right)_{m}^{\operatorname{ord}_{\mathfrak{p}}(\mathfrak{a})}$

where $\mathfrak{p}$ ranges in the set of prime ideals of $S$ dividing
$\mathfrak{a}$ and where $\left(\frac{b}{\mathfrak{p}}\right)_{m}$ is the
unique element of $\mu_{m}$ satisfying

$b^{\frac{q-1}{m}}\equiv\left(\frac{b}{\mathfrak{p}}\right)_{m}\mod\mathfrak{p}$

[BMS67, Appendix on number theory, pages 86 and 89] [Mag02, Theorem 11.33 and
Proposition 15.40]. If $\mathfrak{a}=Sa$ with $a\in S$, we simply write
$\left(\frac{b}{a}\right)_{m}$.

######  Theorem 3.1.

[BMS67, Theorems 3.6 and 4.1] Let $S$ be the ring integers of a totally
imaginary number field. Let $m$ denote the number of roots of unity in $S$. If
$I$ is a non-zero ideal of $S$, define the divisor $r=r(I)$ of $m$ by
$\operatorname{ord}_{p}(r)=j_{p}(I)$ where $j_{p}(I)$ is the nearest integer
in the interval $[0,\operatorname{ord}_{p}(m)]$ to

$\min_{\mathfrak{p}}\left\lfloor\frac{\operatorname{ord}_{\mathfrak{p}}(I)}{\operatorname{ord}_{\mathfrak{p}}(Sp)}-\frac{1}{p-1}\right\rfloor$

and where $\lfloor x\rfloor$ denotes the greatest integer $\leq x$ and
$\mathfrak{p}$ ranges over the prime ideals of $S$ containing $p$.

Then the map $\begin{pmatrix}a&b\\\
*&*\end{pmatrix}\in\operatorname{SL}_{2}(S,I)\mapsto(\frac{b}{a})_{r}$ induces
an isomorphism from $\operatorname{SK}_{1}(S,I)$ onto $\mu_{r}$.

###### Proof of Proposition B.

Let $S=\mathbb{Z}[i],I=4S$ and let $R=\mathbb{Z}+I$. By Theorem 3.1, the power
residue symbol $\binom{12}{1+4i}_{2}=-1$ yields a matrix
$\begin{pmatrix}1+4i&12\\\ *&*\end{pmatrix}\in\operatorname{SL}_{2}(S,I)$
which defines a non-trivial element of $\operatorname{SK}_{1}(S,I)$. The same
matrix can be lifted from $\operatorname{SK}_{1}(S,I)$ to
$\operatorname{SK}_{1}(R,I)$ via (4), and certainly maps to a non-trivial
element of $\operatorname{SK}_{1}(R)$ via the isomorphism (3).

Considering the surjective ring homomorphism from $\mathbb{Z}[x]$ onto $R$
induced by $x\mapsto 4i$, we infer from Theorem 2.2 and Lemma 1.4 that the row
$(1+x,12,x^{2}+16)$ cannot be stable. ∎

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