# Measurement and Simulation of the Magnetic Fields from a 555 Timer
Integrated Circuit using a Quantum Diamond Microscope and Finite Element
Analysis

P. Kehayias<EMAIL_ADDRESS>Sandia National Laboratories, Albuquerque, New
Mexico 87185, USA  E. V. Levine  The MITRE Corporation, Bedford, Massachusetts
01730, USA  Quantum Technology Center, University of Maryland, College Park,
MD 20742, USA  Department of Physics, Harvard University, Cambridge, MA 02138,
USA  L. Basso  Center for Integrated Nanotechnologies, Sandia National
Laboratories, Albuquerque, New Mexico 87123, USA  J. Henshaw  Center for
Integrated Nanotechnologies, Sandia National Laboratories, Albuquerque, New
Mexico 87123, USA  M. Saleh Ziabari  Center for Integrated Nanotechnologies,
Sandia National Laboratories, Albuquerque, New Mexico 87123, USA  University
of New Mexico Department of Physics and Astronomy, Albuquerque, New Mexico
87131  M. Titze  Sandia National Laboratories, Albuquerque, New Mexico 87185,
USA  R. Haltli  Sandia National Laboratories, Albuquerque, New Mexico 87185,
USA  J. Okoro  Sandia National Laboratories, Albuquerque, New Mexico 87185,
USA  D. R. Tibbetts  Sandia National Laboratories, Albuquerque, New Mexico
87185, USA  D. M. Udoni  Sandia National Laboratories, Albuquerque, New Mexico
87185, USA  E. Bielejec  Sandia National Laboratories, Albuquerque, New Mexico
87185, USA  M. P. Lilly  Center for Integrated Nanotechnologies, Sandia
National Laboratories, Albuquerque, New Mexico 87123, USA  T.-M. Lu  Sandia
National Laboratories, Albuquerque, New Mexico 87185, USA  Center for
Integrated Nanotechnologies, Sandia National Laboratories, Albuquerque, New
Mexico 87123, USA  P. D. D. Schwindt  Sandia National Laboratories,
Albuquerque, New Mexico 87185, USA  A. M. Mounce  Center for Integrated
Nanotechnologies, Sandia National Laboratories, Albuquerque, New Mexico 87123,
USA

###### Abstract

Quantum Diamond Microscope (QDM) magnetic field imaging is an emerging
interrogation and diagnostic technique for integrated circuits (ICs). To date,
the ICs measured with a QDM were either too complex for us to predict the
expected magnetic fields and benchmark the QDM performance, or were too simple
to be relevant to the IC community. In this paper, we establish a 555 timer IC
as a “model system” to optimize QDM measurement implementation, benchmark
performance, and assess IC device functionality. To validate the magnetic
field images taken with a QDM, we used a SPICE electronic circuit simulator
and Finite Element Analysis (FEA) to model the magnetic fields from the 555
die for two functional states. We compare the advantages and the results of
three IC-diamond measurement methods, confirm that the measured and simulated
magnetic images are consistent, identify the magnetic signatures of current
paths within the device, and discuss using this model system to advance QDM
magnetic imaging as an IC diagnostic tool.

\begin{overpic}[width=325.215pt]{NVfigCompileV1h.pdf} \put(0.0,97.0){{\Large
a}} \put(100.0,97.0){{\Large b}} \put(100.0,63.0){{\Large c}}
\put(100.0,30.0){{\Large d}} \end{overpic} Figure 1:  (a) Schematic of an NV
epifluorescence microscope setup measuring a 555 die (microwave wire and bias
$B_{111}$ magnetic field not shown). (b) NV center in the diamond lattice,
with the arrow showing the diamond [111] direction. (c) Energy level diagram
of the NV ground-state magnetic sublevels, indicating the Zeeman effect and
the zero-field splitting ($\sim$2870 MHz). (d) Example ODMR spectrum for
diamond Sample A. Each lineshape is split into three peaks due to the 14N
hyperfine interaction.

##  I Introduction

Mapping the magnetic fields from electric current distributions in an
integrated circuit (IC) is a powerful noninvasive probing technique. Magnetic
fields can provide information about the IC components, layout, and materials,
as well as device function, fault locations, secure information leakage, and
possible malicious hardware modifications (Trojans or counterfeits) [1, 2, 3].
Advances in device fabrication and packaging technologies have increased the
IC complexity, requiring diagnostic techniques that can image devices with
weaker electric currents and denser layouts, including devices with multiple
conducting layers and 3D die packaging.

The Quantum Diamond Microscope (QDM) is emerging as a promising IC diagnostic
tool [4, 5, 6], allowing for non-destructive, high-resolution, wide-area
magnetic field imaging of devices that is an alternative to scanning
techniques such as superconducting quantum interference device (SQUID)
microscopy, scanning giant magnetoresistance (GMR) microscopy, magnetic force
microscopy (MFM), or scanning magnetic tunnel junction (MTJ) microscopy [7, 8,
9, 10]. The QDM uses a layer of magnetically-sensitive nitrogen-vacancy (NV)
color centers in diamond to image the magnetic fields from a nearby IC die
[4]. QDM magnetic field imaging has been used to measure state-dependent
magnetic fingerprints of a field-programmable gate array (FPGA), backside
imaging of a flip-chip device, and hardware Trojan detection [11, 12, 13].
Advancement of the QDM technique as an IC diagnostic tool will benefit from a
well-characterized system for benchmarking and optimizing sensor performance.
To date, systems studied by the QDM have either been too simple indicate how
the QDM will perform in operational setting with ICs [14, 15] or too complex
to model the detected magnetic fields and the information they contain [11,
12, 13].

In this paper, we present experimental and computational results that map and
simulate, respectively, the magnetic fields from a commercial bipolar junction
transistor (BJT) 555 timer IC to benchmark and gauge QDM performance (such as
magnetic sensitivity and spatial resolution). The 555 is an ideal “model
system” IC for QDM assessment, since it has $\sim$10-15 $\upmu$m features that
are sufficiently coarse to fully resolve, is simple enough to fully simulate,
and is also among the most widely manufactured ICs [16].

We used a SPICE electronic circuit simulator (PSPICE) and multiphysics Finite
Element Analysis software (COMSOL) to simulate the current densities and
magnetic fields of the 555 die for two functional states. We measured the
magnetic fields of these two states using a QDM, achieving micron-scale
spatial resolution and few-$\upmu$T magnetic sensitivity in a 1$\times$1
$\upmu$m2 pixel after 1 s. Comparing the measured and the simulated magnetic
maps, we confirmed that the QDM measured the expected magnetic field
distributions, and assessed the QDM performance for three IC-diamond
measurement configurations. Analyzing the measured and simulated magnetic
maps, we identified key features and current paths in the magnetic maps that
correspond to the operational subsystems and internal states of the IC,
showing how QDM measurements can characterize the current densities and
internal-state information of an IC. Finally, we discuss how our results apply
to current mapping and failure analysis in other devices.

\begin{overpic}[width=411.93767pt]{fig2revisedV1h.pdf} \put(-1.0,68.0){{\Large
a}} \put(90.0,65.0){{\Large b}} \put(-1.0,33.0){{\Large d}}
\put(53.0,33.0){{\Large c}} \end{overpic} Figure 2:  (a) White-light optical
microscope image of a 555 die, together with external components used to
implement a two-state oscillator. The green box indicates area for cross-
section SEM for subfigures (c)-(d). (b) COMSOL model geometry for simulating a
555 die (silver, black, and green for aluminum, doped silicon, and substrate
silicon, respectively), showing the simulated current density for the output-
off state. (c) SEM image of a contact between the aluminum layer and the
doped-silicon layers. (d) Zoom of subfigure (c), which shows details of the
metal and doped-silicon layers, which are insulated by a glassivation layer
and an interlayer dielectric.
\begin{overpic}[width=390.25534pt]{currentPathsV1b.pdf}
\put(0.0,100.0){{\Large a}} \put(2.5,36.0){{\Large b}} \put(92.0,36.0){{\Large
c}} \end{overpic} Figure 3:  (a) Schematic for the 555 device and external
components used to simulate currents in the two-state oscillator demo circuit.
We implemented $\mathtt{Q19A}$ and $\mathtt{Q19B}$ as the two-collector BJT in
the manufacturer’s schematic, and used BJTs with two leads shorted as the
diodes [17]. (b) For the output-on state, the cyan line indicates current to
$\mathtt{R_{load}}$ and the orange line indicates current through
$\mathtt{R02}$ and $\mathtt{Q07}$. (c) For the output-off state, the magenta
line indicates the capacitor discharge current through $\mathtt{Q14}$ and the
green line shows the current draw to the flip-flop and output networks. In
both states, the dark blue line shows the current through three 5 k$\Omega$
resistors in series. All schematic components are labeled on the die photo in
the supplemental material [18].

##  II Devices and Materials

###  II.1 Experimental setup

In each measurement, we placed a diamond sample on the 555 die NV-side down,
then placed both in a fluorescence microscope (example apparatus shown in Fig.
1a). The diamond was illuminated with 532 nm laser light either from an angle
(side illumination [4]) or through the microscope objective (epifluorescence
illumination). In the presence of a magnetic field $B_{111}$ along the N-V
axis (the $[111]$ crystallographic direction, which is $\sim$35$\degree$ from
the diamond surface), the resonance frequencies between the $m_{s}=\pm 1$
ground-state magnetic sublevels of NVs aligned along the $[111]$ direction are
shifted by $\pm\gamma B_{111}$, where $\gamma\approx 28$ GHz/T is the NV
gyromagnetic ratio (Fig. 1b-c). We performed optically-detected magnetic
resonance (ODMR) spectroscopy by driving microwave transitions between the
$m_{s}=0$ and $m_{s}=\pm 1$ sublevels, which reduces the NV fluorescence
intensity when the microwave frequencies are on resonance. Imaging the NV
fluorescence intensity over a range of probe microwave frequencies, we obtain
an ODMR spectrum for every pixel in camera’s the field of view (Fig. 1d). We
fit the ODMR spectrum in each pixel to extract the frequency splitting between
the $m_{s}=0$ to $\pm 1$ transitions, from which we generate a map of
$B_{111}$. We applied a static 1.5-2.5 mT bias magnetic field along the N-V
axis, and we subtracted the magnetic field maps taken with and without the 555
energized to remove any contributions from sources other than currents in the
555 device.

###  II.2 NV diamond samples

Our magnetic imaging sensors are two single-crystal diamond samples with
shallow NV surface layers. Both samples (Samples A and B) are
4$\times$4$\times$0.5 mm3 electronic-grade diamond substrates with $<$5 ppb
nitrogen density. One surface of Sample A has a 4 $\upmu$m 12C-enriched
diamond layer with 20 ppm of 14N. NV centers were then formed using electron
irradiation and vacuum annealing [19].

The NV layer in Sample B was created by broad-beam 15N ion implantation with
19 energies to form a uniform 1 $\upmu$m 50 ppm nitrogen layer [20, 18]. After
vacuum-annealing to activate NV formation, Sample B was laser-cut into smaller
pieces (1.14$\times$0.84$\times$0.5 mm3) to match the 555 die dimensions for
two of the IC-diamond integration methods, described below. The surfaces of
both diamonds were prepared by triacid cleaning (sulfuric, nitric, and
perchloric), after which we coated the NV surfaces with 5 nm of Ti to provide
adhesion, 150 nm of Ag to prevent photoexcitation of electron-hole pairs in
the device, and 150 nm of Al2O3 to prevent shorting between conductive
elements.

###  II.3 555 timer IC

The 555 timer circuit was designed in 1971 using a BJT architecture, and it
quickly became a best-selling IC used for a wide range of applications [16].
The original design is largely unchanged except for an updated version that
uses complementary metal–oxide–semiconductor (CMOS) technology, which requires
less current for operation, while the BJT version allows for larger current
throughput. Here, we studied the RCA CA555CE BJT 555 timer [17], a BJT version
of the IC with $\sim$10-15 $\upmu$m features that, when carrying current, are
sufficiently large to magnetically detect and resolve with our QDM.

The 555 timer die (Fig. 2) has two conducting layers: a $\sim$1.6 $\upmu$m top
aluminum layer and a $\sim$6.4 $\upmu$m doped-silicon layer, separated by an
interlayer dielectric with contacts between layers. We determined the layer
thicknesses by cross-sectioning a die that was removed from its eight-pin dual
in-line package (DIP). We chemically stained the cross section to reveal the
doped silicon layer, and imaged using scanning electron microscopy [18]. The
IC consists of NPN junctions, PNP junctions, and doped-silicon resistors
arranged into Darlington pairs, current mirrors, voltage comparators, a
voltage divider, and a flip-flop (Fig. 3).

We configured the IC as a two-state oscillator by adding three external
resistors and a capacitor ($\mathtt{R_{ext1}}$, $\mathtt{R_{ext2}}$,
$\mathtt{R_{load}}$, and $\mathtt{C1}$, shown in Fig. 2a) [17, 16, 21, 22,
18]. In this circuit, the voltage across $\mathtt{C1}$ oscillates between
$\mathtt{Vcc}$/3 and 2$\mathtt{Vcc}$/3. Depending on the $\mathtt{C1}$ voltage
and the flip-flop state, the 555 device will either be in an output-on state
(sourcing current to $\mathtt{R_{load}}$) or an output-off state (discharging
$\mathtt{C1}$ through the IC to ground). In Fig. 3 we indicate some of the
primary current paths through the device in each internal state. These current
paths create visible features highlighted in the QDM magnetic images below,
which shows how QDM measurements can collect the internal-state information of
an IC.

The oscillation frequency $f$ and duty cycle $D$ are determined by
$\mathtt{R_{ext1}}$, $\mathtt{R_{ext2}}$, and $\mathtt{C1}$:

$\displaystyle
f=\frac{1}{\ln{2}~{}\left(\mathtt{R_{ext1}}+2\mathtt{R_{ext2}}\right)\mathtt{C1}},$
$\displaystyle
D=\frac{\mathtt{R_{ext2}}}{\mathtt{R_{ext1}}+2\mathtt{R_{ext2}}}.$

We chose $\mathtt{R_{ext1}}$ = 1 k$\Omega$, $\mathtt{R_{ext2}}$ = 330
k$\Omega$, and $\mathtt{C1}=47~{}\upmu$F to get a $\sim$20 s period and a 50%
duty cycle, confirmed with oscilloscope measurements to demonstrate that the
device was in working condition [18]. During the magnetic imaging
measurements, we kept the 555 in the output-on state by connecting
$\mathtt{C1}$ to ground, or in the output-off state by connecting
$\mathtt{C1}$ to $\mathtt{Vcc}$.

\begin{overpic}[width=411.93767pt]{microscopeDrawingsV1c.pdf}
\put(5.0,23.0){{\Large a}} \put(37.5,23.0){{\Large b}} \put(70.0,23.0){{\Large
c}} \end{overpic} Figure 4:  Schematic drawings for the three IC-diamond
integration methods we used: (a) diamond over wire bonds, (b) backside
thinning, (c) and diamond between the wire bonds.
\begin{overpic}[width=368.57964pt]{bigDiamFigV3b.pdf}
\put(0.0,8.0){\includegraphics[width=52.03227pt]{b111Coords.pdf}}
\end{overpic} Figure 5:  Method 1: Diamond over wire bonds. Measurements
(left) and simulations (right) of a 555 die in the output-on (top) and output-
off (bottom) state. The simulation (and black dotted box) dimensions are
1216$\times$1464 $\upmu$m2. This measurement has a
$z_{\textrm{fit}}=71~{}\upmu$m standoff distance from the die surface, and the
simulated magnetic field maps displayed are calculated at $z=71~{}\upmu$m for
comparison. These measurements have a 0.3 $\upmu$T noise floor in a 1$\times$1
$\upmu$m2 pixel. The white arrows show the primary current input/output points
for the device, and the circled regions show magnetic features from currents
in Fig. 3 (with the same colors).
\begin{overpic}[width=368.57964pt]{backthinnedFigV3b.pdf}
\put(0.0,8.0){\includegraphics[width=52.03227pt]{b111Coords.pdf}}
\end{overpic} Figure 6:  Method 2: Backside thinning. Measurements (left) and
simulations (right) of a 555 die in the output-on (top) and output-off
(bottom) state. Note that the die is left-right flipped compared to the other
images. This measurement has a $z_{\textrm{fit}}=26~{}\upmu$m standoff
distance (measured from the surface of the doped-silicon layer), and the
simulated magnetic field maps displayed are calculated at $z=26~{}\upmu$m for
comparison. These measurements have a 3 $\upmu$T noise floor in a 1$\times$1
$\upmu$m2 pixel. The black arrows show the primary current input/output points
for the device, and the circled regions show magnetic features from currents
in Fig. 3 (with the same colors).
\begin{overpic}[width=368.57964pt]{smallDiamFigV2b.pdf}
\put(0.0,8.0){\includegraphics[width=52.03227pt]{b111Coords.pdf}}
\end{overpic} Figure 7:  Method 3: Diamond between the wire bonds.
Measurements (left) and simulations (right) of a 555 die in the output-on
(top) and output-off (bottom) state. This measurement has a
$z_{\textrm{fit}}=4~{}\upmu$m standoff distance from the die surface, and the
simulated magnetic field maps displayed are calculated at $z=4~{}\upmu$m for
comparison. These measurements have a 3 $\upmu$T noise floor in a 1$\times$1
$\upmu$m2 pixel. The black arrows show the primary current input/output points
for the device, and the circled regions show magnetic features from currents
in Fig. 3 (with the same colors).

##  III IC-diamond integration methods

We imaged magnetic fields from the 555 die using three IC-diamond integration
methods: a top approach with a large diamond over the bond wires (Fig. 4a), a
back approach with a small diamond in the slot of a back-thinned die (Fig.
4b), and a top approach with a small diamond between the wire bonds (Fig. 4c),
ordered from largest to smallest IC-diamond standoff distance. Each approach
uses a similar NV imaging apparatus, but they differ in the IC and diamond
preparation steps. During each measurement, we monitored the voltage across
$\mathtt{R_{load}}$ to ensure that there were no bent bond wires causing a
short, light leakage causing photocurrent in the die, or other failure modes.

###  III.1 Method 1: Diamond over wire bonds

Here we used a diamond (Sample A) larger than the die to image the magnetic
fields from the entire 555 die and the bond wires (Fig. 4a). To minimize the
standoff distance between the NV layer and the die surface, we removed the die
from its packaging by placing the IC in 90% fuming nitric acid at 90 °C for 15
minutes. We then rinsed the fully-exposed die with acetone, isopropyl alcohol,
and deionized water and removed the bond wires using tweezers. We then glued
the die to an 8-pin DIP breakout board, electrically connected it with 0.001”
diameter gold wedge wire bonds, and installed it into a perfboard with the
external components.

We affixed Sample A in the QDM setup (glued to a protruding piece of silicon
carbide) with the NV layer at the microscope focal plane, and illuminated the
NV layer using side illumination. We positioned the 555 die under the NV layer
using a stepper-motor translation stage, measuring with decreasing standoff
distances until the device stopped working properly. Using this large
4$\times$4$\times$0.5 mm3 diamond sample facilitates mounting it in the
microscope setup, but since it is larger than the 555 die, the standoff
distance was limited by the bond wires touching the diamond surface.

###  III.2 Method 2: Backside thinning

To prepare the 555 die for this approach, we first cut through the back of the
packaging and the copper ground plane using an Allied X-Prep mill to gain
access to the backside of the die. We then thinned the exposed silicon to
20-30 $\upmu$m, which was the minimum thickness for which the die was still
functional. The DIP pins were bent 180°, then connected to a perfboard with a
mirror-flipped layout compared to the layout for Method 1.

We measured the magnetic map using a 1.14$\times$0.84$\times$0.5 mm3 piece of
Sample B placed on the bottom side of the die (Fig. 4b). To avoid shadows from
the packaging and from the sides of diamond, we illuminated the NV layer using
epifluorescence illumination.

###  III.3 Method 3: Diamond between the wire bonds

For this method, we used the wedge-bonded die from Method 1, as well as die
decapsulation. To expose the die while keeping it in the original packaging,
we decapsulated a 555 DIP IC using an etch tool (RKD Mega Etch) with a custom
gasket and fuming nitric acid (95 °C for 10 seconds to soften the polyimide,
followed by 85 °C for 25 seconds). This removed the packaging material to
expose the die while maintaining functionality.

For the measurement, we used another laser-cut piece of Sample B, and again
used epifluorescence illumination. The main difference in this method is that
we glued the diamond to a cover slip with UV-curing transparent glue (Fig.
4c). Both the diamond and the cover-slip were coated with the Ti-Ag-Al2O3
adhesion-reflection-insulation layers to prevent light from leaking around the
sides of the diamond.

##  IV Computational Methods

We developed a finite element simulation of the magnetic fields from the 555
die for the output-on and output-off states to evaluate the standoff distance
and spatial resolution of each measurement, and to check that the measured
magnetic field maps were consistent with what is expected from simulation.

###  IV.1 Governing equations

For the output-on and output-off states, the magnetic fields generated by the
555 die depend on the current density distributions and the material
characteristics of the device. The dynamics of the $\sim$20 s period two-state
oscillator are slow enough for us to treat the output-on and output-off states
independently in the static limit. The current density in the device is given
by the continuity equation and Ohm’s law:

$\boldsymbol{\nabla}\cdot\mathbf{J}=-\boldsymbol{\nabla}\cdot\left(\sigma\boldsymbol{\nabla}V\right)=0,$
(1)

where $\mathbf{J}$ is the current density in the 555, $\sigma$ is the
electrical conductivity of the 555 materials, and $V$ is the electrical
potential field resulting from the external circuit shown in Fig. 2(a). The
magnetic field is determined by the static Ampère’s law:

$\boldsymbol{\nabla}\times\frac{\boldsymbol{\nabla}\times\mathbf{A}}{\mu}=\mathbf{J},$
(2)

where $\mathbf{A}$ is the vector magnetic potential field and $\mu$ is the
magnetic permeability of the 555 materials.

We sequentially solve the system of governing equations (Eqns. 1 and 2) to
determine the magnetic field maps at different standoff distances. We first
solve Eqn. 1 to determine the scalar electrical potential field,
$V(\mathbf{r})$, with the appropriate boundary conditions for $V$. We use
Ohm’s law to determine $\mathbf{J}$ in the 555, which we then use to solve
Eqn. 2 for $\mathbf{A}$ with appropriate boundary conditions. We then
calculate the magnetic field, $\mathbf{B}(\mathbf{r})$, using
$\mathbf{B}=\boldsymbol{\nabla}\times\mathbf{A}$.

###  IV.2 555 geometry and boundary conditions

The 555 die model consists of a conducting metal layer, a conducting doped
silicon layer, and an interlayer dielectric to electrically insulate the
aluminum from the doped silicon as shown in Fig. 2d. Electrical contact
between the two conducting layers occurs only at specific locations on the die
to connect the internal device components of the 555.

We solved the system of governing equations using the finite element software
COMSOL Multiphysics${}^{\text{\textregistered}}$ version 5.5, implementing the
geometry and material properties of the 555. Finite element modeling allows us
to include the detailed multi-layer geometric features of the 555 die with
dimensions extracted from the optical microscope image in Fig. 2a. Fig. 2b
depicts the the full 3D model geometry built using COMSOL, with the top metal
layer in silver, and the doped silicon layer in black. The un-doped silicon
die is shown in green. The air domain above the die that was also included to
complete the computational domain of the model is not shown.

The boundary conditions for Eqn. 1 are electrically insulating with zero
normal current density everywhere except for the external boundaries
corresponding to the eight pins of the device. These pins are connected to an
external circuit and have voltage determined by the parameters of the
externally connected circuit components in Fig. 2a. The tangential components
of the magnetic vector potential field $\mathbf{A}$ are set to zero for the
boundary condition of Eqn. 2.

###  IV.3 Material properties and SPICE circuit simulation

We estimated the electrical conductivities ($\sigma$) and the magnetic
permeabilities ($\mu$) of the 555 internal components from available
information about the device. The top metal layer was assumed to be aluminum,
based on the SEM analysis in Fig. 2c-d. We modeled the interlayer dielectric
using insulator properties of SiO2. We used the magnetic permeability of
silicon for all domains in the silicon layer and estimated the electrical
conductivity of the resistor and transistor elements. We computed resistor
conductivities using the resistances reported in the 555 circuit schematic in
the datasheet (Fig. 3a) and the geometrical dimensions of the resistor region
approximated from the optical images of the die.

To determine the current in each transistor, we simulated the 555 device in
the two-state oscillator circuit in a SPICE electronic circuit simulator
(PSPICE) by combining the manufacturer’s schematic with the external circuit
components (Fig. 3a) [17]. We measured the 555 external voltages, currents,
and dynamics (frequency and duty cycle) to confirm that the model was
performing as expected [18], giving us confidence that this simulation also
predicted the internal behavior correctly. The SPICE model provided
information about the current in each transistor element, from which we
estimated the electrical conductivities for the relevant parts of the die
model.

###  IV.4 Numerical implementation and output

We solved the governing equations using a steady-state model in COMSOL,
adjusting the mesh resolution such that the solution remained constant when
changing the spatial discretization parameters. A triangular mesh with a
minimum element size approximately 6% of the minimum geometrical feature size
was required in the neighborhood of the metal, insulator, and semiconducting
layers. We coupled the Electric Currents (ec) module with the Electrical
Circuit (cir) module to solve the continuity equation (Eqn. 1) for
$V(\mathbf{r})$. The external circuit elements in Fig. 2(a), constructed using
the cir module, set the boundary conditions of the finite element
computational domain on the pins. The output, $V(\mathbf{r})$, from the
coupled ec and cir modules, determines the current density
$\mathbf{J}(\mathbf{r})$. We use this computed current density as the input to
the Magnetic Fields (mf) module to solve Ampère’s law (Eqn. 2) to determine
$\mathbf{B}(\mathbf{r})$. The full magnetic field solution allowed us to
calculate the $B_{111}(\mathbf{r})$ field component in planes of different
standoff distance above and below the die, which we used in analysis and
comparison with measurements.

##  V Results and Discussion

Figures 5-7 show the magnetic maps measured with each method, together with a
simulated magnetic map for comparison. We used the outputs from the COMSOL
simulations to determine the standoff distance $z_{\textrm{fit}}$ for each
measurement. To do this, we created an interpolating function $B_{111}(x,y,z)$
that is continuous in the spatial coordinate variables $\\{x,y,z\\}$, using a
set of the simulated magnetic field maps generated by the COMSOL simulation
over a range of altitudes $z$ (up to 100 $\upmu$m from the die surface). We
then performed a least-squares fit between the measured $B_{111}$ and the
simulated $B_{111}(x,y,z)$ fit function. The optimal $z_{\textrm{fit}}$ gives
our NV-die standoff distance and the optimal $x_{\textrm{fit}}$ and
$y_{\textrm{fit}}$ are spatial offsets between the measurement and the
simulation. This fit was performed for both the output-on and output-off
device states, and we found close agreement between the measured and the
simulated magnetic maps. This analysis returned
$z_{\textrm{fit}}=\\{71,26,4\\}$ $\upmu$m standoff distances for the three
methods, with a $\\{2,3,0.3\\}$ $\upmu$m uncertainty for each.

Figure 5 shows the resulting magnetic field maps measured with Method 1
(diamond over wire bonds) at the closest possible standoff distance of 71
$\upmu$m. Since the diamond sample is fixed in the QDM apparatus and is large
compared to the die, this approach allows for good heat sinking from pump
laser heating, a large image field of view (including the entire die and the
bond wires), good stray light protection, and variable-altitude measurements.
However, the standoff distance is limited by the bond wires, which reduces the
field strength and the spatial resolution. Furthermore, since the bond wires
are touching the diamond, the fields measured from the bond wires are the
stronger than those of the 555 die.

Figure 6 shows the magnetic field maps measured with Method 2 (backside
thinning) at a 26 $\upmu$m standoff distance from the doped-silicon layer
surface. These magnetic images look different compared to those of the other
methods since we measure from the back of the die (flipped left-right compared
to the other die images) but keep the same $B_{111}$ projection direction.
Method 2 benefits from being able to preferentially detect current in the
doped-silicon layer with greater signal-to-noise ratio and spatial resolution.
However, the necessary die-thinning step may limit the utility of this
magnetic imaging implementation. Furthermore, due to the reduced die thickness
and poor thermal conductivity of the packaging, the die has poor heat sinking
from laser heating. This limits the maximum-allowable laser power and the
magnetic sensitivity for this method.

Figure 7 shows the magnetic field maps measured with Method 3 (diamond between
the wire bonds). Comparing to the simulated magnetic maps, we determined a 4
$\upmu$m standoff distance for these measurements. With this standoff distance
we can image the weaker currents in the die, which are consistent with the
fields predicted by COMSOL. This approach had the best standoff distance,
spatial resolution, and field strength, and is also ideal for imaging dies
where the wire bond spacing is not a limitation to the standoff distance [11].
However, avoiding laser heating is more challenging than with Method 1, since
heat from the diamond can flow to the environment through the silicon carbide
mount in Method 1.

Each measurement method achieved a different standoff distance to the die,
listed in Table 1. The spatial resolution of an NV magnetic imaging
measurement is influenced by the optical diffraction limit ($\sim$1 $\upmu$m),
the standoff distance, and the NV layer thickness [4]. In this work, the
standoff distance (set by the bond wires, silicon thickness, or dust on the
die surface) was the main limitation to the spatial resolution, and is also an
estimate for the minimum spatial resolution for each measurement. As
illustrated in Figs. 5-7, decreasing the standoff distance enhances the
magnetic feature resolution in the 555 magnetic maps. Note that the feature
sizes in the 555 die are coarse enough that measuring with a standoff distance
smaller than 4 $\upmu$m would not reveal more detail.

Table 1 also lists each magnetic noise floor $\delta B_{111}$, which is the
standard deviation of the measured magnetic fields in the field of view when
measuring with the 555 disconnected, after subtracting a background
measurement. These $\delta B_{111}$ values are normalized to a 1$\times$1
$\upmu$m2 pixel size and a 1 s experiment duration. Sample A has a better
magnetic sensitivity than Sample B, due to its NV fluorescence contrast,
resonance linewidth, and fluorescence intensity being better [18]. To quote
the projected best-case $\delta B_{111}$ if using Sample A, we also evaluated
$\delta B_{111}$ for Method 2 and Method 3 when using Sample A in the same
conditions. For comparison, Table 1 also lists the maximum $B_{111}$ at each
standoff distance, from which we can calculate a maximum signal-to-noise ratio
(SNR) by dividing by $\delta B_{111}$. Note that some experiments had an SNR
$<1$ while the images in Figs. 5-7 had an SNR $>1$ because of longer averaging
times (typically 10 minutes to 1 hour). The SNR improves with closer standoff
distance, though the maximum $B_{111}$ improves slower than $1/z$ (as with
current in an infinite straight wire) for $z\lesssim 25~{}\upmu$m because of
the finite size of the conductors. We also convert the $\delta B_{111}$ and
$z_{\textrm{fit}}$ to a current sensitivity $\delta I$ and a surface current
density $\delta K$ in Table 1 assuming an infinite wire or an infinite sheet
of current along the $+x$ direction, respectively [18]. When measuring static
currents, the reported $\delta I$ and $\delta K$ sensitivities can be enhanced
with coarser pixel size and longer experiment duration (compared to 1$\times$1
$\upmu$m2 pixel size and 1 s experiment duration).

|  | Standoff
---
distance $z_{\textrm{fit}}$
| $\delta B_{111}$
---
(1 s, 1$\times$1 $\upmu$m2)
| Max $B_{111}$
---
from device
| Max $B_{111}$ SNR
---
(1 s, 1$\times$1 $\upmu$m2)
| $\delta I$
---
(1 s, 1$\times$1 $\upmu$m2)
| $\delta K$
---
(1 s, 1$\times$1 $\upmu$m2)
Method 1 | 71 $\upmu$m | 8.0 $\upmu$T | 42 uT | 5 | 3.1 mA | 16 A/m
Method 2 | 26 $\upmu$m |  | 160 $\upmu$T
---
(4.0 $\upmu$T)
98 $\upmu$T |  | 0.6
---
(25)
| 23 mA
---
(560 $\upmu$A)
| 310 A/m
---
(7.7 A/m)
Method 3 | 4 $\upmu$m |  | 100 $\upmu$T
---
(3.6 $\upmu$T)
220 $\upmu$T |  | 2
---
(61)
| 2.2 mA
---
(75 $\upmu$A)
| 200 A/m
---
(6.9 A/m)
Table 1:  Performance comparison for the three integration methods, where
$\delta B_{111}$, $\delta I$, and $\delta K$ are the magnetic field, current,
and surface current density noise floors. Method 1 quotes the performance of
Sample A and Methods 2 and 3 quote the performance for Sample B. The numbers
in parentheses are projected for the more sensitive Sample A diamond if used
for Methods 2 and 3.  \begin{overpic}[width=368.57964pt]{calcJxJyV2e.pdf}
\end{overpic} Figure 8:  Output-on and output-off surface current density
$|\vec{K}|$, from Method 1 measurements (Fig. 5) and from COMSOL simulation.
The $|\vec{K}|$ maps calculated from the measurement have broadened features,
since we suppress high spatial frequencies to suppress measurement noise. Here
we show the simulated $|\vec{K}|$ for the metal layer only; the measurement
can not tell the difference between current in the two conducting layers. The
black arrows show the primary current input/output points for the device, and
the circled regions show current paths in Fig. 3 (with the same colors).

##  VI Simulation and Measurement Analysis

Figures 5-7 highlight magnetic features for the key current paths (Fig. 3),
which tell us about the 555 internal behavior in different states. For Method
1 (Fig. 5), the strongest magnetic field signatures come from current flowing
from Vcc to $\mathtt{R_{load}}$ (cyan) and as the capacitor discharges through
the die to ground (magenta). In addition, we see weaker magnetic features as
current flows through $\mathtt{R02}$ and $\mathtt{Q07}$ as part of a current
mirror in the output-on state (orange) and as other components draw current
seen in the output-off state (green). Although these features are coarse due
to the standoff distance, these magnetic images can still be converted to
surface current density maps (Fig. 8) that are consistent with those of Fig.
3. The Method 2 magnetic images (Fig. 6) also have magnetic features from the
above current paths, though with the improved standoff distance, we also see
hints (vertical stripes) of the 0.67 mA of current through the three 5
k$\Omega$ resistors in series ($\mathtt{R07}$, $\mathtt{R08}$, and
$\mathtt{R09}$, dark blue). For comparison, the Method 3 magnetic images (Fig.
7) also show current going to the 5 k$\Omega$ resistors, though the field from
the resistors themselves is less prominent. Furthermore, this method has the
most pronounced magnetic features for currents supplying additional components
on the left side of the die (green), including for the output-on case. By
analyzing these magnetic features, calculating the forward-model and inverse-
model magnetic field and current density maps, and correlating these with the
schematic and die layout, we confirm that the anticipated internal current
paths are present for 555 states.

Using the $B_{111}$ measurements in Fig. 5, and approximating the 555 die
currents as a 2D sheet, we calculated the surface current densities
$\\{K_{x},K_{y}\\}$. To do this, we used a Fourier analysis approach to
compute the magnetic inverse problem needed to reconstruct the surface current
densities from the measured $B_{111}$ maps [23, 24, 25, 26]. We used a 71
$\upmu$m standoff distance (known from the forward model), zero-padded the
original magnetic map to help suppress edge artefacts, and applied a Hann
filter in the frequency domain with $\lambda=1.5\times 71~{}\upmu$m cutoff
wavelength to suppress measurement noise with high spatial frequencies.

Figure 8 shows the resulting $|\vec{K}|$ surface current amplitudes of this
inverse-problem analysis. Comparing with the surface current densities
calculated by the COMSOL simulation, we see good agreement, though the
calculated $|\vec{K}|$ are broadened due to the standoff distance and the
cutoff wavelength. The primary current paths (also drawn in Fig. 3) give rise
to the magnetic features highlighted in the measurements and simulations, and
this inverse-problem analysis shows how we can correlate the magnetic image
information with current paths in the 555 die.

##  VII Conclusion and Outlook

The QDM is a promising magnetic field diagnostic tool for integrated circuits
that has micron-scale spatial resolution, millimeter-scale field of view area,
and can operate at ambient conditions. In this paper we have appraised the
performance of the QDM to measure fields from the die of a 555 timer using
three measurement configurations and compared the measurement outputs to a
finite element simulation. Comparing the experimental and computational
outputs for the magnetic field shows good consistency, demonstrating that the
QDM is capturing the expected magnetic field information available from the IC
without being noticeably affected by artefacts or systematics [27]. We also
identified magnetic features and current paths in the measured and simulated
results, confirming that we can glean accurate information about the internal
current phenomena. Finally, since it has a few-micron feature size, up to 220
$\upmu$T magnetic field strength near the surface, and is feasilble (but also
nontrivial) to simulate, the 555 is an ideal device with which to characterize
and evaluate the QDM performance and techniques when used to sense electric
currents in ICs.

These full-circuit simulation and measurement results establish a foundation
from which to advance and optimize the state of the art of the QDM as a
diagnostic tool for ICs. Continued work will explore how to classify measured
magnetic maps using image analysis techniques (for example, using a structural
similarity index measure or machine learning classification) to quickly
identify working, faulty, and counterfeit ICs [11]. Furthermore, since the QDM
measures the magnetic fields from all pixels in parallel, this instrumentation
can be extended to measuring IC dynamics as a magnetic movie, or can be
modified to measure MHz- or GHz-frequency fields using NV AC magnetometry
techniques [28, 29].

##  VIII Acknowledgements

We thank Mark Gores (Integra Technologies) for help with die cross-section
imaging and analysis, Dan Thompson for help with image preparation, George
Burns for 15N ion implantation, and Shanalyn Kemme for help with optical
microscope troubleshooting. Sandia National Laboratories is a multi-mission
laboratory managed and operated by National Technology and Engineering
Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell
International, Inc., for the DOE’s National Nuclear Security Administration
under contract DE-NA0003525. This work was funded, in part, by the Laboratory
Directed Research and Development Program and performed, in part, at the
Center for Integrated Nanotechnologies, an Office of Science User Facility
operated for the U.S. Department of Energy (DOE) Office of Science. This paper
describes objective technical results and analysis. Any subjective views or
opinions that might be expressed in the paper do not necessarily represent the
views of the U.S. Department of Energy or the United States Government. P.K.
is supported by the Sandia National Laboratories Truman Fellowship Program.
E.V.L. is funded by the MITRE Corporation through the MITRE Innovation
Program.

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NV555suppl.pdf See pages 4 of NV555suppl.pdf


# Hybrid Convolutional and Attention Network for Hyperspectral Image Denoising

Shuai Hu, Feng Gao, Member, IEEE, Xiaowei Zhou, Junyu Dong, Member, IEEE,
and Qian Du, Fellow, IEEE This work was supported in part by the National Key
Research and Development Program of China under Grant 2022ZD0117202 and in
part by the Natural Science Foundation of Qingdao under Grant 23-2-1-222-ZYYD-
JCH. (Corresponding author: Xiaowei Zhou.) Shuai Hu, Feng Gao, Xiaowei Zhou,
and Junyu Dong are with the School of Computer Science and Technology, Ocean
University of China, Qingdao 266100, China. Qian Du is with the Department of
Electrical and Computer Engineering, Mississippi State University, Starkville,
MS 39762 USA.

###### Abstract

Hyperspectral image (HSI) denoising is critical for the effective analysis and
interpretation of hyperspectral data. However, simultaneously modeling global
and local features is rarely explored to enhance HSI denoising. In this
letter, we propose a hybrid convolution and attention network (HCANet), which
leverages both the strengths of convolution neural networks (CNNs) and
Transformers. To enhance the modeling of both global and local features, we
have devised a convolution and attention fusion module aimed at capturing
long-range dependencies and neighborhood spectral correlations. Furthermore,
to improve multi-scale information aggregation, we design a multi-scale feed-
forward network to enhance denoising performance by extracting features at
different scales. Experimental results on mainstream HSI datasets demonstrate
the rationality and effectiveness of the proposed HCANet. The proposed model
is effective in removing various types of complex noise. Our codes are
available at https://github.com/summitgao/HCANet.

###### Index Terms:

Hyperspectral image, image denoising, Transformer, attention mechanism, deep
learning.

##  I Introduction

Hyperspectral imaging is a powerful technique that allows the acquisition of
rich spectral information from an object or a scene. Compared with RGB data,
hyperspectral image (HSI) captures fine-grained spectral information. Hence,
HSIs have been extensively used in many practical applications, such as
unmixing [1] and ground object classification [2]. However, HSI is often
afflicted by inevitable mixed noise generated during the sensor imaging
process, which is caused by insufficient exposure time and reflected energy.
These noise may degrade the image quality and hinder the performance of
subsequent analysis and interpretation. Eliminating these noise could improve
the accuracy of ground object detection and classification. Therefore, HSI
denoising is a critical and indispensable technique in the preprocessing stage
of many remote sensing applications.

Motivated by the spatial and spectral properties of HSI, traditional HSI
denoising methods exploit the optimization schemes with priors, such as low
rankness [3], total variation [4], non-local similarities [5], and spatial-
spectral correlation [6]. Although these methods have achieved appreciable
performance, they commonly depends on the degree of similarity between the
handcrafted priors and the real-world noise model. In recent years,
convolutional neural networks (CNNs) [7] have provided new ideas for HSI
denoising, demonstrating notable performance advancements. Maffei et al. [8]
proposed a CNN-based HSI denoising model by taking the noise-level map as
input to train the network. Wang et al. [9] proposed a convolutional network
based on united Octave and attention mechanism for HSI denoising. Pan et al.
[10] presented a progressively multiscale information aggregation network to
remove the noise in HSI. These CNN-based methods use convolution kernels for
local feature modeling.

More recently, with the emergence of Vision Transformer (ViT) [11],
Transformer-based methods have achieved significant success in various
computer vision tasks. Existing Transformer-based image denoising methods have
achieved great success through learning the global contextual information.
However, if local features are considered and exploited effectively, the HSI
denoising performance may improve further. So, it is important to take account
of both local and global information by combining CNN and Transformers to
enhance denoising performance.

It is commonly non-trivial to build an effective Transformer and CNN hybrid
model for HSI denoising, due to the following two challenges: _1) The optimal
hybrid architecture for local and global feature modeling still remains an
open question._ Convolutional kernels capture local features, which means
losing the long-distance information interaction. The combination of
convolution and attention could offer a viable solution. _2) The single-scale
feature aggregation of the feed-forward network (FFN) in Transformer is
limited._ Some methods use depth-wise convolution to improve local feature
aggregation in FFN. However, due to the larger number of channels in the
hidden layer, single-scale token aggregation can hardly exploit rich channel
representations.

Figure 1: Illustration of our proposed hybrid convolution and attention
network (HCANet) for HSI denoising. (a) Framework of HCANet. (b) Inner
structure of CAMixing block.

To solve the aforementioned two challenges, we proposed a Hybrid Convolution
and Attention Network (HCANet) for HSI denoising, which simultaneously
exploits both the global contextual information and local features, as
illustrated in Fig. 1. Specifically, to enhance the modeling of both global
and local features, we have devised a convolution and attention fusion module
(CAFM) aimed at capturing long-range dependencies and neighborhood spectral
correlations. Furthermore, to improve multi-scale information aggregation in
FFN, we design a multi-scale feed-forward network (MSFN) to enhance denoising
performance by extracting features at different scales. Three parallel dilated
convolutions with different strides are used in MSFN. By conducting
experiments on two real-world datasets, we validate that our proposed HCANet
is superior to other state-of-the-art competitors.

The contributions of this letter can be summarized as follows:

  * •

The promising yet challenging problem of global and local feature modeling for
HSI denoising is explored. To the best of our knowledge, this is the first
work to combine convolution and attention mechanism for the HSI denoising
task.

  * •

We propose multi-scale feed-forward network, which seamlessly extract feature
at different scales, and effectively suppress the noise from multiple scales.

  * •

Extensive experiments are conducted on two benchmark datasets, which
demonstrates the rationality and effectiveness of the proposed HCANet. As a
side contribution, we have released our codes to benefit other researchers.

##  II Methodology

In this section, we present the Hybrid Convolution and Attention Network
(HCANet) for hyperspectral image denoising. As shown in Fig. 1, the main
structure of the model is a U-shaped network with several Convolution
Attention Mixing (CAMixing) blocks. Each CAMixing block includes two parts:
convolution-attention fusion module (CAFM) and multi-scale feed-forward
network (MSFN). For HSI, 3D convolutions capture spatial and spectral features
comprehensively, but increase parameters. To manage complexity, we use 2D
convolutions for channel adjustment, effectively exploiting HSI features.

For a noisy hyperspectral image $\mathbf{I}\in\mathbb{R}^{H\times W\times B}$,
where $H\times W$ denote the spatial resolution and $B$ denotes the channel
dimension, our HCANet first uses $3\times 3\times 3$ convolution to obtain
low-level feature. Then, we use a U-shaped network with several CAMixing
blocks and skip connections to obtain the noise residual map
$\mathbf{I}_{N}\in\mathbb{R}^{H\times W\times B}$, which has the same shape as
the input noisy image. The reconstructed clean HSI can be expressed as:
$\hat{\mathbf{I}}=\mathrm{I}+\mathrm{I}_{N}$. Finally, the reconstruction loss
with a global gradient regularizer is used to train the HCANet.

HSI denoising aims to reconstruct the corresponding clean HSI from a noisy
HSI. Fusing global and local features in HSI is important to enhance the
denoising task. Thus, the CAFM is designed in the CAMixing block. To further
utilize the contextual cues in the feed-forward network for HSI denoising, the
MSFN is designed in the CAMixing block. Next, we present the details of the
CAFM and MSFN.

###  II-A Convolution and Attention Fusion Module

Convolutional operations, limited by their local nature and restricted sensory
field, are insufficient in modeling global features. In contrast, the
Transformer, enabled by the attention mechanism, excels in extracting global
features and capturing long-range dependencies. Convolution and attention are
complementary to each other to model both global and local features. Inspired
by this, we design a convolution and attention fusion module (CAFM), as shown
in Fig. 2. We employ a self-attention mechanism in the global branch to
capture broader hyperspectral data information, while a local branch focuses
on extracting local features for comprehensive denoising.

Figure 2: Illustration of the proposed convolution and attention fusion module
(CAFM). It consists of local and global branches. In the local branch,
convolution and channel shuffling are employed for local feature extraction.
In the global branch, the attention mechanism is used to model long-range
feature dependencies.

The proposed CAFM consists of local and global branches. In the local branch,
to enhance cross-channel interaction and promote information integration, we
first use $1\times 1$ convolution to adjust the channel dimensions. Following
this, a channel shuffling operation is performed to further mix and blend the
channel information. Channel shuffle partitions the input tensor along the
channel dimension into groups, wherein depth-wise separable convolution is
employed within each group to induce channel shuffling. Subsequently, the
resulting output tensors from each group are concatenated along the channel
dimension to generate a novel output tensor. Subsequently, we utilize a
$3\times 3\times 3$ convolution to extract features. The local branch can be
formulated as:

$\textbf{F}_{\rm conv}=W_{3\times 3\times 3}({\rm CS}(W_{1\times
1}(\mathbf{Y}))),$ (1)

where $\textbf{F}_{\rm conv}$ is the output of local branch, $W_{1\times 1}$
denotes $1\times 1$ convolution, $W_{3\times 3\times 3}$ denotes $3\times
3\times 3$ convolution, CS represents channel shuffle operation, and
$\mathbf{Y}$ is the input feature.

In the global branch, we first generates query(Q), key(K) and value(V) via
$1\times 1$ convolution and $3\times 3$ depth-wise convolution, yielding three
tensors with the shape of $\hat{H}\times\hat{W}\times\hat{C}$. Next, Q is
reshaped to $\hat{\textbf{Q}}\in\mathbb{R}^{\hat{H}\hat{W}\times\hat{C}}$, and
$K$ is reshaped to
$\hat{\textbf{K}}\in\mathbb{R}^{\hat{C}\times\hat{H}\hat{W}}$. Then, we
compute the attention map $\mathbf{A}\in\mathbb{R}^{\hat{C}\times\hat{C}}$ via
the interaction of $\hat{\textbf{Q}}$ and $\hat{\textbf{K}}$. The
computational burden is reduced instead of computing the huge regular
attention map of size $\mathbb{R}^{\hat{H}\hat{W}\times\hat{H}\hat{W}}$. The
output $\textbf{F}_{\rm att}$ of global branch is defined as:

$\textrm{$\textbf{F}_{\rm att}$}=W_{1\times
1}\textrm{Attention}\left(\hat{\textbf{Q}},\hat{\textbf{K}},\hat{\textbf{V}}\right)+\textbf{Y},$
(2)
$\textrm{Attention}\left(\hat{\textbf{Q}},\hat{\textbf{K}},\hat{\textbf{V}}\right)=\hat{\textbf{V}}\textrm{Softmax$\left(\hat{\textbf{K}}\hat{\textbf{Q}}/\alpha\right)$},$
(3)

where $\alpha$ is a learnable scaling parameter to control the magnitude of
matrix multiplication of $\hat{\mathbf{K}}$ and $\hat{\mathbf{Q}}$ before
applying the softmax function.

Finally, the output of the CAFM module calculation is computed as:

$\textbf{F}_{\rm out}=\textbf{F}_{\rm att}+\textbf{F}_{\rm conv}.$ (4)

###  II-B Multi-Scale Feed-Forward Network

The original FFN in ViT is composed of two linear layers for single-scale
feature aggregation. However, the information contained in single-scale
feature aggregation of FFN is limited. To enhance the non-linear feature
transformation, we propose a multi-scale feed-forward network (MSFN). After
each CAMixing block, the outputs of CAFM are fed into the MSFN to aggregate
multi-scale features and enhance the non-linear information transformation.
Previous studies have revealed the efficacy of incorporating multi-scale
information in image denoising tasks [10].

Details of the MSFN are illustrated in Fig. 3. Two $1\times 1$ convolutions
are used to expand the feature channels with expanding ratio $\gamma=2$. The
input features are handled in two parallel paths, and gating mechanism is
introduced to enhance the non-linear transformation via element-wise product
of features from both paths. In the lower path, depth-wise convolution is used
for feature extraction. In the upper path, multi-scale dilated convolutions
are employed for multi-scale feature extraction. Two $3\times 3$ dilated
convolutions with dilation rates of 2 and 3 are used.

Given an input tensor
$\mathbf{X}\in\mathbb{R}^{\hat{H}\times\hat{W}\times\hat{C}}$, MSFN is
formulated as:

$\begin{split}\textrm{Gating}(\textbf{X})={\phi}(W_{3\times 3\times
3}W_{1\times
1}(\textbf{X}))~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\\\
~{}~{}~{}~{}\odot(W^{2}_{3\times 3}W_{1\times 1}(\textbf{X})+W^{3}_{3\times
3}W_{1\times 1}(\textbf{X})),\end{split}$ (5)

$\textrm{$\textbf{X}_{out}$}=W_{1\times 1}\textrm{Gating}(\textbf{X}),$ (6)

where $\odot$ denotes element-wise multiplication, $\phi$ represents the GELU
non-linearity, $W^{2}_{3\times 3}$ denotes $3\times 3$ dilated convolution
with dilation rate of 2 and $W^{3}_{3\times 3}$ denotes $3\times 3$ dilated
convolution with dilation rate of 3. MSFN offers a distinct role compared to
CAFM, focusing on enriching features with contextual information.

Figure 3: Illustration of the multi-scale feed-forward network (MSFN).

###  II-C Loss Function

We use the $L_{1}$ loss function as our optimization objective for training
the network, and the reconstruction loss is:

$\mathcal{L}_{\textrm{rec}}=\|\hat{\mathbf{I}}-\mathbf{I}\|_{1},$ (7)

$\hat{\mathbf{I}}$ represents the estimated noise-free Hyperspectral Image
(HSI), while $\mathbf{I}$ denotes the noisy HSI.

To enhance the quality of denoising and curtail redundancy, we incorporate a
global gradient regularizer to constrain $\hat{\mathbf{I}}$,

$\begin{split}\mathcal{L}_{grad}=\|\nabla_{h}\hat{\mathbf{I}}-\nabla_{h}\mathbf{I}\|_{2}^{2}+\|\nabla_{v}\hat{\mathbf{I}}-\nabla_{v}\mathbf{I}\|_{2}^{2}+\\\
\|\nabla_{s}\hat{\mathbf{I}}-\nabla_{s}\mathbf{I}\|_{2}^{2}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\end{split}$
(8)

where $\nabla_{h}$, $\nabla_{v}$, and $\nabla_{s}$ respectively represent the
gradient operator applied along the horizontal, vertical, and spectral axes.
Finally, the total loss function is as follows:

$\mathcal{L}=\mathcal{L}_{rec}+\lambda\mathcal{L}_{grad},$ (9)

where $\lambda$ is the weight parameter governing $\mathcal{L}_{grad}$. Based
on empirical evidence, we set $\lambda$ as 0.01 to maintain a balance between
the various loss terms.

TABLE I: Quantitative evaluation of the proposed HCANet and other comparing
methods in HSI image denoising task with four noise magnitudes
($\sigma=30,50,70$ and $blind$).

Case | Index | Noisy | LRMR | BM4D | LRTV | Restormer | MAFNet | Ours
---|---|---|---|---|---|---|---|---
| PSNR | 17.26 | 31.81 | 39.71 | 36.92 | 42.19 | 43.95 | 45.51
| SSIM | 0.108 | 0.678 | 0.973 | 0.93 | 0.971 | 0.974 | 0.981
30 | SAM | 0.698 | 0.182 | 0.079 | 0.089 | 0.05 | 0.031 | 0.028
| PSNR | 15.43 | 30.04 | 37.22 | 35.99 | 41.51 | 42.13 | 44.54
| SSIM | 0.049 | 0.634 | 0.856 | 0.901 | 0.952 | 0.967 | 0.978
50 | SAM | 0.885 | 0.222 | 0.166 | 0.121 | 0.047 | 0.034 | 0.033
| PSNR | 11.45 | 25.89 | 34.09 | 33.88 | 39.31 | 41.05 | 43.27
| SSIM | 0.03 | 0.565 | 0.781 | 0.858 | 0.947 | 0.951 | 0.968
70 | SAM | 1.01 | 0.275 | 0.189 | 0.155 | 0.041 | 0.036 | 0.037
| PSNR | 14.85 | 30.07 | 37.1 | 37.23 | 40.73 | 42.21 | 43.97
| SSIM | 0.056 | 0.648 | 0.867 | 0.924 | 0.95 | 0.962 | 0.978
blind | SAM | 0.857 | 0.149 | 0.086 | 0.114 | 0.043 | 0.032 | 0.034

##  III Experimental Results and Analysis

###  III-A Experiment Setup

Benchmark datasets. To verify the denoising performance of our model on
hyperspectral images, we trained our model on ICVL dataset and evaluated the
trained model on Pavia dataset. In the ICVL dataset, a total of 31 spectral
bands were utilized to collect images with a resolution of $1392\times 1300$.
To facilitate the training process, the data were randomly cropped and
transformed into cube data with a shape of $128\times 128\times 31$. We
augmented the training dataset with rotation and scaling techniques to improve
the model’s robustness, which resulted in a dataset with 20, 000 new samples.
To test the denoising effect of our model on real remote sensing images, we
conducted experiments on Pavia dataset.

Noise setting. During the testing phase, we conducted experiments under two
settings to demonstrate the effectiveness and generalizability of our proposed
model. For the first setting, we tested our model with varying magnitudes of
Gaussian noise from $\sigma=30$ to $\sigma=70$ and blind Gaussian noise
(random noise magnitude). For the second setting, we evaluated its robustness
against common complex noise types found in hyperspectral data obtained from
real spaceborne sensors, such as Gaussian, impulse, and deadline noise. We
defined five types of complex noise set:

  1. 1.

Case 1: Gaussian noise of varying magnitudes in all spectral channels, with a
randomly chosen standard deviation from 30 to 70.

  2. 2.

Case 2: Gaussian + Stripe noise. In addition to Gaussian noise, we randomly
add strip noise to spectral bands by polluting 5% $\sim$ 15% of columns with
strips.

  3. 3.

Case 3: Gaussian + Deadline noise. On the basis of Case 1, we added the
deadline noise in the third of the spectral bands. 5% to 15% of columns are
conflicted with deadlines in each band.

  4. 4.

Case 4: Gaussian + Impulse noise. On the basis of Case 1. Approximately one
third of the spectral bands were chosen at random to increase the intensity of
impulse noise by a range of 30% to 70%.

  5. 5.

Case 5: Mixture noise. Similar to the above cases, each spectral band is
affected by non-i.i.d Gaussian noise in Case 1. Additionally, each band is
randomly affected by a combination of three other types of noise.

Baseline methods and Implementation Details. We compared HCANet with five
state-of-the-art methods including model-driven methods and deep learning-
based methods. For model-driven methods, we consider BM4D [12], as well as
low-rank methods such as LRMR [13] and LRTV [14]. In terms of deep learning-
based methods, we choose the well-known methods, Restormer [15] and MAFNet
[10] for comparison. We used three evaluation metrics including peak signal-
to-noise ratio (PSNR), structure similarity (SSIM), and spectral angle mapper
(SAM) to quantify the denoising performance. Larger PSNR and SSIM values
indicate better denoising results, while smaller SAM values indicate better
denoising performance. We trained the models with an initial learning rate of
$10^{-4}$ and the learning rate decreased gradually over training epochs. The
HCANet was optimized by the Adam optimizer and it was trained for 100 epochs
on Gaussian noise and 150 epochs on complex noise. We conducted all the
experiments under PyTorch framework on a machine with an NVIDIA GTX 2080Ti
GPU, Intel Xeon E5 CPU and 32GB RAM.

TABLE II: Quantitative evaluation of the proposed HCANet and other comparing
methods in HSI image denoising task with five complex noise cases.

Case | Index | Noisy | LRMR | BM4D | LRTV | Restormer | MAFNet | Ours
---|---|---|---|---|---|---|---|---
| PSNR | 17.79 | 31.91 | 35.62 | 36.89 | 42.03 | 43.48 | 44.11
| SSIM | 0.158 | 0.706 | 0.884 | 0.896 | 0.970 | 0.972 | 0.979
Case1 | SAM | 0.799 | 0.215 | 0.146 | 0.12 | 0.044 | 0.037 | 0.033
| PSNR | 17.41 | 31.07 | 33.94 | 35.64 | 41.13 | 42.43 | 43.58
| SSIM | 0.193 | 0.697 | 0.829 | 0.882 | 0.965 | 0.967 | 0.978
Case2 | SAM | 0.808 | 0.238 | 0.153 | 0.182 | 0.05 | 0.039 | 0.036
| PSNR | 17.42 | 30.04 | 33.77 | 33.82 | 40.16 | 41.74 | 41.79
| SSIM | 0.15 | 0.74 | 0.866 | 0.871 | 0.961 | 0.958 | 0.973
Case3 | SAM | 0.882 | 0.242 | 0.179 | 0.147 | 0.052 | 0.035 | 0.041
| PSNR | 15.12 | 29.34 | 32.61 | 33.25 | 36.89 | 37.71 | 39.99
| SSIM | 0.126 | 0.703 | 0.887 | 0.816 | 0.932 | 0.945 | 0.964
Case4 | SAM | 0.891 | 0.286 | 0.193 | 0.174 | 0.083 | 0.077 | 0.056
| PSNR | 14.15 | 26.9 | 31.02 | 30.91 | 36.15 | 37.27 | 40.57
| SSIM | 0.107 | 0.601 | 0.711 | 0.743 | 0.926 | 0.978 | 0.963
Case5 | SAM | 0.912 | 0.305 | 0.22 | 0.198 | 0.079 | 0.067 | 0.052

###  III-B Experimental Analysis

A comprehensive quantitative comparison of our proposed HCANet and other
benchmarking methods is shown in Table I and II. In the two tables, values
highlighted by Red indicate the best while values highlighted by the Blue
indicate the second best performance.

Table I showcases the results of all methods in different intensities of
Gaussian noise. From this table, it is easy to find that our HCANet
outperforms all other methods in terms of all metrics when the data contains
single-type noise. Additionally, Table II presents the results with complex
noise. We can see that our HCANet achieves the best performance in all noise
cases. In addition to quantitative analysis, we also conducted qualitative
comparisons, as illustrated in Figs. 4 and 5. False-color denoising results
were obtained from three bands (17, 20, 30). It is clear that traditional
denoising methods struggle with complex noise. While MAFNet removes most
noise, it tends to oversmooth, losing some image details. Restormer performs
poorly on complex noise. In contrast, HCANet effectively removes most noise
while preserving local details, successfully restoring original image
features.

To demonstrate the effectiveness of each component in our HCANet, we conducted
ablation study on ICVL dataset. As shown in III, the HCANet (with all
components) performs the best compared with all other variants. Furthermore,
we find the performance constantly becomes better as we add local branch, 3D
convolution, and MSFN upon the base model. It indicates the necessity of each
component in our proposed HCANet.

TABLE III: Ablation studies on ICVL datasets under noise level $\sigma=30$,
epoch = 50

Base model | Local branch | 3D Conv | MSFN | PSNR $\uparrow$ | SSIM $\uparrow$ | SAM $\downarrow$
---|---|---|---|---|---|---
✓ |  |  |  | 39.94 | 0.965 | 0.045
✓ | ✓ |  |  | 40.58 | 0.967 | 0.039
✓ | ✓ | ✓ |  | 42.23 | 0.975 | 0.037
✓ | ✓ | ✓ | ✓ | 42.68 | 0.976 | 0.035

Figure 4: Gaussian noise removal results under noise level $\sigma=50$ on ICVL dataset with bands (17,20,30). Figure 5: Real noise removal results on Pavia dataset with bands (17, 20, 30). TABLE IV: Comparative analysis of computational complexity of models under blind Gaussian noise, epoch = 50 Model | Params | GFLOPS | PSNR
---|---|---|---
MemNet | 1.98M | 32.41 | 35.60
HSID-CNN | 0.63M | 40.80 | 36.72
MAFNet | 20.17M | 20.15 | 40.51
Restormer | 14.88M | 9.55 | 40.27
Ours | 4.75M | 11.5 | 41.21

We also compared the computational complexity of the models, and the results
are shown in Table IV. HCANet achieves optimal denoising performance while
maintaining a relatively moderate number of parameters and computational
complexity.

##  IV Conclusion

In this letter, we propose HCANet, a novel network for HSI denoising. In
particular, we proposed the convolution and attention fusion module, CAFM, to
fuse both global and local features. Furthermore, we propose the multi-scale
feed-forward network, MSFN to extract features from multiple scales and
enhances the denoising performance. Experimental results on challenging HSI
datasets demonstrate the effectiveness of our proposed model in comparison to
the state-of-the-art HSI denoising methods. Our model achieves remarkable
denoising performance in terms of both quantitative metrics and visual quality
of the reconstructed images.

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