final press 8ecm
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1b00a76a17
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@ -146,27 +146,12 @@ def plot_parameters(dat, input_file, deparameterize=False, ref=None):
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for i in range(len(ids)):
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cur_key = ids[i]
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true_level = np.log(true_values[ids[i]]/current_val[i])/np.log(2)
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rec_value = np.round(2**theta[-1, idx]*current_val[i],2)
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#curve = theta[:,idx] + line_split*idx - true_level
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#dash_curve = line_split*idx + t*0
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curve = 2**theta[:, idx]*current_val[i]
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std_down = 2**(-np.sqrt(P[:, idx, idx]))*curve
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std_up = 2**np.sqrt(P[:, idx, idx])*curve
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dash_curve = true_values[ids[i]] + t*0
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if ids_type[i] == 'dirichlet':
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fig3, axes3 = plt.subplots(1,1,figsize=(12,5))
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axes3.plot(t, curve , '-', color=col_,label= legends_ + '= ' + str(rec_value) + '/' + str(true_values[cur_key]) + '$', linewidth = 4)
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axes3.fill_between(t, std_down, std_up, alpha=0.3, color=col_)
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legends_=next(legends)
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axes3.plot(t, dash_curve , color=col_,ls='--')
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axes3.plot(t, dash_curve , color=col_,ls='--' , linewidth = 3)
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axes3.set_ylabel(r'$U$',fontsize=36)
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axes3.legend(fontsize=36,loc='upper right')
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axes3.set_xlim([-0.01,0.81])
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@ -177,28 +162,34 @@ def plot_parameters(dat, input_file, deparameterize=False, ref=None):
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plt.savefig('U.png')
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plt.close(fig3)
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else:
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cur_key = ids[i]
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rec_value = np.round(2**theta[-1, idx]*current_val[i],2)
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curve = 2**theta[:, idx]*current_val[i]
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std_down = 2**(-np.sqrt(P[:, idx, idx]))*curve
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std_up = 2**np.sqrt(P[:, idx, idx])*curve
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dash_curve = true_values[ids[i]] + t*0
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axes1.plot(t, curve , '-', color=col_,label= legends_ + '= ' + str(rec_value) + '/' + str(true_values[cur_key]) + '$', linewidth = 3)
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axes1.fill_between(t, std_down, std_up, alpha=0.3, color=col_)
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axes1.plot(t, dash_curve , color=col_,ls='--')
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axes1.plot(t, dash_curve , color=col_,ls='--',linewidth = 3)
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legends_=next(legends)
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if RC_flag:
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if i<len(current_val_C):
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rec_value_C = np.round(2**theta[-1, idx+1]*current_val_C[idc],6)
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if RC_flag:
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if i<len(current_val_C):
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true_level_C = np.log(true_values_C[ids[i]]/current_val_C[i])/np.log(2)
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rec_value_C = np.round(2**theta[-1, idc]*current_val_C[idc],6)
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curve_C = 2**theta[:, idx+1]*current_val_C[idc]
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dash_curve_C = true_values_C[ids[i]] + t*0
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std_C_down = 2**(-np.sqrt(P[:, idx+1, idx+1]))*curve_C
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std_C_up = 2**np.sqrt(P[:, idx+1, idx+1])*curve_C
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axes2.plot(t, curve_C , '-', color=col_,label= legends_ + '= ' + str(rec_value_C) + '/' + str(true_values_C[cur_key]) + '$', linewidth = 2)
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axes2.fill_between(t, std_C_down, std_C_up, alpha=0.3, color=col_)
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axes2.plot(t, dash_curve_C , color=col_,ls='--')
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legends_=next(legends)
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idx +=1
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idc +=1
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curve_C = 2**theta[:, idx+1]*current_val_C[idc]
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dash_curve_C = true_values_C[ids[i]] + t*0
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std_C_down = 2**(-np.sqrt(P[:, idx+1, idx+1]))*curve_C
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std_C_up = 2**np.sqrt(P[:, idx+1, idx+1])*curve_C
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axes2.plot(t, curve_C , '-', color=col_,label= legends_ + '= ' + str(rec_value_C) + '/' + str(true_values_C[cur_key]) + '$', linewidth = 3)
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axes2.fill_between(t, std_C_down, std_C_up, alpha=0.3, color=col_)
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axes2.plot(t, dash_curve_C , color=col_,ls='--',linewidth = 3)
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legends_=next(legends)
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idx +=1
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idc +=1
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if meas_flag:
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@ -221,10 +212,12 @@ def plot_parameters(dat, input_file, deparameterize=False, ref=None):
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if RC_flag:
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axes2.set_ylabel(r'$C$',fontsize=24)
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axes2.legend(fontsize=18,loc='upper right')
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axes2.set_ylabel(r'$C$',fontsize=36)
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axes2.legend(fontsize=36,loc='upper right')
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axes2.set_xlim([-0.01,0.81])
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axes2.set_xlabel(r'$t (s)$',fontsize=24)
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plt.xticks(fontsize=28)
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plt.yticks(fontsize=28)
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axes2.set_xlabel(r'$t (s)$',fontsize=36)
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fig2.savefig('C.png')
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fig1.savefig('Rd.png')
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@ -8,7 +8,7 @@ fluid:
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implicit_windkessel: True
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io:
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write_path: 'results/HRz_Pb_V70'
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write_path: 'results/impl_1ms'
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restart:
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path: '' # './projects/nse_coa3d/results/test_restart2/'
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time: 0
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@ -27,7 +27,8 @@ boundary_conditions:
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value: ['0','0','-U*sin(DOLFIN_PI*t/Th)*(t<=Th) + (Th<t)*(U*DOLFIN_PI/Th*(t-Th)*exp(-(t-Th)*beta))']
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parameters:
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#U: 75 #REFERENCE
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U: 150 #Pa Pb
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U: 80
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#U: 150 #Pa/Pb
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#U: 40 #Pc
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Th: 0.36
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beta: 70
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@ -55,9 +56,9 @@ boundary_conditions:
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#C: 0.0010 # Pb
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#C: 0.0001 # Pc
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#C: 0.0008 # Pg
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#R_d: 7200 # REFERENCE
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R_d: 7200 # REFERENCE
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#R_d: 8760 #Pa
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R_d: 17520 #Pb x2
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#R_d: 17520 #Pb x2
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#R_d: 4000 #Pc
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p0: 85
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conv: 1333.223874
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@ -71,9 +72,9 @@ boundary_conditions:
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#C: 0.0010 # Pb
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#C: 0.0001 # Pc
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#C: 0.0008 # Pg
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#R_d: 11520 # REFERENCE
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R_d: 11520 # REFERENCE
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#R_d: 8760 #Pa
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R_d: 17520 #Pb x2
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#R_d: 17520 #Pb x2
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#R_d: 4000 #Pc
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p0: 85
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conv: 1333.223874
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@ -86,9 +87,9 @@ boundary_conditions:
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#C: 0.0005 #Pa
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#C: 0.0010 #Pb
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#C: 0.0001 #Pc
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#R_d: 11520 # REFERENCE
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R_d: 11520 # REFERENCE
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#R_d: 8760 #Pa
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R_d: 17520 #Pb x2
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#R_d: 17520 #Pb x2
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#R_d: 4000 #Pc
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p0: 85
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conv: 1333.223874
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@ -131,7 +132,7 @@ fem:
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convection_skew_symmetric: True # aka Temam term
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stabilization:
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forced_normal:
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enabled: True
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enabled: False
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boundaries: [3,4,5,6]
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gamma: 20
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backflow_boundaries: [3,4,5,6]
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@ -162,17 +163,17 @@ estimation:
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-
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id: 4
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type: 'windkessel'
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mode: 'Rd'
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mode: 'RdC'
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initial_stddev: 1
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-
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id: 5
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type: 'windkessel'
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mode: 'Rd'
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mode: 'RdC'
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initial_stddev: 1
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-
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id: 6
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type: 'windkessel'
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mode: 'Rd'
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mode: 'RdC'
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initial_stddev: 1
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-
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id: 2
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@ -186,12 +187,12 @@ estimation:
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mesh: '/home/yeye/NuMRI/kalman/meshes/coaortaH3_leo2.0.h5'
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#mesh: './meshes/coaortaH1.h5'
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fe_degree: 1
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xdmf_file: 'measurements/aorta_zdir/Perturbation/Mg15V70/u_all.xdmf'
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file_root: 'measurements/aorta_zdir/Perturbation/Mg15V70/u{i}.h5'
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#xdmf_file: 'measurements/aorta/u_all.xdmf'
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#file_root: 'measurements/aorta/u{i}.h5'
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#xdmf_file: 'measurements/aorta_dt1ms/Perturbation/Mg15V70/u_all.xdmf'
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#file_root: 'measurements/aorta_dt1ms/Perturbation/Mg15V70/u{i}.h5'
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xdmf_file: 'measurements/aorta_exp_dt1ms/u_all.xdmf'
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file_root: 'measurements/aorta_exp_dt1ms/u{i}.h5'
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indices: 0 # indices of checkpoints to be processed. 0 == all
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velocity_direction: ~
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velocity_direction: [0,0,1]
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noise_stddev: 45 # standard deviation of Gaussian noise
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roukf:
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@ -199,5 +200,5 @@ estimation:
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observation_operator: 'postprocessing' #state or postprocessing
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reparameterize: True
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ODV_functional:
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enable: True
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VENC: 142
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enable: False
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VENC: 138
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@ -0,0 +1,162 @@
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\BOOKMARK [2][]{Outline0.1}{4D flow MRI}{}% 1
|
||||
\BOOKMARK [2][]{Outline0.2}{The mathematical model}{}% 2
|
||||
\BOOKMARK [2][]{Outline0.3}{The inverse problem}{}% 3
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@ -135,14 +135,11 @@ University of Groningen\\[0.5cm]
|
|||
\footnotesize
|
||||
|
||||
\begin{itemize}
|
||||
\item<2-> Full 3D coverage of the region of interest
|
||||
\item<2-> Velocities encoded into the magnetization phase
|
||||
\item<3-> Rich post-proccesing: derived parameters
|
||||
\end{itemize}
|
||||
|
||||
\onslide<4-> Disadvantages:
|
||||
\begin{itemize}
|
||||
\item<5-> Long scan time
|
||||
\end{itemize}
|
||||
|
||||
|
||||
|
||||
\column{.54\textwidth} % Right column and width
|
||||
|
|
@ -232,16 +229,26 @@ P_l = R_{p,l} \ Q_l + \pi_l
|
|||
|
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\begin{frame}
|
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\frametitle{The mathematical model}
|
||||
|
||||
\begin{columns}[c]
|
||||
\column{.5\textwidth} % Left column and width
|
||||
\footnotesize
|
||||
\begin{itemize}
|
||||
\item<1-> $u_{inlet} = -U f(t) \hat{n}$, with $f(t)$ the weaveform.
|
||||
\item<2-> Fractional step scheme.
|
||||
\item<3-> Semi-implicit Windkessel model.
|
||||
\item<4-> Stabilized $\mathbb{P}1/\mathbb{P}1$ finite elements.
|
||||
\item<4-> Implemented in FEniCS.
|
||||
\item Incompressible Navier-Stokes equations:
|
||||
\begin{equation}
|
||||
\begin{cases}
|
||||
\displaystyle \rho \frac{\partial \vec{u}}{\partial t} + \rho \big ( \vec{u} \cdot \nabla \big) \vec{u} - \mu \Delta \vec{u} + \nabla p = 0 \\[0.2cm]
|
||||
\nabla \cdot \vec{u} = 0 \quad \text{in} \quad \Omega \\[0.2cm]
|
||||
\vec{u} = \vec{u}_{inlet} \quad \text{on} \quad \Gamma_{in} \\[0.2cm]
|
||||
\vec{u} = 0 \quad \text{on} \quad \Gamma_{walls}
|
||||
\end{cases}
|
||||
\end{equation}
|
||||
\emph{Three-element} Windkessel coupling at every outlet:
|
||||
\begin{equation}
|
||||
\begin{cases}
|
||||
\displaystyle C_{d,l} \frac{d \pi_l}{dt} + \frac{\pi_l}{R_{d,l}} = Q_l \\[0.2cm]
|
||||
P_l = R_{p,l} \ Q_l + \pi_l
|
||||
\end{cases}
|
||||
\end{equation}
|
||||
\end{itemize}
|
||||
|
||||
|
||||
|
|
@ -258,90 +265,105 @@ P_l = R_{p,l} \ Q_l + \pi_l
|
|||
|
||||
|
||||
|
||||
%\begin{frame}
|
||||
% \frametitle{The mathematical model}
|
||||
%
|
||||
% \begin{columns}[c]
|
||||
%\column{.5\textwidth} % Left column and width
|
||||
%\footnotesize
|
||||
%\begin{itemize}
|
||||
%\item<1-> $u_{inlet} = -U f(t) \hat{n}$, with $f(t)$ the weaveform.
|
||||
%\item<2-> Fractional step scheme.
|
||||
%\item<3-> Semi-implicit Windkessel model.
|
||||
%\item<4-> Stabilized $\mathbb{P}1/\mathbb{P}1$ finite elements.
|
||||
%\item<4-> Implemented in FEniCS.
|
||||
%\end{itemize}
|
||||
%
|
||||
%
|
||||
%\column{.54\textwidth} % Right column and width
|
||||
%\begin{figure}[!hbtp]
|
||||
% \begin{center}
|
||||
% \includegraphics[height=0.9\textwidth]{images/ref.png}
|
||||
% \caption{\footnotesize Reference solution at peak systole}
|
||||
% \end{center}
|
||||
% \end{figure}
|
||||
%\end{columns}
|
||||
%
|
||||
%\end{frame}
|
||||
|
||||
|
||||
|
||||
\begin{frame}
|
||||
\frametitle{The inverse problem}
|
||||
\begin{itemize}
|
||||
\item<1-> Upon this solution $\Longrightarrow$ build a set of measurements
|
||||
\item<2-> Induce typical arifacts via a Magnetization vector: $M(\vec{x},t)= M_0(\vec{x}) exp(i\phi_0 + i \frac{\pi}{venc} u(\vec{x},t))$
|
||||
\item<2-> $M(\vec{x},t)= M_0(\vec{x}) exp(i\phi_0 + i \frac{\pi}{venc} u(\vec{x},t))$
|
||||
\item<3-> reconstructed velocity: $u \in \big ( -venc, + venc \big )$
|
||||
\item<4-> $VNR \sim 1/venc$
|
||||
\end{itemize}
|
||||
\onslide<3->
|
||||
\begin{columns}
|
||||
\column{.3\textwidth}
|
||||
\flushleft
|
||||
\begin{figure}
|
||||
\includegraphics[width=1.1\textwidth]{images/ref_int.png}
|
||||
\caption*{(a) Interpolated reference solution}
|
||||
\end{figure}
|
||||
\column{.67\textwidth}
|
||||
\centering
|
||||
\onslide<4-> \textbf{The measurements:}
|
||||
\onslide<5-> \textbf{The measurements:}
|
||||
\begin{itemize}
|
||||
\item<5-> Gaussian noise into the magnetization
|
||||
\item<6-> Different levels of aliasing varying the $venc$ parameter
|
||||
\item<7-> Only using the dominant component of the velocity: $u_z$
|
||||
\item<8-> Time interpolation $dt = 1 \ ms \Longrightarrow dt = 30 \ ms$
|
||||
\item<6-> Gaussian noise into the magnetization
|
||||
\item<7-> Spatial and temporal interpolation
|
||||
\item<8-> Only using the dominant component of the velocity: $u_z$
|
||||
\item<9-> Different levels of aliasing varying the $venc$ parameter
|
||||
\end{itemize}
|
||||
\end{columns}
|
||||
\end{frame}
|
||||
|
||||
|
||||
|
||||
|
||||
\begin{frame}
|
||||
\frametitle{The inverse problem}
|
||||
\begin{itemize}
|
||||
\item Upon this solution $\Longrightarrow$ build a set of measurements
|
||||
\item $M(\vec{x},t)= M_0(\vec{x}) exp(i\phi_0 + i \frac{\pi}{venc} u(\vec{x},t))$
|
||||
\item reconstructed velocity: $u \in \big ( -venc, + venc \big )$
|
||||
\item $VNR \sim 1/venc$
|
||||
\end{itemize}
|
||||
\begin{figure}
|
||||
\includegraphics[width=0.7\textwidth]{images/supra_venc.png}
|
||||
\caption*{Aliased measurements with different $vencs = 120,70,30 \%$ of $u_{max}$}
|
||||
\hfill
|
||||
\end{figure}
|
||||
\end{frame}
|
||||
|
||||
|
||||
|
||||
\begin{frame}
|
||||
\frametitle{The inverse problem}
|
||||
\begin{itemize}
|
||||
\item Upon this solution $\Longrightarrow$ build a set of measurements
|
||||
\item $M(\vec{x},t)= M_0(\vec{x}) exp(i\phi_0 + i \frac{\pi}{venc} u(\vec{x},t))$
|
||||
\item reconstructed velocity: $u \in \big ( -venc, + venc \big )$
|
||||
\item $VNR \sim 1/venc$
|
||||
\end{itemize}
|
||||
\begin{figure}
|
||||
\includegraphics[width=0.7\textwidth]{images/coartation.png}
|
||||
\caption*{ Aliased measurements with different $vencs = 120,70,30 \%$ of $u_{max}$}
|
||||
\hfill
|
||||
\end{figure}
|
||||
\end{frame}
|
||||
|
||||
|
||||
\section{The inverse problem}
|
||||
|
||||
\begin{frame}
|
||||
\frametitle{The inverse problem}
|
||||
\begin{itemize}
|
||||
\item Upon this solution $\Longrightarrow$ build a set of measurements
|
||||
\item Induce typical arifacts via a Magnetization vector: $M(\vec{x},t)= M_0(\vec{x}) exp(i\phi_0 + i \frac{\pi}{venc} u(\vec{x},t))$
|
||||
\end{itemize}
|
||||
\begin{columns}
|
||||
\column{.3\textwidth}
|
||||
\flushleft
|
||||
\begin{figure}
|
||||
\includegraphics[width=1.1\textwidth]{images/ref_int.png}
|
||||
\caption*{(a) Interpolated reference solution}
|
||||
\end{figure}
|
||||
\column{.67\textwidth}
|
||||
\flushleft
|
||||
\begin{figure}
|
||||
\includegraphics[width=1.1\textwidth]{images/supra_venc.png}
|
||||
\caption*{(b) Aliased measurements with different $vencs = 120,70,30 \%$ of $u_{max}$}
|
||||
\hfill
|
||||
\end{figure}
|
||||
\end{columns}
|
||||
\end{frame}
|
||||
|
||||
|
||||
|
||||
\begin{frame}
|
||||
\frametitle{The inverse problem}
|
||||
\begin{itemize}
|
||||
\item Upon this solution $\Longrightarrow$ build a set of measurements
|
||||
\item Induce typical arifacts via a Magnetization vector: $M(\vec{x},t)= M_0(\vec{x}) exp(i\phi_0 + i \frac{\pi}{venc} u(\vec{x},t))$
|
||||
\end{itemize}
|
||||
\begin{columns}
|
||||
\column{.3\textwidth}
|
||||
\flushleft
|
||||
\begin{figure}
|
||||
\includegraphics[width=1.1\textwidth]{images/ref_int.png}
|
||||
\caption*{(a) Interpolated reference solution}
|
||||
\end{figure}
|
||||
\column{.67\textwidth}
|
||||
\flushleft
|
||||
\begin{figure}
|
||||
\includegraphics[width=1.1\textwidth]{images/coartation.png}
|
||||
\caption*{(b) Aliased measurements with different $vencs = 120,70,30 \%$ of $u_{max}$}
|
||||
\hfill
|
||||
\end{figure}
|
||||
\end{columns}
|
||||
\begin{center}
|
||||
Parameter optimization
|
||||
\end{center}
|
||||
\end{frame}
|
||||
|
||||
|
||||
|
||||
|
||||
\begin{frame}
|
||||
\frametitle{The Kalman Filter}
|
||||
\begin{itemize}
|
||||
\item<1-> We use a Reduced Order Unscendent Kalman Filter (ROUKF) to reconstruct the parameter vector $\theta$ solving the next optimization problem:
|
||||
\item<1-> We use a Reduced Order Unscendent Kalman Filter (ROUKF) to reconstruct the parameter vector $\theta$:
|
||||
|
||||
\onslide<2->
|
||||
\begin{equation*}
|
||||
|
|
@ -350,7 +372,7 @@ P_l = R_{p,l} \ Q_l + \pi_l
|
|||
\begin{equation}
|
||||
J(\theta) = \displaystyle \frac{1}{2} || \theta - \theta_0 ||^2_{P_0^{-1}} + \sum_{k=1}^N \frac{1}{2} || Z_k - \mathbb{H} X_k(\theta) ||^2_{W^{-1}}
|
||||
\end{equation}
|
||||
\onslide<3-> Where:
|
||||
\onslide<4-> Where:
|
||||
\begin{itemize}
|
||||
\item<4-> $Z$ the measurements and $X = (\vec{u} , \pi)$ the state variable
|
||||
\item<5-> $\mathbb{H}$ observation operator
|
||||
|
|
@ -372,32 +394,41 @@ The parameter vector:
|
|||
|
||||
\begin{itemize}
|
||||
\item<1-> Amplitude of the inlet velocity: $U$
|
||||
\item<2-> Since $R_p << R_d$, we only consider an optimization dependent on $\big ( R_{d,l}, C_l \big )$ for $l=1,...,n_l$
|
||||
\item<2-> Only the higher resistence: $R_d$
|
||||
\end{itemize}
|
||||
|
||||
\onslide<3-> $$\theta = (U,\vec{R_d},\vec{C})$$ \\ with $\vec{R_d} = R_{d,l}$, $\vec{C} = C_l$ for $l=1,..., \color{red} n_{l-1}$ \\[0.3cm]
|
||||
\onslide<4-> \color{red} Not all the resistences can be recovered at once $\Longrightarrow$ desc. aorta fixed.
|
||||
\onslide<4-> $$\theta = (U,\vec{R_d})$$ \\ with $\vec{R_d} = R_{d,l}$ for $l=1,..., \color{red} n_{l-1}$ \\[0.3cm]
|
||||
\onslide<5-> \color{red} Not all the resistences can be recovered at once $\Longrightarrow$ desc. aorta fixed.
|
||||
|
||||
\end{frame}
|
||||
|
||||
|
||||
|
||||
|
||||
\section{Numerical Experiments}
|
||||
|
||||
|
||||
\begin{frame}
|
||||
\frametitle{Easy example}
|
||||
\frametitle{Numerical Experiments}
|
||||
\begin{center}
|
||||
Numerical Experiments
|
||||
\end{center}
|
||||
\end{frame}
|
||||
|
||||
|
||||
\begin{frame}
|
||||
\frametitle{Numerical Experiments}
|
||||
\footnotesize
|
||||
\begin{itemize}
|
||||
\item<1-> $\theta_{ref} = (U,\vec{R_d})$ , $U=75$, $\vec{R_d} = (7200,11520,11520)$
|
||||
\end{itemize}
|
||||
\onslide<1-> $\theta_{ref} = (U,\vec{R_d})$ , $U=75$, $\vec{R_d} = (7200,11520,11520)$
|
||||
\begin{columns}
|
||||
\footnotesize
|
||||
\column{.4\textwidth}
|
||||
\column{.45\textwidth}
|
||||
\begin{figure}
|
||||
\onslide<2-> \textbf{Test I:} $U_0 = 150$ $\vec{R_{d,0}}= (8760,8760,8760)$
|
||||
\onslide<3->
|
||||
\includegraphics[width=1.2\textwidth]{images/U_Pb.png}
|
||||
\onslide<2-> \textbf{Test I:} $U_0 = 150$ $\vec{R_{d,0}}= (17520,17520,17520)$
|
||||
\onslide<3-> \includegraphics[width=1.2\textwidth]{images/U_Pb.png}
|
||||
\includegraphics[width=1.2\textwidth]{images/Rd_Pb.png}
|
||||
\end{figure}
|
||||
\column{.4\textwidth}
|
||||
\column{.45\textwidth}
|
||||
\begin{figure}
|
||||
\onslide<2-> \textbf{Test II:} $U_0 = 40$ $\vec{R_{d,0}}= (4000,4000,4000)$
|
||||
\onslide<4->
|
||||
|
|
@ -415,32 +446,48 @@ The parameter vector:
|
|||
\begin{frame}
|
||||
\frametitle{Aliased data}
|
||||
\begin{center}
|
||||
What happend when $venc < u_{max}$ ?
|
||||
\onslide<1-> What happend when $venc < u_{max}$ ?
|
||||
\begin{figure}
|
||||
\onslide<2-> \includegraphics[width=0.45\textwidth]{images/v120.png}
|
||||
\caption{Measurement set with $venc = 120 \% u_{max}$}
|
||||
\end{figure}
|
||||
\end{center}
|
||||
\end{frame}
|
||||
|
||||
|
||||
\begin{frame}
|
||||
\frametitle{Aliased data}
|
||||
\begin{center}
|
||||
What happend when $venc < u_{max}$ ?
|
||||
\begin{figure}
|
||||
\includegraphics[width=0.45\textwidth]{images/v70.png}
|
||||
\caption{Measurement set with $venc = 70 \% u_{max}$}
|
||||
\end{figure}
|
||||
\end{center}
|
||||
\end{frame}
|
||||
|
||||
|
||||
|
||||
|
||||
\begin{frame}
|
||||
\frametitle{Easy example: with $venc = 70 \% u_{max}$}
|
||||
\footnotesize
|
||||
\begin{itemize}
|
||||
\item $\theta_{ref} = (U,\vec{R_d})$ , $U=75$, $\vec{R_d} = (7200,11520,11520)$
|
||||
\end{itemize}
|
||||
$\theta_{ref} = (U,\vec{R_d})$ , $U=75$, $\vec{R_d} = (7200,11520,11520)$
|
||||
\begin{columns}
|
||||
\footnotesize
|
||||
\column{.4\textwidth}
|
||||
\column{.45\textwidth}
|
||||
\begin{figure}
|
||||
\onslide<1-> \textbf{Test I:} $U_0 = 150$ $\vec{R_{d,0}}= (8760,8760,8760)$
|
||||
\onslide<1-> \textbf{Test I:} $U_0 = 150$ $\vec{R_{d,0}}= (17520,17520,17520)$
|
||||
\onslide<2->
|
||||
\includegraphics[width=1.2\textwidth]{images/U_Pb.png}
|
||||
\includegraphics[width=1.2\textwidth]{images/Rd_Pb.png}
|
||||
\includegraphics[width=1.2\textwidth]{images/U_Pb_V70.png}
|
||||
\includegraphics[width=1.2\textwidth]{images/Rd_Pb_V70.png}
|
||||
\end{figure}
|
||||
\column{.4\textwidth}
|
||||
\column{.45\textwidth}
|
||||
\begin{figure}
|
||||
\onslide<1-> \textbf{Test II:} $U_0 = 40$ $\vec{R_{d,0}}= (4000,4000,4000)$
|
||||
\onslide<2->
|
||||
\includegraphics[width=1.2\textwidth]{images/U_Pc.png}
|
||||
\includegraphics[width=1.2\textwidth]{images/Rd_Pc.png}
|
||||
\includegraphics[width=1.2\textwidth]{images/U_Pc_V70.png}
|
||||
\includegraphics[width=1.2\textwidth]{images/Rd_Pc_V70.png}
|
||||
\end{figure}
|
||||
\end{columns}
|
||||
\end{frame}
|
||||
|
|
@ -465,30 +512,70 @@ J(\theta) = \displaystyle \frac{1}{2} || \theta - \theta_0 ||^2_{P_0^{-1}} + \
|
|||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
\section{Numerical Experiments}
|
||||
\begin{frame}
|
||||
\frametitle{Easy example: with $venc = 70 \% u_{max}$}
|
||||
\footnotesize
|
||||
$\theta_{ref} = (U,\vec{R_d})$ , $U=75$, $\vec{R_d} = (7200,11520,11520)$
|
||||
\begin{columns}
|
||||
\footnotesize
|
||||
\column{.45\textwidth}
|
||||
\begin{figure}
|
||||
\onslide<1-> \textbf{Test I:} $U_0 = 150$ $\vec{R_{d,0}}= (17520,17520,17520)$
|
||||
\onslide<2->
|
||||
\includegraphics[width=1.2\textwidth]{images/HU_Pb_V70.png}
|
||||
\includegraphics[width=1.2\textwidth]{images/HRd_Pb_V70.png}
|
||||
\end{figure}
|
||||
\column{.45\textwidth}
|
||||
\begin{figure}
|
||||
\onslide<1-> \textbf{Test II:} $U_0 = 40$ $\vec{R_{d,0}}= (4000,4000,4000)$
|
||||
\onslide<3->
|
||||
\includegraphics[width=1.2\textwidth]{images/HU_Pc_V70.png}
|
||||
\includegraphics[width=1.2\textwidth]{images/HRd_Pc_V70.png}
|
||||
\end{figure}
|
||||
\end{columns}
|
||||
\end{frame}
|
||||
|
||||
|
||||
\begin{frame}
|
||||
\frametitle{Numerical Experiments}
|
||||
\frametitle{Aliased data}
|
||||
\begin{center}
|
||||
Numerical Experiments
|
||||
Or even higher aliasing...
|
||||
\begin{figure}
|
||||
\includegraphics[width=0.45\textwidth]{images/v30.png}
|
||||
\caption{Measurement set with $venc = 30 \% u_{max}$}
|
||||
\end{figure}
|
||||
\end{center}
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}
|
||||
\frametitle{Numerical Experiments}
|
||||
|
||||
|
||||
|
||||
\begin{frame}
|
||||
\frametitle{Easy example: with $venc = 30 \% u_{max}$}
|
||||
\footnotesize
|
||||
$\theta_{ref} = (U,\vec{R_d})$ , $U=75$, $\vec{R_d} = (7200,11520,11520)$
|
||||
\begin{columns}
|
||||
\footnotesize
|
||||
\column{.45\textwidth}
|
||||
\begin{figure}
|
||||
\onslide<1-> \textbf{Test I:} $U_0 = 150$ $\vec{R_{d,0}}= (17520,17520,17520)$
|
||||
\onslide<2->
|
||||
\includegraphics[width=1.2\textwidth]{images/HU_Pb_V30.png}
|
||||
\includegraphics[width=1.2\textwidth]{images/HRd_Pb_V30.png}
|
||||
\end{figure}
|
||||
\column{.45\textwidth}
|
||||
\begin{figure}
|
||||
\onslide<1-> \textbf{Test II:} $U_0 = 40$ $\vec{R_{d,0}}= (4000,4000,4000)$
|
||||
\onslide<3->
|
||||
\includegraphics[width=1.2\textwidth]{images/HU_Pc_V30.png}
|
||||
\includegraphics[width=1.2\textwidth]{images/HRd_Pc_V30.png}
|
||||
\end{figure}
|
||||
\end{columns}
|
||||
\end{frame}
|
||||
|
||||
|
||||
|
||||
|
||||
\section{Conclusions}
|
||||
|
||||
\begin{frame}
|
||||
|
|
@ -499,8 +586,32 @@ Conclusions
|
|||
|
||||
|
||||
|
||||
\begin{frame}
|
||||
\frametitle{Conclusions}
|
||||
\footnotesize
|
||||
\begin{itemize}
|
||||
\item<1-> 4D Flow measurements are promising for extracting data via inverse problems
|
||||
\item<2-> Noise and aliasing are the typical artifacts involved.
|
||||
\item<3-> Using a suitable Kalman filter, have shown to bypass aliasing defining the funcional in terms of the frequencies.
|
||||
\end{itemize}
|
||||
|
||||
\onslide<4-> Future Work
|
||||
|
||||
\begin{itemize}
|
||||
\item<5-> To include the capacitancies in the inverse problem (adding some pressure meas.)
|
||||
\item<6-> Real data!
|
||||
\end{itemize}
|
||||
|
||||
\end{frame}
|
||||
|
||||
|
||||
\begin{frame}
|
||||
\begin{center}
|
||||
Thank you for your time!
|
||||
\end{center}
|
||||
|
||||
|
||||
\end{frame}
|
||||
|
||||
|
||||
|
||||
|
|
|
|||
|
|
@ -0,0 +1,6 @@
|
|||
\babel@toc {english}{}
|
||||
\beamer@sectionintoc {1}{4D flow MRI}{3}{0}{1}
|
||||
\beamer@sectionintoc {2}{The mathematical model}{6}{0}{2}
|
||||
\beamer@sectionintoc {3}{The inverse problem}{23}{0}{3}
|
||||
\beamer@sectionintoc {4}{Numerical Experiments}{37}{0}{4}
|
||||
\beamer@sectionintoc {5}{Conclusions}{56}{0}{5}
|
||||
Loading…
Reference in New Issue