
\newcommand\resultsTwoDimensional{
\begin{table*}[tb]
\centering
\caption{\small Quantitative evaluation on three 2D datasets.
For each dataset, we compare results of multiple baselines (rows) to results obtained with our proposed pipeline (\EmbedSeg) highlighted in gray.
First results column shows the required GPU-memory (training) of the respective method. 
The remaining columns show the Mean Average Precision ($\text{AP}_{\text{dsb}}$, see main text) for selected IoU thresholds.
Best and second best performing methods per column are indicated in bold and underlined, respectively.
}\label{tab:results2d}
%\vspace{.7em}
\small
\resizebox{\columnwidth}{!}{%
\begin{tabular}{crrrrrrrrrr}
    \toprule
    & $\text{GPU}_{\text{GB}}$ &
    $\text{AP}_{0.50}$ & $\text{AP}_{0.55}$ & $\text{AP}_{0.60}$ & $\text{AP}_{0.65}$ & $\text{AP}_{0.70}$ & $\text{AP}_{0.75}$ & $\text{AP}_{0.80}$ & $\text{AP}_{0.85}$ & $\text{AP}_{0.90}$ \\ \midrule
    %
    \multicolumn{11}{c}{\textit{BBBC010}} \\ \midrule
     \multicolumn{1}{l}{3-Class Unet} & 5.6 &
     0.521 & 0.466 & 0.451 & 0.440 & 0.427& 0.407 &0.377 & 0.332 & 0.243 \\
    \multicolumn{1}{l}{Cellpose (public)} &  & 
    0.225 &  0.204 & 0.184 & 0.155 &  0.097 & 0.043 & 0.013 & 0.002 & 0.000 \\
     \multicolumn{1}{l}{Harmonic Emb.} &      &
     0.900 & & & & & 0.723 & & & \\
    \multicolumn{1}{l}{PatchPerPix} &      &
    0.930 & & 0.905 & &\underline{0.879} & & \textbf{0.792} & &\textbf{ 0.386}\\
        % \cdashline{1-1}
    \multicolumn{1}{l}{Neven ~\etal} & \textbf{$<$1} & 
    \underline{0.953} & \underline{0.941} & \underline{0.927} & \underline{0.904} & 0.878 & \underline{0.830} & 0.731 & \underline{0.563} & 0.297 \\
    \rowcolor{gray!30}
    \multicolumn{1}{l}{\EmbedSeg (\textit{Ours})} &  \textbf{$<$1} & 
    \textbf{0.965} & \textbf{0.954} & \textbf{0.934} & \textbf{0.917} & \textbf{0.896} & \textbf{0.854} & \underline{0.762} & \textbf{0.596} & \underline{0.326} \\ \midrule
    %
    \multicolumn{11}{c}{\textit{Usiigaci}} \\ \midrule
    %
   \multicolumn{1}{l}{3-Class Unet} & 5.6 &
   0.245 & 0.188 & 	0.133 &	0.090 &	0.049 &	0.016 &	0.008 &	0.000 &	0.000 \\
   \multicolumn{1}{l}{Cellpose (public)} &  &
   0.291 &	0.237 & 0.169 &	0.128 &	0.066 & 0.031 & 0.010 &	0.000 &	0.000 \\
   \multicolumn{1}{l}{Cellpose (\textit{Usiigaci})} & \underline{3.6} &
    \textbf{0.704} &	\underline{0.600} & \underline{0.499} &	\underline{0.370} &	\underline{0.258} &	\underline{0.138} & 0.040 & \underline{0.005} & 0.000\\
    \multicolumn{1}{l}{Mask R-CNN} & 6.9 & 0.583 & 0.520 & 0.439 & 0.365 & 0.235 & 0.130 & \textbf{0.045} & \textbf{0.008} & 0.000 \\
   \multicolumn{1}{l}{StarDist} & 6.9 & 
   0.510 & 	0.427 & 0.337 & 0.235 & 0.143 &	0.076 &	0.019 &	0.002 &	0.000 \\
    % \cdashline{1-1}
    \multicolumn{1}{l}{Neven ~\etal} & \textbf{2.9} &
    0.648 & 0.570 & 0.463 & 0.343 & 0.233 & 0.115 & 0.035 &  0.004 & 0.000 \\
   \rowcolor{gray!30}
    \multicolumn{1}{l}{\EmbedSeg \textit{(Ours)}} &     \textbf{2.9}&
    \textbf{0.704} &  \textbf{0.643} &   \textbf{0.535} &  \textbf{0.414} & \textbf{0.273} &   \textbf{0.140} & \underline{0.044} &  \underline{0.005} & 0.000 \\ 
    \midrule
    %
    \multicolumn{11}{c}{\textit{DSB}} \\ \midrule
    \multicolumn{1}{l}{3-Class Unet} & 5.6 &
    0.806 & 0.775 & 0.743 & 0.701 & 0.654 & 0.578 & 0.491 & 0.374 & 0.226 \\
    \multicolumn{1}{l}{Cellpose (public)} &  &
    0.868 & \underline{0.852} & 0.829 & \underline{0.802} & 0.755 & 0.676 & 0.563 & 0.418 & 0.234 \\
    \multicolumn{1}{l}{Cellpose (\textit{DSB})} & \underline{3.6} &
    0.853 & 0.826 & 0.812 & 0.792 & \textbf{0.768} & \textbf{0.716} & \textbf{0.645} & \textbf{0.536} & \textbf{0.402} \\
    \multicolumn{1}{l}{Mask R-CNN}& 6.9 &
    0.832 & 0.805 & 0.773 & 0.730 & 0.684 & 0.597 & 0.489 & 0.353 & 0.189 \\
    \multicolumn{1}{l}{PatchPerPix} &       &
    0.868 & &   0.827 & &   0.755 & &   0.635 & &   0.379\\
    \multicolumn{1}{l}{StarDist} & 6.9 &
    0.864 &  0.836 &  0.804 &  0.755 & 0.685 & 0.586 &  0.450 &  0.287 & 0.119 \\
    % \cdashline{1-1}
    \multicolumn{1}{l}{Neven~\etal} & \textbf{1.3}  &
    \underline{0.873} &
    \underline{0.852} &	
    \underline{0.830} &
    0.799 &	
    0.762 &
    0.704 & 
    0.623 &
    0.511 &	
    0.373 \\
    \rowcolor{gray!30}
    \multicolumn{1}{l}{\EmbedSeg (\textit{Ours})} & \textbf{1.3}    & 
    \textbf{0.876} & 
    \textbf{0.858} & 
    \textbf{0.834} & 
    \textbf{0.806} & 
    \textbf{0.768} & 
     \underline{0.715} & 
    \textbf{0.645} & 
     \underline{0.530} & 
     \underline{0.399} \\
    \bottomrule
\end{tabular}%
}
\end{table*}
}






\newcommand\resultsThreeDimensional{
\begin{table*}[tb]
\centering
\caption{\small Quantitative Evaluation on four 3D datasets.
For each dataset, we compare results of multiple baselines (rows) to results obtained with our proposed pipeline (\EmbedSeg) highlighted in gray.
First results column shows the required GPU-memory (training) of the respective method. 
The remaining columns show the Mean Average Precision ($\text{AP}_{\text{dsb}}$) for selected IoU thresholds.
Best and second best performing methods per column are indicated in bold and underlined, respectively.
}\label{tab:results3d}
%\vspace{.7em}
\small
\resizebox{\columnwidth}{!}{%
\begin{tabular}{crrrrrrrrrr}
    \toprule
    & $\text{GPU}_{\text{GB}}$ &
    $\text{AP}_{0.1}$ & $\text{AP}_{0.2}$ & $\text{AP}_{0.3}$ & $\text{AP}_{0.4}$ & $\text{AP}_{0.5}$ & $\text{AP}_{0.6}$ & $\text{AP}_{0.7}$ & $\text{AP}_{0.8}$ & $\text{AP}_{0.9}$ \\ \midrule
    %
    \multicolumn{11}{c}{\textit{\Organoid}} \\ \midrule
    \multicolumn{1}{l}{Cellpose (\Organoid)} & 3.6  & 0.217 & 0.214 & 0.212 & 0.210 & 0.203& 0.197& 0.183 & 0.146 &0.042\\
     \multicolumn{1}{l}{StarDist-3D} & 20     & \textbf{0.988} &	\textbf{0.982} & 	\textbf{0.982} & 	\textbf{0.982} & 	\textbf{0.973} & 	\underline{0.970} & 	\underline{0.958} & 	\underline{0.774} & 	\underline{0.052} \\
    \rowcolor{gray!30}
    \multicolumn{1}{l}{\EmbedSeg (Ours)} & 7  & \textbf{0.988} & \textbf{0.982} &  \textbf{0.982} & \textbf{0.982} & \textbf{0.973} & \textbf{0.973} & \textbf{0.973} & \textbf{0.970} & \textbf{0.929} \\ \midrule
   %
   \multicolumn{11}{c}{\textit{\PlatynereisLive}} \\ \midrule
    \multicolumn{1}{l}{Cellpose (\PlatynereisLive)} & 3.6 & 0.971 & 	\underline{0.971} &	\underline{0.966} & 	0.957 & 	0.931 & 	0.872 & 	0.700 &	\underline{0.299} & 	\textbf{0.009}\\
     \multicolumn{1}{l}{StarDist-3D} & 20     & \underline{0.973} & 0.969 & \underline{0.966} & \underline{0.966} & \underline{0.937} & \underline{0.910} & \underline{0.736} & 0.246 & 0.002\\
    \rowcolor{gray!30}
    \multicolumn{1}{l}{\EmbedSeg (Ours)} &  7 & \textbf{0.982} & \textbf{0.982} &  \textbf{0.982} & \textbf{0.975} & \textbf{0.964} &   \textbf{0.932} & \textbf{0.804} & \textbf{0.361} & \underline{0.004}\\ \midrule
    \multicolumn{11}{c}{\textit{\MouseSkull}} \\ \midrule
    \multicolumn{1}{l}{Cellpose (\MouseSkull)} & 3.6  & \underline{0.613} & \underline{0.587} & \underline{0.587} & \underline{0.563} & \underline{0.515} & \underline{0.471} & \underline{0.389} & \underline{0.316} & \textbf{0.064} \\
    \multicolumn{1}{l}{StarDist-3D} & 20 &  0.468 & 0.468 & 0.400 & 0.358 & 0.264 & 0.138 & 0.034 & 0.000 & 0.000\\
    \rowcolor{gray!30}
    \multicolumn{1}{l}{\EmbedSeg (Ours)}  & 7  & \textbf{0.837} & 	\textbf{0.837} & 	\textbf{0.837} & 	\textbf{0.837} & 	\textbf{0.795} & 	\textbf{0.646} & 	\textbf{0.549} & 	\textbf{0.362} & 	\underline{0.053}\\ 
    \midrule
    \multicolumn{11}{c}{\textit{\PlatynereisFixed}} \\ \midrule
    \multicolumn{1}{l}{Cellpose (\PlatynereisFixed)} & 3.6 & \underline{0.731} & \underline{0.674} & \underline{0.629} & \underline{0.554} & \underline{0.493} & \underline{0.390} & \underline{0.247} & \underline{0.038} & 0.000\\
    \multicolumn{1}{l}{StarDist-3D} & 20 & 0.599 & 0.587 & 0.545 & 0.442 & 0.280 & 0.114 & 0.010 & 0.000 & 0.000 \\
    \rowcolor{gray!30}
    \multicolumn{1}{l}{\EmbedSeg (Ours)}  & 7  & \textbf{0.884} & \textbf{0.884} & \textbf{0.874} & \textbf{0.852} & \textbf{0.781} & \textbf{0.655} & \textbf{0.482} & \textbf{0.120} & 0.000 \\ 
    \bottomrule
\end{tabular}%
}
\end{table*}
}




\newcommand\threeDimensionalDataDescription{
\begin{table*}[tb]
\centering
\caption{\small Used 3D datasets. 
In this work we introduce four new volumetric microscopy datasets, covering various practically relevant imaging conditions and microscopy modalities. All datasets come with high quality ground truth labels for training and are publicly available at \url{https://github.com/juglab/EmbedSeg}.
}\label{tab:data3d}
\small
\resizebox{\columnwidth}{!}{%
\begin{tabular}{llccl}
    \toprule
    \textbf{Name} & \textbf{Description} &
    \textbf{Pixel Size} (Z,Y,X) $[\mu m^{3}]$ & \textbf{Bit Depth} & \textbf{Used Microscope}   \\ \midrule
    %
    \multirow{2}*{\Organoid}
            & Mouse Embryonic Stem Cells,
                        & \multirow{2}*{(1.0, 0.1733, 0.1733)}    & \multirow{2}*{uint16}    & Selective Plane\\
               &R1 cell line, labeled membrane &&&Illumination Microscopy\\\midrule
    \multirow{3}*{\PlatynereisLive}
& Nuclei of a developing \textit{Platynereis dumerilii} embryo
    & \multirow{3}*{(2.031, 0.406, 0.406)}    & \multirow{3}*{uint16}    & Simultaneous Multi-view  \\
    & at stages between 0 to 16 hours post fertilization, &&&Light-Sheet 
    Microscopy\\
    &injected  with  a  fluorescent  nuclear  tracer &&&  \\
    \midrule
    \multirow{2}*{\MouseSkull}
& Nuclei of the skull region of developing mouse  
    & \multirow{2}*{(0.200, 0.073, 0.073)}    & \multirow{2}*{uint16}    & Inverted Zeiss  \\
    &embryos, labeled with DAPI&&&LSM 880 Microscope  \\
    \midrule
    \multirow{3}*{\PlatynereisFixed}
& Nuclei of whole-mount \textit{Platynereis dumerilli}  
    & \multirow{3}*{(0.4501, 0.4499, 0.4499)}    & \multirow{3}*{uint8}    & Laser Scanning  \\
    & specimens at stage of 16 hours post fertilization, &&&Confocal Microscopy\\
    & labeled with DAPI&&&\\
    \bottomrule
\end{tabular}%
}
\end{table*}

}

\newcommand\ablationResults{
\begin{table*}[tb]
\centering
\caption{\small Ablation studies.
For the BBBC010, Usiigaci and \PlatynereisLive datasets, we show how using the centroid instead of the medoid during training and/or removing test-time augmentation negatively impacts overall performance.
Like before, columns show $\text{AP}_{\text{dsb}}$ (first row) or $\Delta\text{AP}_{\text{dsb}}$ (rows 2-4) at selected IoU thresholds. 
Individual rows show: results obtained with \EmbedSeg; using centroids instead of medoids during training; using medoids but without test-time augmentation; using centroids during training and no test-time augmentation.
}\label{tab:ablation}
\small
\vspace{-2mm}
\resizebox{\columnwidth}{!}{%
\begin{tabular}{crrrrrrrrr}
    \toprule
     $\text{AP}_{\text{dsb}}$ & $\text{AP}_{0.50}$ & $\text{AP}_{0.55}$ & $\text{AP}_{0.60}$ & $\text{AP}_{0.65}$ & $\text{AP}_{0.70}$ & $\text{AP}_{0.75}$ & $\text{AP}_{0.80}$ & $\text{AP}_{0.85}$ & $\text{AP}_{0.90}$ \\
     \midrule
    \multicolumn{10}{c}{\textit{BBBC010} (2D)} \\ 
    \midrule
    \multicolumn{1}{l}{\EmbedSeg (ours)} & 
    0.965 & 0.954 & 0.934 & 0.917 & 0.896 & 0.854 & 0.762 & 0.596 & 0.326\\
    \multicolumn{1}{l}{$\drsh$ medoid $\Rightarrow$ centroid} & 
    -0.002 & -0.002 & -0.000 & -0.002 & -0.001 & -0.004 & +0.004 & +0.001 & +0.003\\   
    \multicolumn{1}{l}{$\drsh$ no test-time augm.} & 
    -0.007 & -0.008 & -0.003 & -0.008 & -0.014 &
       -0.020 & -0.028 & -0.033 & -0.025\\
    \multicolumn{1}{l}{$\drsh$ both ($=$Neven~\etal)} &     -0.011 & -0.013 & -0.007 & -0.012 & -0.018 &
       -0.024 & -0.032 & -0.033 & -0.029\\
    \midrule
    \multicolumn{10}{c}{\textit{Usiigaci} (2D)} \\ \midrule
    \multicolumn{1}{l}{\EmbedSeg (ours)} & 
    0.704 & 0.643 & 0.535 & 0.414 & 0.273 & 0.140 & 0.044 & 0.005 & 0.000\\
    \multicolumn{1}{l}{$\drsh$ medoid $\Rightarrow$ centroid} & 
    -0.014 & -0.013 & -0.006 & -0.005 & +0.006 & +0.009 & +0.002 & -0.001 & 0.000\\
    \multicolumn{1}{l}{$\drsh$ no test-time augm.} & 
    -0.028 & -0.048 & -0.050 & -0.052 & -0.040 & -0.030 & -0.008 & -0.001 & 0.000 \\
    \multicolumn{1}{l}{$\drsh$ both ($=$Neven~\etal)} & 
    -0.038 & -0.053 & -0.053 & -0.055 & -0.028 & -0.020 & -0.006 & 0.000 & 0.000  \\ 
    \midrule
    $\text{AP}_{\text{dsb}}$ & $\text{AP}_{0.10}$ & $\text{AP}_{0.20}$ & $\text{AP}_{0.30}$ & $\text{AP}_{0.40}$ & $\text{AP}_{0.50}$ & $\text{AP}_{0.60}$ & $\text{AP}_{0.70}$ & $\text{AP}_{0.80}$ & $\text{AP}_{0.90}$ \\
     \midrule
        \multicolumn{10}{c}{\PlatynereisLive (3D)} \\ \midrule
    \multicolumn{1}{l}{\EmbedSeg (ours)} & 0.982 &   0.982 &  0.982 &     0.975 &     0.964 &     0.932 &     0.804 &     0.361 &     0.004\\
    \multicolumn{1}{l}{$\drsh$ medoid $\Rightarrow$ centroid} & 
    -0.006 & -0.006 & -0.008 & -0.005 &
       -0.010 & -0.007 & +0.018 & +0.026 & 0.000\\
    \multicolumn{1}{l}{$\drsh$ no test-time augm.} & 
    -0.012 & -0.012 & -0.012 & -0.012 & -0.013 &
       -0.019 & -0.023 & -0.037 & -0.003\\
    \multicolumn{1}{l}{$\drsh$ both} & 
    -0.013 & -0.014 & -0.016 & -0.018 &
       -0.022 &-0.013 & -0.033 & -0.019 & 0.000
        \\ 
    
\bottomrule
\end{tabular}%
}
\end{table*}
}


\newcommand\resultsStarConvex{
\begin{table*}[h]
\centering
\caption{\small Quantitative results on the \Parhyale data.
% This image dataset  shows nuclei from the regenerating legs of \textit{Parhyale hawaiensis} expressing Histone-EGFP from the PhHS-H2B-EGFP transgene.
% These were imaged with the Microscope - cLSM Zeiss 780 inverted confocal microscope, using the Objective lens - EC Plan-Neofluar 20x/0.5 (without Immersion) and a voxel size (microns) $0.2768 \times 0.2768 \times 2.1605$.
We compare results of multiple baselines (rows) to results obtained with our proposed pipeline (\EmbedSeg), highlighted in gray.
First results column shows the required GPU-memory (training) of the respective method. 
The remaining columns show the Mean Average Precision ($\text{AP}_{\text{dsb}}$, see main text) for selected IoU thresholds.
Best and second best performing methods per column are indicated in bold and underlined, respectively.
}\label{tab:resultsstarconvex3d}
\vspace{.7em}
\small
\resizebox{\columnwidth}{!}{%
\begin{tabular}{crrrrrrrrrr}
    \toprule
    & $\text{GPU}_{\text{GB}}$ &
    $\text{AP}_{0.1}$ & $\text{AP}_{0.2}$ & $\text{AP}_{0.3}$ & $\text{AP}_{0.4}$ & $\text{AP}_{0.5}$ & $\text{AP}_{0.6}$ & $\text{AP}_{0.7}$ & $\text{AP}_{0.8}$ & $\text{AP}_{0.9}$ \\ \midrule
    %
    \multicolumn{11}{c}{\textit{\Parhyale}~\cite{alwes2016, weigert2020}} \\ \midrule
    \multicolumn{1}{l}{U-Net } &   & 
    \underline{0.592} & 0.552 &   0.481 &   0.372 &  0.280 &   0.198 &  0.097 &  0.010 &  0.000\\
    \multicolumn{1}{l}{Cellpose (\Parhyale)} & 3.6  & 0.545 & 	0.498 & 0.456 & 	0.384 & 0.285 & 0.154 & 0.040 & 	0.006 & 	0.000\\
     \multicolumn{1}{l}{StarDist-3D} & 20     & \textbf{0.766} &  \textbf{0.757} &  \textbf{0.741} & \textbf{0.698} & \textbf{0.593} & \textbf{0.443} & \textbf{0.224} & \textbf{0.038} & 0.000  \\
    \rowcolor{gray!30}
    \multicolumn{1}{l}{\EmbedSeg (Ours)} & 7  & 0.581 & 	\underline{0.581} & 	\underline{0.579} & 	\underline{0.543} & 	\underline{0.472} & 	\underline{0.359} & 	\underline{0.185} &	\textbf{0.038} & 	0.000 \\ \bottomrule
\end{tabular}%
}
\end{table*}
}
