The traditional shape from shading (SFS) algorithms inaccurately estimate the initial information of the original direction of the light source

thereby causing the reconstructed surface to be smooth in several objects with a rough surface.The ideal reflection model

i.e.

Lambertian surface

is too simple to meet the conditions in real applications.Therefore

the reconstructed shape of the object presents serious errors.In this study

a reflection model based on the radial basis function (RBF) neural network is established to solve this problem.Moreover

the fixed learning rate in the traditional algorithm slows down the training process.Thus

the neural network is improved in the aspect of learning rate to accelerate the training speed and avoid being trapped in a local minimum at the same time. The reflection model

which is based on the RBF

replaces the ideal Lambertian reflection model used in the traditional SFS algorithms.The excellent local mapping and function approximation capabilities of the RBF are suitable for dealing with the classic SFS problems.The original information of the light source is replaced by the weight in the training process of the RBF neural network.Then

the SFS problem is transformed to obtain the optimal solution for its energy equations.In this manner

the limitation that the direction of the light source parameters must be known in the beginning is removed.Only one single image is needed in the improved algorithm to restore the 3D surface of the target object.However

the fixed learning rate in the traditional neural network easily causes the local minimum when the parameters are inappropriate in the beginning and slows down the training speed in practice.Therefore

the adaptive learning rate algorithm is added to the network to accelerate the convergence and training speeds.The learning rate can be automatically adjusted from the iterations. In the experiments

two synthetic images for the classic SFS problems are Vase and Mozart and two real images are Map and Antique.The four images are used to test the improved algorithms.Then

the 3D visual effects and the quantitative comparison of the results are conducted.The experimental results show that the results of the improved algorithm in 3D visual effects are better than those of the traditional SFS algorithms.The reconstruction of the 3D shape information is apparently superior to that of the traditional algorithm.The traditional SFS algorithms always show serious errors in real images.The traditional algorithms show some errors in details in synthetic images.The improved algorithm using RBF performs well when dealing with the synthetic images and yields improved results when dealing with the real images.The min-max normalization function is introduced to deal with the height information to compare the absolute errors clearly.According to the comparison of the normalized 128 pixels height error from the map

the result of the normalized 3D height error using the improved algorithm decreases by more than 60% compared with the traditional SFS algorithms.The entire calculation process of the improved algorithm takes 50% less time than that of the traditional algorithms.Moreover

the improved algorithm is appropriate for synthetic and real images simultaneously. The ideal Lambertian reflection model cannot easily meet the needs in practice because the assumption that the Lambertian surface reflects light in all directions is inaccurate in real application.Therefore

the results of the objects with rough surface using the traditional SFS algorithm exhibit many errors.The Lambertian reflection model is only suitable for some synthetic images with a smooth surface.However

for the real images

the reconstructed 3D surface cannot be precise when using this model as it causes serious errors in the shape of the restored images and the 3D visual effect is poor.With some real images

such as Map

the basic shape of mountain can be barely distinguished.The RBF is introduced to the SFS problems to establish a new reflection model to solve this problem.The reflection model based on the RBF replaces the original direction of the light source by the weight in the neural network.By training the relevant parameters in the network

the improved algorithm overcomes the shortcomings of traditional methods

e.g.

the inaccurate estimation of the original light source.The new reflection model is suitable for synthetic and real images and shows excellent 3D recovery capability in real images with rough surface.Moreover

the adaptive learning rate algorithm accelerates the training speed

thereby obviously improving the work efficiency.The training time of the new reflection model is 50% less than that of the traditional algorithm.The maximum absolute error value of the new reflection model is reduced by more than 60% compared with the traditional SFS algorithm.In this study

the improved algorithm is applicable to the 3D reconstruction of surfaces in most non-high-precision situations

without the need for the initial information of the direction of the light source.